Nested Hilbert schemes on surfaces: Virtual fundamental class

Advances in Mathematics 365 (2020) 107046

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Nested Hilbert schemes on surfaces: Virtual fundamental class Amin Gholampour a , Artan Sheshmani b,c,d,∗ , Shing-Tung Yau e a

University of Maryland, College Park, MD 20742-4015, USA Center for Mathematical Sciences and Applications, Harvard University, Department of Mathematics, 20 Garden Street, Room 110, Cambridge, MA 02139, USA c Aarhus University, Department of Mathematics, Ny Munkegade 118, building 1530, 319, 8000 Aarhus C, Denmark d National Research University Higher School of Economics, Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow, 119048, Russia e Department of Mathematics, Harvard University, Cambridge, MA 02138, USA b

a r t i c l e

i n f o

Article history: Received 21 July 2019 Accepted 3 February 2020 Available online xxxx Communicated by L. Katzarkov Keywords: Nested Hilbert scheme Projective surfaces Virtual fundamental class

a b s t r a c t We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [15]. We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov’s vertex operator formulas. © 2020 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (A. Gholampour), [email protected] (A. Sheshmani), [email protected] (S.-T. Yau). https://doi.org/10.1016/j.aim.2020.107046 0001-8708/© 2020 Elsevier Inc. All rights reserved.

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

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Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Nested Hilbert schemes on surfaces . . . . . . 1.2. Punctual nested Hilbert schemes . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Notation and conventions . . . . . . . . . . . . 2. Nested Hilbert schemes on surfaces . . . . . . . . . . . 2.1. 2-Step nested Hilbert schemes . . . . . . . . . 2.2. Gillam’s construction . . . . . . . . . . . . . . . 2.3. Reduced perfect obstruction theory . . . . . 2.4. Invariants . . . . . . . . . . . . . . . . . . . . . . . 2.5. Proof of Theorem 1 (Proposition 2.2) . . . . 3. Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Punctual nested Hilbert schemes . . . . . . . . . . . . 4.1. Toric surfaces . . . . . . . . . . . . . . . . . . . . . 4.2. Proof of Theorem 3 . . . . . . . . . . . . . . . . 4.3. Relative nested Hilbert schemes . . . . . . . . 4.4. Double point relation . . . . . . . . . . . . . . . 5. Vertex operator formulas and proof of Theorem 4 . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Hilbert scheme of points on a nonsingular surface S have been vastly studied. They are nonsingular varieties with rich geometric structures some of which have applications in physics (see [38] for a survey). We are mainly interested in the enumerative geometry of Hilbert schemes of points [17,29,3,12]. This has applications in curve counting problems on S [33,42]. The first two authors of this paper have studied the relation of some of these enumerative problems to the Donaldson-Thomas theory of 2-dimensional sheaves in threefolds and to S-duality conjectures [11]. In contrast, Hilbert scheme of curves on S can be badly behaved and singular. They were studied in detail by Dürr-KabanovOkonek [6] in the context of Poincaré invariants (algebraic Seiberg-Witten invariants [4]). More recently, the stable pair invariants of surfaces have been employed in the context of curve counting problems [41,37,26,27]. The moduli space of stable pairs on S is identified with the Hilbert scheme of points on curves on S, and so is a nested Hilbert scheme on S. This paper studies general nested Hilbert schemes of points and curves on S. The geometry of nested Hilbert scheme of points was studied in [5]. We construct a reduced and a non-reduced perfect obstruction theories for the nested Hilbert scheme of curves and points on S. The reduced one is an extension of the reduced perfect obstruction theory of stable pairs worked out in [26], and its construction closely follows their technique. Our main application of the non-reduced perfect obstruction theory is in the study of (reduced!) Donaldson-Thomas theory of local surfaces that is carried out in [15]. The non-reduced perfect obstruction theory appears in the study of Vafa-Witten theory of S [44], and also in the work of A. Negut [40].

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1.1. Nested Hilbert schemes on surfaces Suppose that r ∈ Z>0 , n := n1 , n2 , . . . , nr ∈ Z≥0 , and β := β1 , . . . , βr−1 ∈ H 2 (S, Z). [n] We denote the corresponding nested Hilbert scheme Sβ , whose underlying set of closed points consists of tuples of subschemes of S (Z1 , Z2 , . . . , Zr ),

(C1 , . . . , Cr−1 )

where Zi is a 0-dimensional of length ni , and Ci a divisor with [Ci ] = βi , and for any i < r, IZi (−Ci ) ⊆ IZi+1 ,

(1) [n]

where IZi ⊆ OS is the ideal sheaf of Zi .1 In other words, a closed point of Sβ corresponds to a chain of subschemes of S given by the ideals IZ1 (−C1 − · · · − Cr−1 ) ⊆ IZ2 (−C2 − · · · − Cr−1 ) ⊆ · · · ⊆ IZr ⊆ OS . [n]

To define invariants of S arising from Sβ

(see Section 2.4), we construct a virtual

[n] fundamental class [Sβ ]vir . More precisely, we construct a natural (non-reduced) perfect [n] obstruction theory over Sβ . This is done by studying the deformation/obstruction the-

ory of the maps of coherent sheaves given by the natural inclusions (1) following Illusie [21] and Joyce-Song [22].2 As we will see, this in particular provides a uniform way of studying all known perfect obstruction theories on the Hilbert schemes of points and curves, as well as the stable pair moduli spaces on S. The main result of the paper is (Propositions 2.2, 2.5 and Corollary 2.6): Theorem 1. Let S be a nonsingular projective surface over C and ωS be its canonical [n] bundle. The nested Hilbert scheme Sβ with r ≥ 2 carries a natural perfect obstruction theory with the virtual fundamental class 1 βi · (βi − c1 (ωS )). 2 i=1 r−1

[n]

[n]

[Sβ ]vir ∈ Ad (Sβ ),

d = n1 + nr + [n]

For r ≥ 2 and βi = 0, S [n] := S(0,...,0) is the nested Hilbert scheme of points on S parameterizing flags of 0-dimensional subschemes Zr ⊆ · · · ⊆ Z2 ⊆ Z1 ⊂ S. S [n] is in general singular of actual dimension 2n1 . 1 Taking double dual shows that an inclusion of ideals IZi (−Di ) ⊆ IZi+1 (−Di+1 ) ⊂ OS is equivalent to (1) with Ci = Di − Di+1 . So we are not losing information by starting with r − 1 effective divisors C1 , . . . , Cr−1 instead of r of them. 2 Another approach would be to study the deformation/obstruction theory of quotients IZi+1 /IZi (−Ci ) following the work of Gillam [16], which in turn also uses a great deal of [21]. We give a brief sketch of this approach in Subsection 2.2.

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[n]

[n1 ,n2 ]

We are specifically interested in the case r = 2 in this paper: Sβ = Sβ [0,0] Sβ

for some

[0,n ] Sβ 2

β ∈ H (S, Z). The cases Sβ := and (when β = 0) are respectively isomorphic to the Hilbert scheme of divisors and the moduli space of stable pairs in classes β. The comparison of the virtual class of Theorem 1 to other known virtual classes is carried out in Proposition 3.1. In certain cases, we construct a reduced virtual fundamental class 2

[n1 ,n2 ] vir ]red

[Sβ

[n1 ,n2 ]

∈ An1 +n2 + 12 β·(β−c1 (ωS ))+pg (S) (Sβ

)

by reducing the perfect obstruction theory in Theorem 1 (Propositions 2.7, 2.10). The [0,n ] reduced virtual fundamental class [Sβ 2 ]vir red match with the reduced virtual fundamental class of stable pair theory constructed in [26] (Proposition 3.1). 1.2. Punctual nested Hilbert schemes Sections 4 and 5 are devoted to study of the nested Hilbert schemes of points 1 S [n1 ≥n2 ] := Sβ=0

[n ,n2 ]

in more detail. Let ι : S [n1 ≥n2 ] → S [n1 ] × S [n2 ] be the natural inclusion. If S is toric with the torus T and the fixed set S T , in Section 4.1 we provide a purely combinatorial formula for computing [S [n1 ≥n2 ] ]vir by torus localization along the lines of [35]. Let d be a positive integer, by a partition μ of d, denoted by μ  d, we mean a finite sequence of positive integers μ = (μ1 ≥ μ2 ≥ μ3 ≥ . . . )

such that d =



μi .

i

The number of μi ’s is called the length of the partition μ, and is denoted by (μ). If μ  d with d ≤ d, we say μ ⊆ μ if (μ ) ≤ (μ) and μi ≤ μi for all 1 ≤ i ≤ (μ ). Theorem 2. For a toric nonsingular surface S the T-fixed set of S [n1 ,n2 ] is isolated and in bijective correspondence by the tuples of nested partitions:  (μP

⊆ μP )P ∈S T |

μP



dP ,

μP  dP ,

n2 =

 P

dP ,

n1 =



 dP

.

P

Moreover, the T-character of the virtual tangent bundle T vir of S [n1 ≥n2 ] at the fixed point Q = (μP ⊆ μP )P ∈S T is given by

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

trTQvir (t1 , t2 ) =



5

VP ,

P ∈S T

where t1 , t2 are the torus characters and VP is a Laurent polynomial in t1 , t2 that is completely determined by the partitions μP and μP and is given by the right hand side of formula (30). The bijection in Theorem 2 is deduced from the standard bijection between the set of the fixed points of punctual Hilbert schemes on the affine plane and the set of monomial ideals of finite colengths in the polynomial rings in two variables. Let I1 , I2 be the universal ideal sheaves on S ×S [n1 ] ×S [n2 ] , and let π be the projection to the last two factors S [n1 ] × S [n2 ] and p be the projection to S. Following [3], for any line bundle M on S, we define a K-theory class on S [n1 ] × S [n2 ] by EnM1 ,n2 :=

2 

  (−1)i H i (S, M ) ⊗ O − Extiπ (I1 , I2 ⊗ p∗ M ) .

i=0

If M is trivial we drop it from the notation (see Definition 4.2). When S is toric, by torus localization, we can express [S [n1 ≥n2 ] ]vir in terms of the fundamental class of the product of Hilbert schemes S [n1 ] × S [n2 ] (Proposition 4.4): Theorem 3. If S is a nonsingular projective toric surface, then, ι∗ [S [n1 ≥n2 ] ]vir = [S [n1 ] × S [n2 ] ] ∩ cn1 +n2 (En1 ,n2 ). Theorem 3 holds in particular for S = P 2 , P 1 × P 1 , which are the generators of the cobordism ring of nonsingular projective surfaces. Similar formulas was worked by A. Negut and others (see [39,40] and the references within). We use a refinement of this fact together with a degeneration formula developed for [S [n1 ≥n2 ] ]vir (Proposition 4.5) to prove that for any nonsingular projective surface S, certain integrals against [S [n1 ≥n2 ] ]vir can be expressed as integrals against S [n1 ] × S [n2 ] (Corollary 4.13, Proposition 4.14). A generalization of such formulas have been recently worked out in [13,14] using degeneracy loci techniques. This last result (Proposition 4.14) is used in [15] to express some of the reduced localized DT invariants of S as sums of integrals over the product of Hilbert schemes of points on S, when S is one of five types generic complete intersections (5) ⊂ P 3 , (3, 3) ⊂ P 4 , (4, 2) ⊂ P 4 , (3, 2, 2) ⊂ P 5 , (2, 2, 2, 2) ⊂ P 6 . Such integrals also have applications in evaluating Vafa-Witten invariants recently defined in [44].

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The operators  − ∪ cn1 +n2 (EnM1 ,n2 ) S [n1 ] ×S [n2 ]

were studied by Carlsson-Okounkov in [3]. They expressed these operators in terms of explicit vertex operators. Using this, we prove the following explicit formula (Proposition 5.2): Theorem 4. Let S be a nonsingular projective surface, ωS be its canonical bundle, and KS = c1 (ωS ). Then, 

 n1 +n2

(−1)

n1 ≥n2 ≥0

ι∗ c(EnM1 ,n2 )q1n1 q2n2 =

[S [n1 ≥n2 ] ]vir



1 − q2n−1 q1n

KS ,KS −M

KS −M,M −e(S)

(1 − q1n q2n )

,

n>0

where −, − is the Poincaré paring on S. This is the only situation that we have been able to find a closed formula for the complete generating series of invariants. It would be interesting to seek similar formulas for the more involved integrals over nested Hilbert schemed that appear in Vafa-Witten theory or reduced local DT theory of S. Acknowledgment We are grateful to Richard Thomas for providing us with many valuable comments. We would like to thank Eric Carlsson, Andrei Negut, Davesh Maulik, Hiraku Nakajima, Takur¯ o Mochizuki, Alexey Bondal and Mikhail Kapranov for useful discussions. A.G. was partially supported by NSF grant DMS-1406788. A.S. was partially supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, as well as NSF DMS-1607871, NSF PHY-1306313 and Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001.S.-T.Y. was partially supported by NSF DMS-0804454, NSF PHY-1306313, and Simons 38558. A.S. would like to further sincerely thank the center for Quantum Geometry of Moduli Spaces at Aarhus University, the Center for Mathematical Sciences and Applications at Harvard University and the Laboratory of Mirror Symmetry in Higher School of Economics,Russian federation, for the great help and support over the period that this paper was being completed.

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1.3. Notation and conventions 1) We will use the symbol L• for the (full) cotangent complex, and L•,gr for the cotangent complex for the sheaves of graded algebras. Our sheaves of graded algebras are always of the form A0 ⊕ A1 where Ai is in degree i. If M is a complex of graded (A0 ⊕ A1 )-modules then ki (M ) takes the degree i part in the grading. If A0 ⊕ A1 → B0 ⊕ B1 ,

id ⊕s

A0 ⊕ A1 −−−→ A0 ⊕ C1

are graded homomorphisms of sheaves of graded algebras in which s is injective then, we will use the following isomorphisms proven in [21, IV.2.2.4), IV.2.2.5, IV.3.2.10]

∼ • k0 L•,gr (B0 ⊕B1 )/(A0 ⊕A1 ) = LB0 /A0 ,



∼ k1 L•,gr (A0 ⊕C1 )/(A0 ⊕A1 ) = coker(s).

Associated, to graded homomorphisms of graded sheaves of algebras A0 ⊕ A1 → B0 ⊕ B1 → C0 ⊕ C1 is a natural exact triangle •,gr •,gr L•,gr (B0 ⊕B1 )/(A0 ⊕A1 ) ⊗(B0 ⊕B1 ) (C0 ⊕ C1 ) → L(C0 ⊕C1 )/(A0 ⊕A1 ) → L(C0 ⊕C1 )/(B0 ⊕B1 )

that is referred to as the transitivity triangle [21, IV.2.3]. [m] [m] 2) We will denote the universal ideal sheaves of Sβ , S [m] , and Sβ respectively by I−β , [m]

I [m] , and I−β , and the corresponding universal subschemes respectively by Zβ , [m]

Z [m] , and Zβ . We will also write Iβ

for I [m] ⊗ O(Zβ ). Using the universal property [m] of the Hilbert scheme, it can be seen that I−β ∼ = I [m] ⊗ O(−Zβ ). [n]

[n]

3) Let π : S × Sβ → Sβ be the projection, we denote the derived functor Rπ∗ RHom by RHomπ and its i-th cohomology sheaf by Extiπ . π is a smooth morphism of relative dimension 2 and hence by Grothendieck-Verdier duality π ! (−) := π ∗ (−) ⊗ ωπ [2] is a right adjoint of Rπ∗ . 4) Throughout the paper, we slightly abuse notation and suppress many of the symbols f ∗ and f −1 for the pullback of sheaves on Y via a given morphism f : X → Y of schemes. This makes most of the formulas notationally lighter and hence more readable. 5) For any line bundle L on S we define LD := L−1 ⊗ ωS . Similarly, for any class β ∈ H 2 (S, Z) we define β D := KS − β. 2. Nested Hilbert schemes on surfaces Let S be a nonsingular projective surface over C. We denote the canonical line bundle on S by ωS and KS := c1 (ωS ). For any nonnegative integer m and effective curve class

8

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

[m]

β ∈ H 2 (S, Z), we denote by Sβ such that

the Hilbert scheme of 1-dimensional subschemes Z ⊂ S

[Z] = β,

c2 (IZ ) = m.

If β = 0 we drop it from the notation and denote by S [m] the Hilbert scheme of m points on S. Similarly, in the case m = 0 but β = 0 we drop m from the notation and use Sβ to denote the Hilbert scheme of curves in class β. There are natural morphisms [m]

div : Sβ

→ Sβ ,

[m]

det : Sβ

[m]

→ Pic(S),

pts : Sβ

→ S [m] ,

where div sends a 1-dimensional subscheme Z ⊂ S to its underlying divisor on S, det(Z) := O(div(Z)), and pts(Z) is the 0-dimensional subscheme of S defined by the ideal IZ (div(Z)). In fact div(−) is well-behaved with respect to basechange (see [9,24]) and sends a flat family of 1-dimensional subschemas of S to a flat family of divisors on S, and so using the universal property of the Hilbert schemes, these maps are morphisms [m] of schemes. There is a natural isomorphism of schemes Sβ ∼ = S [m] × Sβ defined by mapping Z → (pts(Z), div(Z)). Under this, we have the following relation among the [m] universal ideal sheaves: I−β ∼ = I [m]  O(−Zβ ). [m] It is well known that S is a nonsingular variety of dimension 2m. The tangent bundle of S [m] is identified with       TS [m] ∼ = RHomπ I [m] , I [m] 0 [1] ∼ = Homπ I [m] , OZ [m] ∼ = Ext1π I [m] , I [m] 0 ,

(2)

where the index 0 indicates the trace-free part. The main object of study in this paper is the following Definition 2.1. Suppose that n := n1 , n2 , . . . , nr is a sequence of r ≥ 1 nonnegative integers, and β := β1 , . . . , βr−1 is a sequence of classes in H 2 (S, Z). The nested Hilbert scheme is the closed subscheme [n]

[n ]

[n

]

r−1 ι : Sβ → Sβ11 × · · · × Sβr−1 × S [nr ]

(3)

naturally defined by the r-tuples (Z1 , . . . , Zr ) of subschemes of S such that pts(Zi ) ⊂  Zi−1 is a subscheme for all 1 < i ≤ r. We drop β or n from the notation respectively [n] when βi = 0 for all i or ni = 0, βj = 0 for all i, j. The scheme Sβ represents the functor that takes a scheme U to the set of flat families of ideals I1 , . . . , Ir ⊆ OS×U and flat families of Cartier divisors D1 . . . , Dr−1 ⊂ S × U such that I1 (−D1 − D2 − · · · − Dr−1 ) ⊆ I2 (−D2 − · · · − Dr−1 ) ⊆ · · · ⊆ Ir−1 (−Dr−1 ) ⊆ Ir , and on restriction to any closed fiber Ii |S×{u} has colength ni and [Di |S×{u} ] = βi .

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[n]

As a set we can think of Sβ as given by the tuples of subschemes (Z1 , . . . , Zr ) ∈ S [n1 ] × · · · × S [nr ] ,

(C1 , . . . , Cr−1 ) ∈ Sβ1 × · · · × Sβr−1

together with the nonzero maps φi : IZi → IZi+1 (Ci ), up to multiplication by scalars, for all 1 ≤ i < r. Note that each φi is necessarily injective, and taking double dual, it gives (up to a scalar) the tautological section of O(Ci ). In this correspondence, Zr = Zr , and for 1 ≤ i < r the ideal sheaf of the subscheme Zi is IZi (−Ci ). By the construction of nested Hilbert schemes, the maps φi above are induced from the universal maps [n

Φi : I [ni ] → Iβi i+1

]

1≤i
[n]

defined over S × Sβ .

   [n ] Applying the functors RHomπ −, Iβi i+1 and RHomπ I [ni ] , − to the universal map Φi , we get the following morphisms in derived category   Ξi  [n ] RHomπ I [ni+1 ] , I [ni+1 ] −→ RHomπ I [ni ] , Iβi i+1 ,   Ξi  [n ] RHomπ I [ni ] , Iβi i+1 . RHomπ I [ni ] , I [ni ] −→ Consider the map ⎡

r 

⎢ ⎢ ⎢ ⎣

−Ξ1 0 · · · 0

Ξ1 −Ξ2 · · · 0

0 Ξ2 ... ... ... ...

... 0 · · · 0

0 ... · · · −Ξr−1

⎤

0 0 · · ·

⎥ ⎥ ⎥ ⎦

r−1     Ξr−1 [n ] RHomπ I [ni ] , I [ni ] −−−−−−−−−−−−−−−−−−−−−−−−→ RHomπ I [ni ] , Iβi i+1 .

i=1

i=1

(4) We will show that this map factors through the trace free part 

r 



RHomπ I

[ni ]

,I

i=1

[ni ]





 : = Cone 0



r 



RHomπ I

[ni ]

,I

[ni ]



 [tr ... tr]

−−−−−→ Rπ∗ O [−1]

i=1 [id ... id]t

∼ = Cone Rπ∗ O −−−−−−→

r 



RHomπ I [ni ] , I

 [ni ]

 .

i=1

The following proposition that implies Theorem 1 is proven in Section 2.5: [n]

Proposition 2.2. Sβ

is equipped with a perfect absolute obstruction theory F • → L• [n]

whose virtual tangent bundle is given by

Sβ

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F

•∨

∼ = Cone

 r 

  r−1   [ni ] [ni ]   [ni ] [ni+1 ]  RHomπ I , I → RHomπ I , Iβi ,

i=1

i=1

0

where the map is the one defined above.3 2.1. 2-Step nested Hilbert schemes [n ,n ]

[n]

In this section we study Sβ 1 2 := Sβ in the case r = 2. Recall from Definition 2.1 that for a pair of nonnegative integers n1 , n2 and an effective curve class β ∈ H 2 (S, Z), [n ,n ] we defined the projective scheme Sβ 1 2 , whose set of closed points is given by   (Z1 , C, Z2 , φ) | Zi ∈ S [ni ] , C ∈ Sβ , φ ∈ P (Hom(IZ1 , IZ2 (C))) . [n1 ,n2 ]

There are universal objects defined over S × Sβ

[n2 ]

Φ : I [n1 ] → Iβ

as before:

.

[n ]

Applying the functors RHomπ (−, Iβ 2 ) and RHomπ (I [n1 ] , −) to the universal map Φ, we get the following morphisms in derived category   Ξ  [n ]  RHomπ I [n2 ] , I [n2 ] − → RHomπ I [n1 ] , Iβ 2

(5)

  Ξ  [n ]  RHomπ I [n1 ] , I [n1 ] −→ RHomπ I [n1 ] , Iβ 2 . Let T be any scheme over C-scheme U , and let T

T

a a

U

be a square zero extension over U with the ideal J. As J 2 = 0, J can be considered as an OT -module. Suppose we have a Cartesian diagram g

T a

U

[n1 ,n2 ]

Sβ

(6)

p:=pts ◦ pr1

S [n1 ] ,

3 The negative signs on the diagonal of the matrix in (4) were missing in the first draft of the paper. We were notified of the corrected form of the matrix by Richard Thomas.

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[n ]

11

[n ]

where pr1 is the composition of ι with the projection Sβ 1 × S [n2 ] → Sβ 1 . The bottom row of (6) corresponds to a flat U -family Z1,U ⊂ S × U of subschemes of length n1 , and the top row corresponds to the data (Z1,T , CT , Z2,T , φT ),

(7)

consisting of Z1,T = Z1,U ×S×U,(id,a) S × T , a T -flat family Z2,T ⊂ S × T of subschemes of length n2 , a T -flat family CT ⊂ S × T of effective Cartier divisors in class β, and (up to a scalar) a homomorphism φT : IZ1,T → IZ2,T (CT ). Let πT be the projections from S × T → T . By [21, Prop. IV.3.2.12] and [22, Thm 12.8], there exists an element   ob := ob(φT , J) ∈ Ext2S×T coker(φT ), πT∗ J ⊗ IZ2,T (CT ) , whose vanishing is necessary and sufficient to extend the T -family (7) to a T -family (Z1,T , CT , Z2,T , φT )

(8)

such that Z1,T = Z1,U ×S×U,(id,a) S × T . In fact by [21, Prop. IV.3.2.12], ob is the obstruction to deforming the morphism φT while the deformation IZ1,T of IZ1,T is given. Suppose that φT : IZ1,T → F is such a deformation, where F is a flat family of rank 1 torsion free sheaves with F|S×T = IZ2,T (CT ). Then by [25, Lemma 6.13] the double dual F ∗∗ is an invertible sheaf. Now φ∗∗ : OS×T → F ∗∗ is injective when restricted to closed T fibers of S ×T → T , and hence by [19, Lemma 2.1.4], coker(φ∗∗ ) is also flat over T . Thus, T there exists a T -flat subscheme CT ⊂ S × T (i.e. the Cartier divisor cut out by φ∗∗ ) that T ∗∗ ∼ restricts to CT and F ∗∗ ∼ O(C ). We conclude from F ⊆ F that F I (C = Z2,T T ) for = T some T -flat subscheme Z2,T ⊂ S × T restricting to Z2,T . If ob = 0 then by [21, Prop. IV.3.2.12] again, the set of isomorphism classes of deformations forms a torsor under   Ext1S×T coker(φT ), πT∗ J ⊗ IZ2,T (CT ) . Lemma 2.3. ob = Atred (φT ) ∪ (πT∗ e(T ) ⊗ id), where   Atred (φT ) ∈ Ext1S×T coker(φT ), πT∗ L•a ⊗ IZ2,T (CT ) , is a certain reduction of the Atiyah class AtOS×T /OS×U (φT ) of φT [21, IV.2.3], and πT∗ e(T ) ⊗ id ∈ Ext1S×T (πT∗ L•a ⊗ IZ2,T (CT ), πT∗ J ⊗ IZ2,T (CT )), in which

(9)

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

12

e(T ) ∈ Ext1T (L•a , J)

(10)

is the Kodaira-Spence class associated to the square zero extension T → T . Proof. The proof is the same as the proof of [22, Thm 12.9]: in diagram (12.17) of [22], the first vertical arrow needs to be replaced by

AtOS×T /OS×U (φT ) : coker(φT ) → k1 L•,gr )/OS×U ⊗ (OS×T ⊕ IZ2,T (CT )) [1]. (OS×T ⊕IZ 1,T

This is the degree 1 part of the transitivity triangle (see item 1 in Subsection 1.3) associated to natural homomorphism of sheaves of graded algebras on S × T [id φT ]

OS×U → OS×T ⊕ IZ1,T −−−−→ OS×T ⊕ IZ2,T (CT ). By flatness of IZ1,U over OU , we see that OS×U ⊕ IZ1,U is flat over OU and hence L

(OS×U ⊕ IZ1,U ) ⊗OS×U OS×T ∼ = (OS×U ⊕ IZ1,U ) ⊗OS×U OS×T ∼ = OS×T ⊕ IZ1,T , so [21, II.2.2] L•,gr (OS×T ⊕IZ

1,T

)/OS×U

is isomorphic to

L•OS×T /OS×U ⊗ (OS×T ⊕ IZ1,T ) ⊕ L•,gr (OS×U ⊕IZ

1,U

)/OS×U

⊗ (OS×T ⊕ IZ1,T ) ,

and hence as in the proof of [22, Thm 12.9], composing AtOS×T /OS×U (φT ) with the projection L•,gr (OS×T ⊕IZ

1,T

)/OS×U

→ L•OS×T /OS×U ∼ = πT∗ L•a ,

we arrive at the definition of Atred (φT ) in (9).  Now the idea is that from the deformation/obstruction theory of the universal map Φ : [n ] • → I [n1 ] → Iβ 2 reviewed above, we construct a relative perfect obstruction theory Frel • [n1 ] L [n1 ,n2 ] [n1 ] (Proposition 2.4). Using the fact that S is nonsingular, we will deduce Sβ

/S

• the absolute perfect obstruction theory F • → L• [n1 ,n2 ] from Frel (Proposition 2.5). This Sβ

will prove Proposition 2.2 for r = 2. Proposition 2.4. The complex

∨ [n ] • Frel := Cone(Ξ)∨ ∼ [−1] = RHomπ coker(Φ), Iβ 2 [n ,n ]

defines a relative perfect obstruction theory for the morphism p : Sβ 1 2 → S [n1 ] . In • other words, Frel is perfect with amplitude contained in [−1, 0], and there exists a mor• → L•p , such that h0 (α) and h−1 (α) are respectively phism in the derived category α : Frel • isomorphism and epimorphism. The rank of Frel is equal to

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

• Rank [Frel ] = n2 − n1 −

13

β · βD . 2

•∨ Proof. Step 1: (perfectness) We show that the complex Frel is perfect with amplitude [0, 1]. By basechange and the same argument as in the proof of [20, Lemma 4.2], it suffices [n ,n ] •∨ to show that hi (Lt∗ Frel ) = 0 for i = 0, 1, where t : P → Sβ 1 2 is the inclusion of an [n1 ,n2 ]

arbitrary closed point P = (Z1 , C, Z2 , φ) ∈ Sβ we get the exact sequence

•∨ . Therefore, by the definition of Frel

•∨ · · · → ExtiS (IZ1 , IZ2 (C)) → hi (Lt∗ Frel ) → Exti+1 S (IZ2 , IZ2 ) → . . . . •∨ All the ExtiS for i = 0, 1, 2 vanish, so we deduce easily that hi (Lt∗ Frel ) = 0 for i = −1, 0, 1, 2. From the sequence above we see that

  •∨ h−1 (Lt∗ Frel ) = ker HomS (IZ2 , IZ2 ) → HomS (IZ1 , IZ2 (C)) . But by definition this map is induced by applying HomS (−, IZ2 (C)) to the map φ : •∨ IZ1 → IZ2 (C), so it takes id to φ, and hence it is injective. Therefore, h−1 (Lt∗ Frel ) = 0. 2 ∗ •∨ To prove h (Lt Frel ) = 0, we show that the map Ext2S (IZ2 , IZ2 ) → Ext2S (IZ1 , IZ2 (C)) in the exact sequence above is surjective, or equivalently by Serre duality, the dual map HomS (IZ2 , IZ1 ⊗ ωS (−C)) → HomS (IZ2 , IZ2 ⊗ ωS ) is injective. But this follows after applying the left exact functor HomS (IZ2 , −) to the injection IZ1 ⊗ ωS (−C) → IZ2 ⊗ ωS that is induced by tensoring the map φ above by ωS (−C). Step 2: (map to the cotangent complex) We construct a morphism in derived category [n ,n ] • α : Frel → L•p . Consider the reduced Atiyah class (9) in the case T = Sβ 1 2 and U = S [n1 ] . It defines an element in  [n ]  Ext1S×S [n] coker(Φ), π ∗ L•p ⊗ Iβ 2 ∼ = β

Ext1S×S [n] β



 [n ]   RHom Iβ 2 , coker(Φ) , π ∗ L•p ∼ = (by the definition of π ! )

  [n ]   Ext1S×S [n] RHom Iβ 2 , coker(Φ) ⊗ ωπ [2] ,π ! L•p ∼ = (by Grothendieck-Verdier duality) β

Ext1S [n] β

  [n ]   RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [2] , L•p ∼ =

  [n ]   HomS [n] RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] , L•p . β

So under the identification above, the reduced Atiyah class defines a morphism in derived category

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

14

 [n ]  α : RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] → L•p . But by Grothendieck-Verdier duality again,  [n ]   [n ] ∨ • RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] ∼ , = RHomπ coker(Φ), Iβ 2 [−1] ∼ = Frel and hence we are done. Step 3: (obstruction theory) We show h0 (α) and h−1 (α) are respectively isomorphism and epimorphism. We use the criterion in [1, Theorem 4.5]. Suppose we are in the situation of the diagram (6). Define [n1 ,n2 ]

f := (id, g) : S × T → S × Sβ

.

Composing e(T ) (given in (10)) and the natural morphism of cotangent complexes Lg ∗ L•p → L•a gives the element (g) ∈ Ext1T (Lg ∗ L•p , J) whose image under α is denoted by • α∗ (g) ∈ Ext1T (Lg ∗ Frel , J) .

For i = 0, 1, we will use the following identifications:

    [n ] • ExtiT Lg ∗ Frel ,J ∼ = ExtiT Lg ∗ RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] , J     [n ] ∼ = ExtiS [n] RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] , Rg∗ J β

∼ =

ExtiS×S [n] β



   [n ] RHom Iβ 2 , coker(Φ) ⊗ ωπ [1] , π ! Rg∗ J

    [n2 ] ∼ , coker(Φ) , π ∗ Rg∗ J . = Exti+1 [n] RHom Iβ S×S β

Here we have used the fact that Lg ∗  Rg∗ i.e. Lg ∗ is the left adjoint of Rg∗ , and Grothendieck-Verdier duality. Now using Rf∗ πT∗ = π ∗ Rg∗ in the last Ext above, we get Exti+1

[n] S×Sβ



 [n ]   RHom Iβ 2 , coker(Φ) , Rf∗ πT∗ J ∼ =

(by Lf ∗  Rf∗ )

 ∗  [n2 ]  ∗  ∼ Exti+1 S×T Lf RHom Iβ , coker(Φ) , πT J =   ∗ [n2 ]  ∗  ∗ ∼ Exti+1 S×T RHom Lf Iβ , Lf coker(Φ) , πT J =  ∗  [n ] ∗ ∗ [n2 ] ∼ Exti+1 = (by flatness of coker(Φ) and Iβ 2 ) S×T Lf coker(Φ), πT J ⊗ Lf Iβ   ∗ Exti+1 S×T coker(φT ), πT J ⊗ IZ2,T (CT ) . Similar to the Step 2 it can be seen that the composition g∗ α

• Lg ∗ Frel −−→ Lg ∗ L•p → L•a

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

15

arises from Atred (φT ) over S × T (see (9)). Therefore, ob(φT , J) = Atred (φT ) ∪ (πT∗ e(T ) ⊗ id), is identified with the element α∗ (g) via the identifications above for i = 1. By the definition of the obstruction class ob(φT , J), this means that α∗ (g) vanishes if and only if there exists an extension g of g corresponding to (8). Using the identifications above, this time for i = 0, we can see that if α∗ (g) = 0, then the set of extensions forms a torsor under HomT (Lg ∗ F • , J). Now by [1, Theorem 4.5] α is an obstruction theory. • Step 4: (rank of Frel ) The claim about the rank follows from • Rank [Frel ] = Rank [Cone(Ξ)]       [n ]  = Rank RHomπ I [n1 ] , Iβ 2 − Rank RHomπ I [n2 ] , I [n2 ]

= χ(IZ1 , IZ2 (C)) − χ(IZ2 , IZ2 ) = −n1 − n2 + χ(OS (C)) + 2n2 − χ(OS ) = n2 − n1 − β · KC /2 + β 2 /2, [n1 ,n2 ]

where (Z1 , C, Z2 , φ) is a closed point of Sβ

.



Proposition 2.5. Sβ 1 2 is equipped with the absolute perfect obstruction theory F • → L• [n1 ,n2 ] . Its virtual tangent bundle is given by [n ,n ]

Sβ

F

•∨

∼ = Cone



    RHomπ I [n1 ] , I [n1 ] ⊕ RHomπ I [n2 ] , I [n2 ]

0

  [−Ξ Ξ] [n ]  −−−−−→ RHomπ I [n1 ] , Iβ 2 . Proof. Since S [n1 ] is nonsingular, by the standard techniques (see [37, Section 3.5])

• θ F • := Cone Frel − → p∗ ΩS [n1 ] [1] [−1]

(11)

[n ,n ]

gives a perfect absolute obstruction theory over Sβ 1 2 , where θ is the composition of • α : Frel → L•p and the Kodaira-Spencer map c : L•p → p∗ ΩS [n1 ] [1]. We claim that θ is given by  [n ]    • ∼ Frel =RHomπ Iβ 2 , coker(Φ) ⊗ ωπ [1] → RHomπ I [n1 ] , coker(Φ) ⊗ ωπ [1]   → RHomπ I [n1 ] , I [n1 ] ⊗ ωπ 0 [2] ∼ = p∗ ΩS [n1 ] [1], [n ]

where the first and second maps are respectively induced by Φ : I [n1 ] → Iβ 2 and the natural map c : coker(Φ) → I [n1 ] [1]. To see the claim consider the commutative diagram

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

16

[n1 ,n2 ]

of graded sheaves of algebras on S × Sβ natural maps4 :

with all unlabelled arrows are the obvious

OS×S [n1 ]

OS×S [n1 ,n2 ] ⊕ I [n1 ]

OS

OS×S [n1 ,n2 ]

(id,Φ)

β

[n2 ]

OS×S [n1 ,n2 ] ⊕ Iβ

(12)

β

[n2 ]

OS×S [n1 ,n2 ] ⊕ Iβ

β

,

β

(id,Φ)

OS

OS×S [n1 ,n2 ]

OS×S [n1 ,n2 ] ⊕ I [n1 ]

β

β

Taking the transitivity triangles (see item 1 in Subsection 1.3) of the top two rows and applying the same construction leading to the definition of Atred (Φ) in the proof of [n ,n ] Lemma 2.3, we obtain the commutative diagram of sheaves on S × Sβ 1 2 Atred (Φ)

coker(Φ)

L•p ⊗ Iβ

[n2 ]

[1]

L• [n1 ,n2 ] ⊗ Iβ

[n2 ]

[n2 ]

Iβ

Sβ

[1]

in which both vertical arrows are the natural maps, the bottom horizontal arrow is the pullback of the Atiyah class [n2 ]

At(Iβ

[n2 ]

) ∈ Ext1 (Iβ

, L• [n2 ] ⊗ Iβ

[n2 ]

Sβ

)

composed with q∗ L• [n2 ] → L• [n1 ,n2 ] induced by the natural morphism q : Sβ

[n1 ,n2 ]

Sβ

Sβ

→

[n ]

Sβ 2 . Taking cones over the vertical arrows in turn induces the commutative diagram p∗ ΩS [n1 ] ⊗ Iβ

[n2 ]

I [n1 ] [1] c

coker(Φ)

[2]

(13)

c⊗id Atred (Φ)

L•p ⊗ Iβ

[n2 ]

[1].

On the other hand, the transitivity triangles of the bottom two rows of (12) imply that the top row in (13) is in fact the composition 4

Following our convention we have suppressed the symbols for pullbacks via natural morphisms.

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046 At(I [n1 ] )

id ⊗Φ

I [n1 ] −−−−−−→ p∗ ΩS [n1 ] ⊗ I [n1 ] [1] −−−→ p∗ ΩS [n1 ] ⊗ Iβ

[n2 ]

17

[1],

where At(I [n1 ] ) is the (pullback of the) usual Atiyah class. The claim now follows from this and the construction of the map α using the class Atred (Φ) in Step 2 of proof of Proposition 2.4. To ease the notation let      [n ]  A• := RHomπ I [n1 ] , I [n1 ] , B • := RHomπ I [n2 ] , I [n2 ] , C • := RHomπ I [n1 ] , Iβ 2 , and denote by E • the right hand side of the expression in the proposition. By Proposi•∨ tion 2.4, Frel = Cone (B • → C • ), so by (2) and the claim above, (11) can be rewritten as

θ∨ F •∨ = Cone A•0 −−→ Cone (B • → C • ) . Consider the commutative diagram Rπ∗ O

Rπ∗ O ,

[id id]t

B•

A• ⊕ B •

id

A•

in which the bottom row is the natural exact triangle. Taking the cone of the diagram one gets the exact triangle B • → [A• ⊕ B • ]0 → A•0 .

(14)

Next consider the commutative diagram

B•

[id 0]t

A• ⊕ B • [−Ξ Ξ]

B•

Ξ

C•

A•0 ⊕ Rπ∗ O [θ ∨ 0]

Cone (B • → C • )

in which both rows are exact triangles, and in the rightmost vertical arrow we use the splitting A• ∼ = A•0 ⊕ Rπ∗ O given by the trace map (the left square is obviously commutative, and the right square is commutative by the claim we proved above). Now by construction the vertical arrows in the above diagram factor through the exact triangle (14), and hence we arrive at the following commutative diagram in which all the rows and columns are exact triangle.

18

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

B•

[A• ⊕ B • ]0

A•0

[−Ξ Ξ]

B

•

Ξ

C

•

θ∨ •

Cone (B → C • )

E•

F •∨ .

In fact the columns and the top and middle rows are exact triangles with commutative top squares, and the bottom row is induced from the rest of the diagram by taking the cone. Therefore, the bottom row must also be an exact triangle which means that E• ∼ = F •∨ as desired.  This finishes the proof of Proposition 2.2 in the case r = 2. Propositions 2.4 and 2.5 imply Corollary 2.6. The perfect obstruction theory of Proposition 2.5 defines a virtual funda[n ,n ] mental class on Sβ 1 2 denoted by [n1 ,n2 ] vir

[Sβ

]

[n1 ,n2 ]

∈ Ad (Sβ

d = n1 + n2 −

),

β · βD . 2



2.2. Gillam’s construction • The relative obstruction theory Frel is obtained from the deformation/obstruction [n ] [n1 ] theory of the universal map Φ : I → Iβ 2 . As mentioned in the introduction, following [16], one can instead use deformation/obstruction theory of the quotient coker(Φ) to construct a relative perfect obstruction theory • Frel → L• [n1 ,n2 ] Sβ

[n1 ,n2 ]

This amounts to identifying Sβ

/S [n2 ]

.

with a component of the relative quot scheme

QuotS×S [n2 ] /S [n2 ] (I [n2 ] ) of quotients of I [n2 ] and applying [16, Theorem 4.6]. By a similar argument as Proposition 2.5 (using the smoothness of S [n2 ] this time), one can then deduce an absolute perfect obstruction theory F  • → L• [n1 ,n2 ] . By comparing the K-theory classes of F • Sβ

and F  • the reader can verify that the resulting virtual class from this approach is the same as that of Corollary 2.6 (see [43]). 2.3. Reduced perfect obstruction theory In this section we assume that for any effective line bundle L on S with c1 (L) = β, we have

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

|LD | = ∅

19

or equivalently H 2 (L) = 0.

(15)

Recall that the relative virtual tangent bundle of Proposition 2.4 is given by    [n ] [n ]   [n ]  •∨ Frel = Cone RHomπ Iβ 2 , Iβ 2 → RHomπ I [n1 ] , Iβ 2 . We get a natural map  [n ] [n ]  •∨ μ : Frel → RHomπ Iβ 2 , Iβ 2 [1], that induces  [n ] [n ]  •∨ ∼ h1 (μ) : h1 (Frel ) = Ext2π Iβ 2 /I [n1 ] , Iβ 2 →  [n ] [n ]  p Ext2π Iβ 2 , Iβ 2 ∼ = R2 π∗ OS×S [n1 ,n2 ] ∼ = O g[n1 ,n2 ] . Sβ

β

We claim that h1 (μ) is surjective. To see this, by basechange, it suffices to prove that [n ,n ] h1 (μ) is fiberwise surjective. Let t : P → Sβ 1 2 be the inclusion of an arbitrary closed [n1 ,n2 ]

point P = (Z1 , C, Z2 , φ) ∈ Sβ sequence5

. Then, by basechange we have the natural exact

h1 (μ)P

•∨ · · · → h1 (Lt∗ Frel ) −−−−→ Ext2S (IZ2 (C), IZ2 (C)) − → Ext2S (IZ1 , IZ2 (C)) → 0. u

The surjectivity of the map u was established in Step 1 of the proof of Proposition 2.4. We have Ext2S (IZ2 (C), IZ2 (C)) ∼ = Ext2S (IZ2 , IZ2 ) ∼ = H 2 (OS ), Ext2S (IZ1 , IZ2 (C))∗ ∼ = HomS (IZ2 , IZ1 (C)D ) ⊆ HomS (IZ2 , OS (C)D ) ∼ = H 0 (OS (C)D ). By assumption (15), H 0 (OS (C)D ) = 0, and hence h1 (μ)P is surjective and the claim follows. We now have the diagram Ext1S (IZ1 , IZ1 )0

h1 (θ ∨ )P

Ext2S (IZ2 (C)/IZ1 , IZ2 (C))

h1 (Lt∗ F •∨ )

h1 (μ)P

Ext2S (IZ2 (C), IZ2 (C)) where the first row is exact by the proof Proposition 2.5. But since 5

Note that Ext3S (coker(φ), IZ2 (C)) = Ext3S (IZ2 (C), IZ2 (C)) = 0.

0

20

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

h1 (μ)P ◦ h1 (θ∨ )P = 0, the surjection h1 (μ)P factors through h1 (Lt∗ F •∨ ). Therefore, by basechange again there p exists a surjection h1 (F •∨ ) → O g[n1 ,n2 ] . Sβ

Proposition 2.7. If the condition (15) is satisfied and pg (S) > 0, then, [n1 ,n2 ] vir

[Sβ

]

= 0.

Proof. Under the assumptions of the proposition, we showed above that the obstruction sheaf admits a surjection h1 (F •∨ ) → O

pg [n1 ,n2 ]

Sβ

[1 ... 1]

−−−−→ OS [n1 ,n2 ] , β

and hence the associated virtual class vanishes by [23, Theorem 1.1].  Definition 2.8. The map h1 (μ) induces the morphism in derived category h1 (μ)

•∨ F •∨ → h1 (F •∨ )[−1] → h1 (Frel )[−1] −−−→ O

pg [n]

Sβ

Dualizing gives a map O

pg [n]

Sβ

[−1].

• [1] → F • . Define Fred to be its cone.

• We will show that under a slightly stronger condition than (15), Fred gives rise to a [n1 ,n2 ] perfect obstruction theory over Sβ . First note that the curve class β ∈ H 1,1 (S) ∩ H 2 (S, Z) defines an element of H 1 (ΩS ) and consider the natural pairing TS ⊗ ΩS → OS . This condition is6 ∗∪β

H 1 (TS ) −−→ H 2 (TS ⊗ ΩS ) → H 2 (OS )

is surjective.

(16)

• To show Fred gives rise to a perfect obstruction theory, we use the beautiful idea of [26]. We sketch their method here and make some necessary changes; the reader can find the missing detail in [26]. S is embedded as the central fiber of an algebraic twistor family S → B, where B is a first order Artinian neighborhood of the origin in a certain pg -dimensional family of the first order deformations of S.7 Explicitly, let

V ⊂ H 1 (TS ) be a subspace over which ∗ ∪ β in (16) restricts to an isomorphism, and let m denote the maximal ideal at the origin 0 ∈ H 1 (TS ). Then, 6 7

This is condition (3) in [26]. To simplify notation, we have used S instead of SB that was used in [26].

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

21

B := Spec(OV /m2 ), and S is the restriction of a tautological flat family of surfaces with Kodaira-Spencer class the identity in H 1 (TS )∗ ⊗ H 1 (TS ) ∼ = Ext1 (ΩS , OS ⊗ H 1 (TS )). The Zarisiki tangent space TB is naturally identified with V . By [26, Lemma 2.1], S is transversal to the Noether-Lefschetz locus of the (1, 1)-class β, and as a result, β does not deform outside of the central fiber of the family. Using this fact, as in [26, Proposition 2.3], one can show that [n1 ,n2 ]

Sβ

[n ,n ] ∼ = (S/B)β 1 2 ,

(17)

where the right hand side is the relative nested Hilbert scheme of the family S → B. We use the same symbols [n2 ]

Φ : I [n1 ] → Iβ

[n ,n2 ]

as before to denote the universal objects over S ×B (S/B)β 1 [n ,n ] ×B (S/B)β 1 2 .

projection to the second factor of S be adapted with no changes to prove that

, and we let π be the

The arguments of Section 2.1 can

   [n1 ] [n1 ]   [n2 ] [n2 ]   [n1 ] [n2 ]  Cone RHomπ I , I ⊕ RHomπ I , I → RHomπ I , Iβ 0

is the virtual tangent bundle of a perfect B-relative obstruction theory G•rel → L• , and [n1 ,n2 ] (S/B)β

/B

G• := Cone (G•rel → ΩB [1]) [−1] → L•

[n1 ,n2 ]

(S/B)β

is the associated absolute perfect obstruction theory. By the definitions of F • and G•rel , and the isomorphism (17), we see that F • ∼ = G•rel . Now we claim that the composition • G• → G•rel ∼ = F • → Fred • is an isomorphism. By the definitions of G•rel and Fred , to prove the claim, it suffices to show that

O

pg [n]

Sβ

→ F • [−1] ∼ = G•rel [−1] → ΩB

(18)

is an isomorphism. By the Nakayama lemma we may check this at a closed point P = [n ,n ] (Z1 , C, Z2 , φ) ∈ Sβ 1 2 . Define the reduced Atiyah class corresponding to P as follows. Consider the natural homomorphisms of sheaf of graded algebras on S

A. Gholampour et al. / Advances in Mathematics 365 (2020) 107046

22

OSpec C → OS ⊕ IZ1 − → OS ⊕ IZ2 (C). The degree 1 part of the transitivity triangle (see item 1 in Subsection 1.3) associated to the graded cotangent complexes gives the first arrow in   project coker(φ) → k1 L•,gr OS ⊕IZ ⊗ (OS ⊕ IZ2 (C) ) [1] −−−−→ ΩS ⊗ I2 (C)[1]. 1

Define Atred (φ) to be the composition of these two arrows. After dualizing and using the identifications above the pullback of (18) to P becomes Atred (φ)

TB = V ⊂ H 1 (TS ) −−−−−→ Ext2 (coker(φ), IZ2 (C)) h1 (μ)P

tr

−−−−→ Ext2 (IZ2 (C), IZ2 (C)) − → H 2 (OS ). Here as in [26], one needs to use a similar argument as [37, Proposition 13] to deduce that the composition of G•rel → L• and the Kodaira-Spencer map [n1 ,n2 ] (S/B)β

L•

[n ,n2 ]

[n1 ,n2 ]

(S/B)β

/B

→ ΩB [1] for (S/B)β 1

/B

coincides with the cup product of Atred (φ)

and the Kodaira-Spencer class for S. Similar to [37], this is achieved by relating the re[n ,n ] duced Atiyah class of S ×B (S/B)β 1 2 arising (as in Lemma 2.3) from the transitivity triangle associated to the natural maps of sheaves of graded algebras OB → OS×

[n1 ,n2 ] B (S/B)β

⊕ I [n1 ] → OS×

[n1 ,n2 ] B (S/B)β

[n2 ]

⊕ Iβ

to the reduced Atiyah classes of each factors. Lemma 2.9. h1 (μ)P ◦ Atred (φ) = At(IZ2 (C)), where At(IZ2 (C)) ∈ Ext1 (IZ2 (C), ΩS ⊗ IZ2 (C)) is the usual Atiyah class. Proof. Consider the commutative diagram of sheaves of graded algebras with all unlabelled arrows are the obvious natural maps:

C

OS ⊕ I1

C

OS

(id,φ)

OS ⊕ I2 (C) OS ⊕ I2 (C).

Taking the degree 1 part of the transitivity triangles of the rows followed by a projection as in the definition of Atred (φ) above we get the commutative diagram

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coker(φ)

Atred (φ)

23

ΩS ⊗ I2 (C)[1]

h1 (μ)P

I2 (C)

At(I2 (C))

ΩS ⊗ I2 (C)[1]

proving the lemma.  But by [2, Prop 4.2], tr ◦ At(IZ2 (C)) = − ∗ ∪β, which by condition (16) is an isomorphism when restricted to V ⊂ H 1 (TS ), and hence the claim is proven. We have shown • Proposition 2.10. If the condition (16) is satisfied, then, Fred is a perfect obstruction [n] theory on Sβ , and hence defines a reduced virtual fundamental class

[n1 ,n2 ] vir ]red

[Sβ

[n1 ,n2 ]

∈ Ad (Sβ

),

d = n1 + n2 −

β · βD + pg (S). 2



2.4. Invariants Let P1 , . . . , Ps be polynomials in the Chern classes of I [n1 ] , I [n2 ] , I−β , π ∗ TS [n1 ] , π TS [n2 ] , etc. cupped with the pullback of a cohomology classes from S then, we can define the invariant ∗



s 

NS (n1 , n2 , β; P1 , . . . , Ps ) :=

π∗ Pi .

i=1

[n1 ,n2 ] vir ]

[Sβ

If the condition (16) is satisfied, we can define the reduced invariants  Nred S (n1 , n2 , β; P1 , . . . , Ps )

s 

:= [n ,n ] [Sβ 1 2 ]vir red

π∗ Pi .

i=1

Let u := c1 (O(Zβ ))|x×S [n1 ,n2 ] , where x ∈ S is an arbitrary closed point. Define β

⎛ ⎞  [n ,n ] PS (n1 , n2 , β) := det∗ ⎝ [Sβ 1 2 ]vir ∩ ui ⎠ ∈ H ∗ (Pic(S)). i≥0 [0,n ]

In Section 3 we will see that virtual classes [Sβ ]vir , [Sβ 2 ]vir red coincide respectively with the virtual classes constructed in [6] and [26]. Therefore, by suitable choices of the

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integrands π∗ Pi , the invariants Nred S (0, n2 , β; P1 , . . . , Ps ) recover the reduced stable pair invariants of [26]. Similarly, the invariants PS (0, 0, β) recover Poincaré invariants of [6]. In [15], we express certain contributions to the reduced localized DT invariants of S in terms of the invariants NS (n1 , n2 , β; P) by making suitable choices of the integrand P.8 We will study some of the invariants NS (n1 , n2 , 0; P) in Sections 4 and 5. 2.5. Proof of Theorem 1 (Proposition 2.2) In Section 2.1 we proved Proposition 2.2 in the case r = 2. We now use induction on r to prove the theorem in general. For the simplicity of the notation, we show in detail how the result of Section 2.1 can be used to prove Proposition 2.2 in the case r = 3. Other induction steps are completely similar and hence are omitted. Suppose that n := n1 , n2 , n3 is a sequence of nonnegative integers, and β := β1 , β2 is a sequence of effective curve classes in H 2 (S, Z). Define n := n1 , n2 . Our goal is to prove the expression in Proposition 2.2 for r = 3 is a perfect obstruction theory. Consider the chain of natural forgetful morphisms and the associated exact triangle of cotangent complexes [n] f2

[n ] f1

Sβ −→ Sβ1 −→ S [n1 ] ,

j2

j1

j3

L•f2 [−1] −→ Lf2∗ (L•f1 ) −→ L•f −→ L•f2

(19)

where f := f1 ◦ f2 = pts ◦ pr1 , using the notation at the beginning of Section 2. Proposition 2.4, provides the relative perfect obstruction theory for the morphism f1, that we denote by 1 Ff•1 −→ L•f1 .

α

(20) α

2 Lemma 2.11. There exists a relative perfect obstruction theory Ff•2 −→ L•f2 , where

 [n ]  Ff•2 = RHomπ Iβ23 , coker(Φ2 ) ⊗ ωS [1]

(21)

Proof. The proof is along the line of the proof of Proposition 2.4 (see Step 2 of that proof for the corresponding expression in RHS of (21)). This time the obstruction theory is obtained by the deformation/obstruction theory of the universal map [n ]

Φ2 : I [n2 ] → Iβ23

while the data (I [n1 ] , I [n2 ] , Zβ1 , Φ1 ) is kept fixed.  8 To clarify the potential confusion for the reader, we emphasize that the reduced localized DT invariants [n ,n ] of [15] uses the non-reduced virtual class [Sβ 1 2 ]vir .

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Lemma 2.12. The complexes Ff•1 and Ff•2 fit into the following commutative diagram: r

Ff•2 [−1]

Lf2∗ (Ff•1 )

(22)

f2∗ (α1 )

α2 [−1] j2

L•f2 [−1]

Lf2∗ (L•f1 ).

Proof. Step 1: (Define the map r) All the maps in diagram (22) except r are already defined above (see (19), (20), and Lemma 2.11). By the universal properties of the Hilbert schemes and using our convention in suppressing the pullback symbols from the universal ideal sheave, we can write  [n ]  Lf2∗ (Ff•1 ) ∼ = RHomπ Iβ12 , coker(Φ1 ) ⊗ ωS [1]

(23)

Twisting by O(Zβ1 ), we get [n ]

[n ]

Φ2 (Zβ1 ) : Iβ12 → Iβ23 (Zβ1 ), and hence (21) can be written as   [n ] Ff•2 ∼ = RHomπ Iβ23 (Zβ1 ), coker(Φ2 (Zβ1 )) ⊗ ωS [1] . Φ

[n ] Φ2 (Zβ )

(24)

[n ]

1 The chain of maps I [n1 ] −−→ Iβ12 −−−−−1→ Iβ23 (Zβ1 ) induces the natural exact triangle

i

i2 [1]

3 coker(Φ2 (Zβ1 ) ◦ Φ1 ) −→ coker(Φ2 (Zβ1 )) −−−→ coker(Φ1 )[1].

(25)

The maps i2 [1] and Φ2 (Zβ1 ) induce  [n ]  Ff•2 [−1] ∼ = RHomπ Iβ23 (Zβ1 ), coker(Φ2 (Zβ1 )) ⊗ ωS →  [n ]  RHomπ Iβ12 , coker(Φ2 (Zβ1 )) ⊗ ωS →  [n ]  RHomπ Iβ12 , coker(Φ1 ) ⊗ ωS [1] ∼ = Lf2∗ (Ff•1 ).

(26)

The map r in diagram (22) is then defined by composition of two maps in (26). Step 2: (Commutativity of diagram (22)) We start with the following diagram in which the columns are the exact triangles (25) and (19):

26

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coker(Φ1 )[1]

(id ⊗Φ2 (Zβ1 ))◦Atred (Φ1 )[1]

  [n ] π ∗ Lf2∗ L•f1 [1] ⊗ Iβ23 (Zβ1 )[1]

(27)

(π ∗ j2 [1]⊗id)[1]

i2 [1] Atred (Φ2 (Zβ1 ))

coker(Φ2 (Zβ1 ))

π ∗ L•f2 ⊗ Iβ23 (Zβ1 )[1] [n ]

(π ∗ j3 ⊗id)[1]

i3 Atred (Φ2 (Zβ1 )◦Φ1 )

coker(Φ2 (Zβ1 ) ◦ Φ1 )

π ∗ L•f ⊗ Iβ23 (Zβ1 )[1] [n ]

(π ∗ j1 ⊗id)[1]

i1 (id ⊗Φ2 (Zβ1 ))◦Atred (Φ1 )

coker(Φ1 )

  [n ] π ∗ Lf2∗ L•f1 ⊗ Iβ23 (Zβ1 )[1]

We prove diagram (27) is commutative. For this, consider the following natural com[n] mutative diagrams of sheaf of graded algebras over S × Sβ (following our convention we have suppressed the symbols for pullbacks of theses via natural morphisms): [n ]

OS×S [n1 ]

OS×S [n] ⊕ I [n1 ]

OS×S [n] ⊕ Iβ23 (Zβ1 )

OS×S [n1 ]

OS×S [n ] ⊕ I [n1 ]

OS×S [n ] ⊕ Iβ12 ,

OS×S [n ]

OS×S [n] ⊕ Iβ12

OS×S [n] ⊕ Iβ23 (Zβ1 )

OS×S [n1 ]

OS×S [n] ⊕ I [n1 ]

OS×S [n] ⊕ Iβ23 (Zβ1 ).

β

β1

β

[n ]

β1

and

β1

[n ]

β

β

[n ]

β

[n ]

β

Applying k1 (−) to the resulting commutative diagrams of the transitivity (see item 1 in Subsection 1.3) triangle of each row, we get the commutativity of the following two squares: coker(Φ2 (Zβ1 ) ◦ Φ1 ) i1

coker(Φ1 )



[n ] [n ] π ∗ L•f ⊗ Iβ23 (Zβ1 )[1] ⊕ Iβ12 [1] (π ∗ j1 ⊗id ⊕Φ1 )[1]





[n ] π ∗ Lf2∗ L•f1 ⊗ Iβ23 (Zβ1 )[1] ⊕ I [n1 ] [1],

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and coker(Φ2 (Zβ1 ))

[n ] [n ] π ∗ L•f2 ⊗ Iβ23 (Zβ1 )[1] ⊕ Iβ23 (Zβ1 )[1] (π ∗ j3 ⊗id ⊕Φ2 (Zβ1 ))[1]

i3

coker(Φ2 (Zβ1 ) ◦ Φ1 )



[n ] [n ] π ∗ L•f ⊗ Iβ23 (Zβ1 )[1] ⊕ Iβ12 [1].

Now projecting to the first factors in the second columns of the last two diagrams, and using the definition of Atred (−) (from the proof of Lemma 2.3), we obtain the commutativity of the bottom and middle squares of diagram (27). Since in diagram (27) both columns are exact triangles, the commutativity of the top square follows, and hence we have proven that the whole diagram (27) commutes. Recall from Step 2 in the proof of Proposition 2.4, that the maps αi : Ff•i → L•fi are induced from the classes Atred (Φ1 ) and Atred (Φ2 (Zβ1 )). Therefore, by the definition of the map r in Step 1 of the proof, the commutativity of diagram (22) is equivalent to the commutativity of the top square in diagram (27) proven above, and hence the proof of lemma is complete.  As a result of Lemma 2.12 we get a commutative digram r

Ff•2 [−1]

Lf2∗ (Ff•1 ) f2∗ (α1 )

α2 [−1]

L•f2 [−1]

Cone(r) =: Ff•

j2

Lf2∗ (L•f1 )

(28)

α3 j1

L•f ,

in which both rows are exact triangles (the commutativity of the right square was established in Lemma 2.12). Proposition 2.13. α3 : Ff• → L•f is a relative perfect obstruction theory. Proof. Since α1 and α2 are perfect obstruction theories, we know Ff•2 [−1] is of perfect amplitude contained in [0, 1] and Lf2∗ (Ff•1 ) is of perfect amplitude contained in [−1, 0], therefore Ff• = Cone(r) is of perfect amplitude contained in [−1, 0]. It remains to show that h0 (α3 ) is an isomorphism and h−1 (α3 ) is surjective. From diagram (28) and the fact that α1 and α2 are perfect obstruction theories, we get the following commutative diagram in which both rows are exact: h−1 (Ff•1 )

h−1 (Ff• )

h−1 (Ff•2 )

h0 (Ff•1 )

h−1 (α3 )

h−1 (L•f1 )

h−1 (L•f )

h0 (Ff• )

h0 (Ff•2 )

0

h0 (L•f2 )

0.

h0 (α3 )

h−1 (L•f2 )

h0 (L•f1 )

h0 (L•f )

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Applying 4-lemma once to the leftmost three squares and once to the rightmost three squares above prove the desired properties for h0 (α3 ) and h−1 (α3 ).  Proof of Proposition 2.2 (for r = 3). First note that by construction, for i = 1, 2,  [ni+1 ] [ni+1 ]  Ξi  [ni ] [ni+1 ] 

Ff•∨ = Cone RHom , I − → RHom I I , Iβi . π π i Now define   A•i := RHomπ I [ni ] , I [ni ] ,

 [n ] Bj• := RHomπ I [nj ] , Iβj j+1

i = 1, 2, 3, j = 1, 2,

and consider the following two commutative diagrams A•3

A•2 ⊕ A•3 

A•3

Ξ1 0 −Ξ2 Ξ2

[0 Ξ2 ]t

Cone(Ξ1 )[−1] r ∨ [−1]

Cone(Ξ2 )

A•2



[Ξ1 −q◦Ξ2 ]t

B1• ⊕ B2•



id 0 0 q



A•2

B1• ⊕ Cone(Ξ2 ) Ξ1

B1•

[Ξ1 −q◦Ξ2 ]t

B1• ⊕ Cone(Ξ2 )

B1•

in which all four rows are natural exact triangles and q : B2• → Cone(Ξ2 ) is the natural map. Taking cones of the columns of the right diagram gives   Ff•∨ ∼ = Cone [Ξ1 − q ◦ Ξ2 ]t . = Cone(r∨ [−1]) ∼ Therefore, taking cones of the columns of the left diagram 

Ff•∨



1

  −Ξ2 Ξ2 ∼ = Cone [Ξ1 − q ◦ Ξ2 ]t ∼ = Cone A•2 ⊕ A•3 −−−−−−−→ B1• ⊕ B2• .

Ξ

0

As in the proof of Proposition 2.5, the fact that S [n1 ] is nonsingular can be used to show that   F • := Cone Ff• → f ∗ ΩS [n1 ] [1] [−1] [n]

is an absolute perfect obstruction theory for Sβ , and then (using the expression above for Ff•∨ ) to prove that

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  F •∨ ∼ = Cone [A•1 ⊕ A•2 ⊕ A•3 ]0 → B1• ⊕ B2• , where the arrow is as in Proposition 2.2.  3. Special cases In this section, we show that the virtual fundamental classes arising from the perfect • obstruction theories F • and Fred of Propositions 2.5 and 2.10 specialize to several interesting and important cases such as the ones arising from the algebraic Seiberg-Witten theory and the reduced stable pair theory of surfaces. For the sake of brevity we do not try to match our perfect obstruction theories with these other cases, but rather we only match K-group classes of the underlying virtual tangent bundles. Since the virtual fundamental class only depends on the K-theory class of the virtual tangent bundle [43], this is sufficient for the purpose of the following proposition: Proposition 3.1. The virtual fundamental class of Theorem 1 recovers the following known cases: [n,n] [n,n] 1. If β = 0 and n1 = n2 = n then Sβ=0 ∼ = S [n] and [Sβ=0 ]vir = [S [n] ] is the fundamental class of the Hilbert scheme of n points. 2. If β = 0 and n = n2 = n1 − 1, as it is known [n+1,n]

Sβ=0

∼ = P (I [n] ) := Proj Sym(I [n] ) → S × S [n]

is nonsingular [29, Section 1.2], then, [n+1,n] vir

[Sβ=0

]

[n+1,n]

= [Sβ=0

] ∩ c1 (H)

for a line bundle H on P (I [n] ). [n,0] 3. If β = 0 and n2 = 0, then Sβ=0 ∼ = S [n] and [n,0]

[n]

[Sβ=0 ]vir = (−1)n [S [n] ] ∩ cn (ωS ), [n]

where ωS is the rank n tautological vector bundle over S [n] associated to the canonical bundle ωS of S. [0,0] 4. If n1 = n2 = 0 and β = 0, then Sβ = Sβ is the Hilbert scheme of divisors in class [0,0]

[0,0]

β, and [Sβ ]vir and [Sβ ]vir red (in case β satisfies condition (16)) coincide with the virtual cycles constructed in [6]. [0,n ] 5. If n1 = 0 and β = 0, then Sβ 2 is the relative Hilbert scheme of points on the [0,0]

universal divisor over Sβ [0,n ] [Sβ 2 ]vir

, and by [41] is isomorphic to a moduli space of stable

[0,n ] [Sβ 2 ]vir red

pairs; and (in case β satisfies condition (16)) are the same as the virtual fundamental classes of [26].

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1 Proof. If β = 0, the nested Hilbert scheme of points S [n1 ≥n2 ] := Sβ=0 fundamental class

[n ,n2 ]

carries a virtual

[S [n1 ≥n2 ] ]vir ∈ An1 +n2 (S [n1 ≥n2 ] ). Note that by [5], S [n1 ≥n2 ] is nonsingular only in the following two cases: • n1 = n2 . In this case S [n1 ≥n2 ] ∼ = S [n1 ] by definition, and [S [n1 ] ]vir = [S [n1 ] ], because • ∼ by Proposition 2.4, Frel = 0, and so by Proposition 2.5, F • ∼ = ΩS [n1 ] . This gives part 1. [n2 +1,n2 ] • n1 = n2 + 1. In this case, since S is nonsingular of dimension 2n2 + 2 (see [5,29]). The virtual dimension is 2n2 + 1, and hence the obstruction sheaf H := h1 (F •∨ ) is an invertible sheaf. Then, we can write [S [n2 +1,n2 ] ]vir = [S [n2 +1,n2 ] ] ∩ c1 (H) , where we have used [1, Proposition 5.6] to write [S [n1 ≥n2 ] ]vir as the fundamental class capped with the Euler class of the obstruction bundle. We can express c1 (H) in terms of other classes. We know that S [n2 +1,n2 ] ∼ = P (I [n2 ] ) (see [29, Section 1.2]), so in the K-group of P (I [n2 ] ) we can write H − TP (I [n2 ] ) = [RHomπ (I [n1 ] , I [n1 ] ) ⊕ RHomπ (I [n2 ] , I [n2 ] )]0 − RHomπ (I [n1 ] , I [n2 ] ). In taking the Chern class, we can ignore the trivial terms and hence we have   c1 (H) = c1 TP (I [n2 ] ) − TS [n1 ] − TS [n2 ] − RHomπ (I [n1 ] , I [n2 ] ) . This proves part 2. If n2 = 0 and β = 0, then we get a perfect obstruction theory over the nonsingular Hilbert scheme of points S [n1 ] that is arising from the natural obstruction theory of the Hilbert scheme. In fact in this case         RHomπ I [n1 ] , I [n1 ] ⊕ RHomπ O, O F •∨ ∼ → RHomπ I [n1 ] , O = Cone 0



    ∼ = Cone RHomπ I [n1 ] , I [n1 ] → RHomπ I [n1 ] , O



  ∼ = RHomπ I [n1 ] , OZ [n1 ] . Note that   TS [n1 ] = h0 (F •∨ ) ∼ = Homπ I [n1 ] , OZ [n1 ] ,

  h1 (F •∨ ) ∼ = Ext1π I [n1 ] , OZ [n1 ] .

Since S [n1 ] is nonsingular of dimension 2n1 , we see that the obstruction sheaf h1 (F •∨ ) is a vector bundle of rank n1 , and hence by [1, Proposition 5.6]  

[S [n1 ] ]vir = [S [n1 ] ] ∩ cn1 Ext1π I [n1 ] , OZ1 .

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  We were notified by Richard Thomas that the obstruction bundle Ext1π I [n1 ] , OZ1   [n] can be identified with the dual of the tautological bundle ωS := π∗ ωS |OZ1 . This can   be seen by applying Hom OZ1 , − to the short exact sequence 0 → I [n1 ] ⊗ ωS → ωS → ωS |OZ1 → 0 over S × S [n1 ] to get the isomorphism     ωS |OZ1 ∼ = Hom OZ1 , ωS |OZ1 ∼ = Ext1 OZ1 , I [n1 ] ⊗ ωS . Now pushing forward, we prove the claim      ∗ [n] ωS ∼ = π∗ Ext1 OZ1 , I [n1 ] ⊗ ωS ∼ = Ext1π OZ1 , I [n1 ] ⊗ ωS ∼ = Ext1π I [n1 ] , OZ1 , where the second isomorphism is because of local to global spectral sequence (as Z1 is fiberwise 0-dimensional) and the third one is by Grothendieck-Verdier duality. This completes the proof of part 3. [0,0] If n1 = n2 = 0, and β = 0, the perfect obstruction theory F •∨ on Sβ = Sβ specializes to F

•∨

∼ = Cone



 [RHomπ (O, O) ⊕ RHomπ (O, O)]0 → RHomπ (O, O(Zβ ))

∼ = Rπ∗ OZβ (Zβ ), studied by Dürr-Kabanov-Okonek [6] in the course of algebraic Seiberg-Witten invariants (Poincaré invariants). Moreover, one can see by inspection that under condition (16), the • ∨ K-group class of (Fred ) coincides with the K-group class of the reduced virtual tangent bundle over Sβ constructed in [6]. This is because by Definition 2.8 in the K-group p • ∨ (Fred ) = F •∨ + OSgβ and the same is true for the reduced virtual tangent constructed in [6]. This proves part 4. Finally, if n1 = 0, n2 = 0 and β = 0, then by [41, Prop B.8], [0,n2 ]

Sβ

= Hilbn2 (Zβ /Sβ ) ∼ = Pn2 −β·(β+KS )/2 (S, β),

where Hilbn2 (Zβ /Sβ ) is the relative Hilbert scheme of points on the universal curve Zβ , and P− (S, −) is the moduli space of stable pairs on S. Let OS×P → F be the universal stable pair over S × Pn2 −β·(β+KS )/2 (S, β), and let I • be the associated complex. In this case F •∨ is given by F

•∨

∼ = Cone

   [n2 ] [n2 ]   [n2 ]  → RHomπ O, Iβ RHomπ (O, O) ⊕ RHomπ I , I 

0



∼ = Cone RHomπ I [n2 ] , I

 [n2 ]



→ RHomπ O(−Zβ ), I [n2 ]

 

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    ∼ = RHomπ I • , F . = RHomπ IZ [n2 ] ⊂Zβ , I [n2 ] [1] ∼ Here by IZ [n2 ] ⊂Zβ we mean the push-forward of the ideal sheaf of Z [n2 ] as a subscheme [0,n ]

of Zβ ⊂ S ×Sβ 2 . The last isomorphism above follows from (91) in [26]. We have shown that in this case F •∨ coincides with virtual tangent bundle of the stable pair moduli space Pn2 −β·(β+KS )/2 (S, β). Moreover, by the same reasoning given for the proof of part 4, one • ∨ can see by inspection that under condition (16), the K-group class of (Fred ) coincides with the K-group class of the reduced virtual tangent bundle over Pn2−β·(β+KS )/2 (S, β) constructed in [26]. This finishes the proof of part 5.  4. Punctual nested Hilbert schemes We will discuss a few tools for evaluating the virtual fundamental class [S [n1 ≥n2 ] ]vir constructed in Corollary 2.6. We first develop a localization formula (30) in the case that S is toric along the lines of [35]. When S is toric we express ι∗ [S [n1 ≥n2 ] ]vir as the top Chern class of a vector bundle over the product of Hilbert schemes S [n1 ] × S [n2 ] (see Proposition 4.4). We have not been able to prove such a formula for general projective surfaces. Instead, we prove a weaker statement for general projective surfaces in which the integral of certain cohomology classes against [S [n1 ≥n2 ] ]vir is expressed in terms of integrals over S [n1 ] × S [n2 ] . This is done by using degeneration and the double point relations (see Corollary 4.13, Proposition 4.14). Such integrals arise in all the applications that we have in mind, particularly, they are related to the localized DT invariants of S discussed in [15]. In Section 5 we express some of these integrals against [S [n1 ≥n2 ] ]vir in terms of Carlsson-Okounkov’s vertex operators and as a result obtain explicit product formulas for their generating series. Recall that   S [n1 ≥n2 ] = (Z1 , Z2 ) | Zi ∈ S [ni ] , Z1 ⊇ Z2 ⊂ S [n1 ] × S [n2 ] . For simplicity in this section, we denote by Ii the ideal sheaf IZi of Zi . Hence for any closed point (Z1 , Z2 ) ∈ S [n1 ≥n2 ] we have I1 ⊆ I2 . Sometimes, we denote the closed point above by the pair (I1 , I2 ), or by I1 ⊆ I2 , when we want to emphasize the inclusion of subschemes. As before, we have the universal objects over S × S [n1 ≥n2 ] : Φ : I [n1 ] → I [n2 ] . We will use the following simple lemma in Section 4.1: Lemma 4.1. 1. If (I1 ⊆ I2 ) ∈ S [n1 ,n2 ] is a closed point, then HomS (I1 , I2 ) = HomS (I1 , I1 ) = HomS (I2 , I2 ) = H 0 (OS ) ∼ = C.

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33

2. If (I1 , I2 ) ∈ S [n1 ] × S [n2 ] \ S [n1 ,n2 ] is a closed point then HomS (I1 , I2 ) = 0. 3. If pg (S) = 0 and if (I1 , I2 ) ∈ S [n1 ] × S [n2 ] is a closed point then Ext2S (Ii , Ij ) = 0. Proof. Applying the functor Hom(I1 , −) to the short exact sequence 0 → I2 → OS → OZ2 → 0, we get the exact sequence u 0 → Hom(I1 , I2 ) ⊆ Hom(I1 , OS ) ∼ → Hom(I1 , OZ2 ), = H 0 (OS ) = C −

where u composes any map I1 → OS with the natural map OS → OZ2 . In part 1 the inclusion I1 ⊆ I2 gives a nonzero element of Hom(I1 , I2 ) and hence the claim follows. In part 2, u(I1 ⊂ OS ) = 0 because I1 ⊂ I2 , and so the claim is proven. For part 3, applying the functor Hom(Ij , −) to the short exact sequence 0 → Ii ⊗ ωS → ωS → OZi → 0, we get Hom(Ij , Ii ⊗ ωS ) ⊆ HomS (Ij , ωS ) = H 0 (ωS ) = 0, and so the claim follows by Serre duality.  As will become clear shortly, the following K-group element plays an important role in the rest of the paper: Definition 4.2. For any line bundles M on S, let EnM1 ,n2 ∈ K(S [n1 ] × S [n2 ] ) be the element of rank n1 + n2 defined by   EnM1 ,n2 := [Rπ∗ p∗ M ] − RHomπ (I [n1 ] , I [n2 ] ⊗ p∗ M ) , where p and π are respectively the projections from S × S [n1 ] × S [n2 ] to the first and the product of the last two factors. Similarly, we define the twisted tangent bundle as the rank 2ni element of K(S [n1 ] × S [n2 ] )   ∗ [ni ] [ni ] ∗ TM := [Rπ p M ] − RHom (I , I ⊗ p M ) . [n ] ∗ π S i S Note that TO is the class of (pullback of) the usual tangent bundle of S [ni] . If M = OS , S [ni ] we sometimes drop it from the notation.

4.1. Toric surfaces Let (C 2 )[n1 ≥n2 ] be the nested Hilbert scheme of points on C 2 = Spec(R), where R = C[x1 , x2 ]. The 2-dimensional torus T acts on C 2 . We denote by t1 , t2 the torus −1 characters, such that the tangent space at 0 ∈ C 2 has the T-character t−1 1 + t2 . The T-fixed set (C 2 )[n1 ≥n2 ],T ⊂ (C 2 )[n1 ],T × (C 2 )[n2 ],T

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34

is isolated, and is given by the inclusion of the monomial ideals I1 ⊆ I2 or equivalently the corresponding nested partitions μ ⊆ μ. By Proposition 2.5 and Lemma 4.3, the virtual tangent space at the T-fixed point I1 ⊆ I2 is given by9 TIvir = −χ(I1 , I1 ) − χ(I2 , I2 ) + χ(I1 , I2 ) + χ(R, R), 1 ⊆I2

(29)

2

where χ(−, −) = i=0 (−1)i ExtiR (−, −). By the exact method as in [35, Section 4.6] can be exusing Taylor resolutions and Čech complexes, the T-representation of TIvir 1 ⊆I2 plicitly written down as a Laurent polynomial in t1 and t2 . For the T-fixed 0-dimensional subschemes Z2 ⊆ Z1 ⊂ C 2 corresponding to the monomial ideals I1 ⊆ I2 define Z1 :=



tk11 tk22 =

(k1 ,k2 )∈μ

1 − P1 (t1 , t2 ) , (1 − t1 )(1 − t2 )

Z2 :=



tk11 tk22 =

(k1 ,k2 )∈μ

1 − P2 (t1 , t2 ) . (1 − t1 )(1 − t2 )

Here, P1 , P2 are the Poincaré polynomials associated to the monomial ideals I1 and I2 (defined using their Taylor resolutions, see [35, Section 4.7]), respectively. Also, define −1 Pi := Pi (t−1 1 , t2 ),

−1 Zi := Zi (t−1 1 , t2 ),

i = 1, 2.

Putting these expressions into (29) and simplifying, we get

trTIvir⊆I = 1

2

−P 1 P1 − P 2 P2 + P 1 P2 + 1 (1 − t1 )(1 − t2 )

= Z1 +

(30)

  (1 − t1 )(1 − t2 ) Z2 + Z1 · Z2 − Z1 · Z1 − Z2 · Z2 . t1 t2 t1 t2

Now if S is a toric surface, then the set of T-fixed points of S [n1 ≥n2 ] ⊂ S [n1 ] × S [n2 ] is again isolated (Lemma 4.3), and the T-character of the virtual tangent space at any fixed point is obtained by summing over the expression (30) for all the T-invariant open subsets of S. This finishes the proof of Theorem 2. Lemma 4.3. Suppose that S is a nonsingular projective toric surface, and Z2 ⊆ Z1 is a T-fixed point of S [n1 ≥n2 ] , then Ext2S (I1 , I1 ) = Ext2S (I2 , I2 ) = Ext2S (I1 , I2 ) = 0, the T-representations Ext1S (I1 , I1 ),

Ext1S (I2 , I2 ),

Ext1S (I1 , I2 )

contain no trivial sub-representations. 9 This is obtained by taking the derived restriction of the complex F • to the point I1 ⊆ I2 , and then taking the K-group class of the resulting complex. Also note that by slightly modifying the proof of part 1 of Lemma 4.1, Hom(I1 , I1 ) = Hom(I1 , I2 ) = Hom(I2 , I2 ) = R.

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Proof. The vanishings in the lemma follow from the fact that pg (S) = 0 for toric surfaces, and part 3 of Lemma 4.1. For any fixed point α ∈ S, let Uα ∼ = C 2 be the T-invariant open neighborhood of α, and let Ii,α := Ii |α , and Oi,α := OZi |α . By [7, Lemma 3.2], HomUα (Ii,α , Oi,α ) contains no trivial subrepresentations. Therefore, Ext1S (Ii , Ii ) ∼ = HomS (Ii , OZi ) =



HomUα (Ii,α , Oi,α )

α

contains no trivial representations either (in the first isomorphism we used the vanishing H 1 (OS ) = 0 for toric surfaces). Next, applying HomS (I1 , −) to the natural short exact sequence 0 → I2 → OS → OZ2 → 0, we obtain the exact sequence HomS (I1 , OZ2 ) → Ext1S (I1 , I2 ) → Ext1S (I1 , OS ).

(31)

To finish the proof it suffices to show that the 1st and the 3rd terms in (31) contain no trivial representations. The claim for the 1st term in (31) follows from the fact that for each α, T acts with different weights on the C-basis elements for I1,α and O2,α that are given by the monomials (because of the inclusion I1,α ⊆ I2,α ). The claim for the 3rd term in (31) also follows because, applying HomS (−, OS ) to the natural short exact sequence 0 → I1 → OS → OZ1 → 0, and using equivariant Serre duality, we get Ext1S (I1 , OS ) ∼ = Ext2S (OZ1 , OS ) ∼ = H 0 (OZ1 ⊗ ωS )∗ . But since Z1 is zero dimensional and T-fixed H 0 (OZ1 ⊗ ωS ) =



H 0 (Uα , OZ1 ⊗ ωS ).

α

For each α, let μα be the partition corresponding to Z1 |Uα , and suppose that the T−1 character of Tα S is t−1 1 + t2 for some T-characters t1 and t2 , then, the fiber of ωS at α has the T-character t1 t2 , and therefore, H 0 (Uα , OZ1 ⊗ ωS ) = t1 t2



tk11 tk22

(k1 ,k2 )∈μα

has no trivial representations.  4.2. Proof of Theorem 3 Suppose that S is a toric surface, and (I1 , I2 ) ∈ S [n1 ] × S [n2 ] is a closed point. By Lemma 4.3 Ext2S (Ii , Ij ) = 0.

(32)

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Therefore by basechange, the sheaves Ext1π (I [n1 ] , I [n1 ] ),

Ext1π (I [n2 ] , I [n2 ] ),

Ext1π (I [n1 ] , I [n2 ] )

are vector bundles over S [n1 ≥n2 ] of ranks 2n1 , 2n2 , n1 + n2 , respectively. Moreover, the virtual tangent bundle of Proposition 2.5, simplifies to the 2-term complex   F •∨ = Ext1π (I [n1 ] , I [n1 ] ) ⊕ Ext1π (I [n2 ] , I [n2 ] ) → Ext1π (I [n1 ] , I [n2 ] ) . (33) Recall that the rank of F •∨ is equal to n1 + n2 , and recall the K-group elements En1 ,n2 and En1 ,n2 |(I1 ,I2 ) of ranks n1 + n2 from Definition 4.2. We have

E

n1 ,n2

|(I1 ,I2 ) =

 Ext1S (I1 , I2 )

(I1 , I2 ) ∈ S [n1 ≥n2 ] ,

H 0 (OS ) ⊕ Ext1S (I1 , I2 )

(I1 , I2 ) ∈ / S [n1 ≥n2 ] .

(34)

This is true because of basechange, the vanishing (32), the vanishing H 1 (OS ) = H 2 (OS ) = 0, and that by Lemma 4.1,  HomS (I1 , I2 ) =

H 0 (OS )

I1 ⊆ I2 ,

0

I1  I2 .

Note that this is consistent with the fact that the dimension of Ext1 (I1 , I2 ) jumps by 1 on S [n1 ,n2 ] ⊂ S [n1 ] × S [n2 ] , and that the dimension of En1 ,n2 |(I1 ,I2 ) is constant over S [n1 ] × S [n2 ] . Now we are ready to express the main result of this section relating the push forward of [S [n1 ≥n2 ] ]vir to the products of the fundamental classes of Hilbert scheme of points. The following proposition proves Theorem 3. Proposition 4.4. Suppose that S is a nonsingular projective toric surface, then, ι∗ [S [n1 ≥n2 ] ]vir = cn1 +n2 (En1 ,n2 ) ∩ [S [n1 ] × S [n2 ] ], where ι is the natural inclusion S [n1 ≥n2 ] → S [n1 ] × S [n2 ] . Proof. Let i and j be inclusion of the fixed point set in S [n1 ≥n2 ] and S [n1 ] × S [n2 ] , respectively. By (33) and Lemma 4.3, the virtual localization formula (see [18]) gives [S [n1 ≥n2 ] ]vir =

 (I1 ⊆I2 )∈S [n1 ≥n2 ],T

=

 (I1 ⊆I2 )∈S [n1 ≥n2 ],T

i∗ [(I1 ⊆ I2 )] e(TIvir ) 1 ⊆I2 e(Ext1S (I1 , I2 )) i∗ [(I1 ⊆ I2 )], e(Ext1S (I1 , I1 ))e(Ext1S (I2 , I2 ))

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where the sum is over the isolated T-fixed points, and e(−) indicates the equivariant Euler class. By Lemma 4.3, the coefficient of i∗ [(I1 ⊆ I2 )] in the last sum is the product of the pure nontrivial torus weights. On the other hand, by Lemma 4.3 and the AtiyahBott localization formula 

cn1 +n2 (En1 ,n2 ) ∩ [S [n1 ] × S [n2 ] ] =

(I1 ,I2 )∈S [n1 ],T ×S [n2 ],T

=

 (I1 ,I2 )∈S [n1 ],T ×S [n2 ],T

=

 I1 ⊆I2 ∈S [n1 ≥n2 ],T

e(En1 ,n2 |(I1 ,I2 ) ) j∗ [(I1 , I2 )] e(T(I1 ,I2 ) (S [n1 ] × S [n2 ] ))

e(En1 ,n2 |(I1 ,I2 ) ) j∗ [(I1 , I2 )] 1 e(ExtS (I1 , I1 ))e(Ext1S (I2 , I2 ))

e(Ext1S (I1 , I2 )) ι∗ ◦ i∗ [I1 ⊆ I2 ], e(Ext1S (I1 , I1 ))e(Ext1S (I2 , I2 ))

∼ C is the where the last equality is because of (34), and the fact that since H 0 (OS ) = 0 trivial T-representation, we have e(H (OS )) = 0. The proposition is proven by comparing the outcomes of both localization formulas above, and taking the non-equivariant limit at the end.  4.3. Relative nested Hilbert schemes In this section we sketch how the degeneration formula of Li and Wu can be applied to the case of nested Hilbert scheme of points. Let (S, D) be a pair of nonsingular projective surface and a nonsingular effective divisor. Li and Wu [34] introduced the notion of a stable relative ideal sheaf. I ∈ S [n] is said to be relative to D if the natural map I ⊗ OD → OS ⊗ OD

(35)

is injective (see also [36]). This is equivalent to OS /I having support disjoint from D. Relativity is an open condition in S [n] . Li and Wu constructed a relative Hilbert scheme, denoted by (S/D)[n] , by considering the equivalence classes of the stable relative ideal sheaves on the k-step semistable models S[k] for 0 ≤ k ≤ n. Let D0 , . . . , Dk−1 be the singular locus of S[k] and Dk ⊂ S[k] be the proper transform of D. S[k] consists of k + 1 irreducible components Δ0 , . . . , Δk with Δ0 = S and Di = Δi ∩Δi+1 for i = 0, . . . , k − 1. A relative ideal sheaf I on S[k] satisfies (35) for D = D0 , . . . , Dk . Two relative ideal sheaves I and I  on S[k] are equivalent if the quotients OS[k] /I and OS[k] /I  differ by an automorphism of S[k] covering the identity on Δ0 = S. The stability of a relative ideal sheaf means that it has finitely many automorphisms as described above. (S/D)[n] is a smooth proper Deligne-Mumford stack of dimension 2n. Since by the relativity condition for any relative ideal sheaf I, I|Dj ∼ = ODj , the generalization of Li-Wu Hilbert schemes to the set up of the nested Hilbert schemes is straightforward. In other words, we can construct a proper Deligne-Mumford stack

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(S/D)[n1 ≥n2 ] as the moduli space of relative ideal sheaves I1 and I2 with I1 stable and I1 ⊆ I2 .10 Notation. Following [34, Section 2.3], let A be the Artin stack of expanded degenerations for the pair (S, D), and let S → A be the universal family of surfaces over it. They fit into the fibered diagram S

S

A

Spec C.

Let (S/D)[n1 ≥n2 ] → A be the natural morphism; it factors through the substack A ⊂ A corresponding to the numerical data n (see [34, Section 2.5] for the construction [n] of these substacks11 ). A is a smooth Artin stack of dimension 0. We use the same notation as in the absolute case to denote the inclusion of the universal objects over S ×A[n] (S/D)[n] : [n]



0 = Φ : I [n1 ] → I [n2 ] . Let π be the projection to the second factor of S ×A[n] (S/D)[n] , and p be the projection  to its first factor followed by the natural map S → S. By the method of [37, Section 3.9] and [34], one can see, after modifying our argument for the usual nested Hilbert schemes (Proposition 2.5), that there is a relative perfect • obstruction theory Frel → L• [n] with the relative virtual tangent bundle [n1 ≥n2 ] (S/D)

•∨ Frel := Cone

/A

     RHomπ I [n1 ] , I [n1 ] ⊕ RHomπ I [n2 ] , I [n2 ] 

→ RHomπ I [n]

Since the Artin stack A fundamental class

[n1 ]

,I

[n2 ]



0



(36)

.

is smooth of dimension 0 by [1, Section 7] there is a virtual

[(S/D)[n1 ≥n2 ] ]vir ∈ An1 +n2 ((S/D)[n1 ≥n2 ] ) 10 Note that if Zi ⊂ S[k] is the 0-dimension subscheme corresponding to Ii , then the number of the autoequivalences of Z2 ⊆ Z1 ⊂ S[k] is less than or equal to that of Z1 ⊂ S[k], which is finite by the stability of I1 . 11 Since we are dealing with zero dimensional subschemes their Hilbert polynomials (used in [34]) are simply the nonnegative integers n1 , n2 .

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associated to this relative perfect obstruction theory. Let X : S  S0 := S1 ∪D S2 be a good degeneration of the surface S along D over a pointed curve (C, 0),12 and let S→C→C

(37)

be the universal family of surfaces over the stack of expanded degenerations C (see [31,32], [34, Sections 2.1-2.2]). They fit into the fibered diagram S

X

C

C.

Following the construction of Li and Wu [34], one can construct the nested Hilbert scheme of points, denoted by S[n1 ≥n2 ] on the fibers of S. The Hilbert scheme S[n1 ≥n2 ] is a proper Deligne-Mumford stack over C and its structure morphism factors through the substack C[n] ⊂ C corresponding to the numerical data n (see [34, Section 2.5] for the construction of these substacks). C[n] is a smooth Artin stack of dimension 1. Let [n] C0 ⊂ C[n] be the substack corresponding to 0 ∈ C. A non-special fiber of S[n1 ≥n2 ] is [n ≥n ] isomorphic to S [n1 ≥n2 ] , whereas the special fiber of S[n1 ≥n2 ] , denoted by S0 1 2 , can be written as the (non-disjoint) union [n1 ≥n2 ]

S0

!

=









(S1 /D)[n1 ≥n2 ] × (S2 /D)[n1 ≥n2 ] ,

(38)

n = n + n

where n = (n1 , n2 ) and n = (n1 , n2 ) with n1 ≥ n2 and n1 ≥ n2 . Each component 







S0,n ,n := (S1 /D)[n1 ≥n2 ] × (S2 /D)[n1 ≥n2 ] [n]

[n]

[n]

is the pull-back of a divisor Cn ,n ⊂ C[n] . Let Ln ,n be the corresponding line bundle. We then have "

[n] Ln ,n ∼ = L0

n=n +n

where L0 is the line bundle associated to the pull back of the divisor {0} ⊂ C. We denote the universal objects over S ×C[n] S[n1 ≥n2 ] by 0 = Φ : I[n1 ] → I[n2 ] . 12 It means we have a nonsingular threefold X over C, whose general fibers are isomorphic to S, and whose fiber over 0 ∈ C is a normal crossing divisor S1 ∪D S2 consisting of two nonsingular surfaces S1 , S2 glued along a nonsingular divisor isomorphic to D and contained in S1 and S2 .

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    The restriction of Φ to the component S ×C[n] (S1 /D)[n1 ≥n2 ] × (S2 /D)[n1 ≥n2 ] is identified with the pair of universal maps 







(I [n1 ] → I [n2 ] , I [n1 ] → I [n2 ] ). We denote by π the projection to the second factor of S ×C[n] S[n1 ≥n2 ] , and by p the projection to its first factor followed by the natural morphism to the total space of the good degeneration of S over C. Again by the method of Section 2.1 and [37,34], one can construct a relative perfect obstruction theory F•rel → L•S[n1 ≥n2 ] /C[n] with the relative virtual tangent bundle: F•∨ rel =         Cone RHomπ I[n1 ] , I[n1 ] ⊕ RHomπ I[n2 ] , I[n2 ] → RHomπ I[n1 ] , I[n2 ] . 0

Since C[n] is a smooth Artin stack (h−1 (L•C[n] ) = 0 but h1 (L•C[n] ) = 0) there exists an absolute perfect obstruction theory F• → L•S[n1 ≥n2 ] associated to F•rel (see for example the argument after diagram (49) in [37]). The restriction of F•rel to S0 [n] S0,n ,n induce relative perfect obstruction theories

[n]

F•0 → L•S [n] /C[n] 0

0

and

F•0,n ,n → L•S [n]

0,n ,n

and its components

[n]

/C0,n ,n

,

respectively. As in [37], they satisfy the following compatibilities: F• |S [n] → F•0 → L∨ 0 [1], 0

F• |S [n]

0,n ,n

→ F•0,n ,n → Ln ,n [1], [n]∨

(39)

where each sequence is an exact triangle. A decomposition S0 [k1 , k2 ] := S1 [k1 ] ∪D S2 [k2 ] yields the natural exact sequence 0 → OS0 [k1 ,k2 ] → OS1 [k1 ] ⊕ OS2 [k2 ] → OD → 0. Suppose that I1 ⊆ I2 is a nested pair of relative ideal sheaves on S0 [k1 , k2 ], and let Ii := Ii |S1 [k1 ] and Ii := Ii |S2 [k2 ] . Tensoring the short exact sequence above with the perfect complexes RHom(I1 , I1 ), RHom(I2 , I2 ), and RHom(I1 , I2 ) and applying RΓ we get the commutative diagram ⊕2i=1 R Hom(Ii , Ii )

⊕2i=1 R Hom(Ii , Ii ) ⊕ ⊕2i=1 R Hom(Ii , Ii )

RΓOD ⊕ RΓOD [−1 1]

RHom(I1 , I2 )

R Hom(I1 , I2 ) ⊕ R Hom(I1 , I2 )

RΓOD

where each row is an exact triangle and the first two vertical maps are induced from the natural inclusions I1 ⊆ I2 , I1 ⊆ I2 and I1 ⊆ I2 as in Section 2.1, and in the third

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column we have used the relativity condition of ideal sheaves i.e. Ii |D = OD . As before the vertical maps factor through the trace free parts and hence, using the natural exact triangle RΓOS → RΓOS1 ⊕ RΓOS2 → RΓOD , this induces the following commutative diagram of the exact triangles [⊕2i=1 R Hom(Ii , Ii )]0

[⊕2i=1 R Hom(Ii , Ii )]0 ⊕ [⊕2i=1 R Hom(Ii , Ii )]0

RΓOD

RHom(I1 , I2 )

R Hom(I1 , I2 ) ⊕ R Hom(I1 , I2 )

RΓOD .

Taking the cones we get the isomorphism  Cone

 [R Hom (I1 , I1 ) ⊕ R Hom (I2 , I2 )]0 → R Hom (I1 , I2 )

∼ =





Cone

[R Hom (I1 , I1 ) 

Cone

⊕

R Hom (I2 , I2 )]0

[R Hom (I1 , I1 )

⊕

R Hom (I1 , I2 )

→

R Hom (I2 , I2 )]0

→

R Hom (I1 , I2 )

 .

One of the upshots is that following the construction of [37,34], we are led by the isomorphism above to the following degeneration formula for the virtual integration over S [n1 ≥n2 ] ([37, Thm. 16], [34, Prop. 6.5, Thm. 6.6]). This is done by using the compatibilities (39) and relating the relative prefect obstruction theory F•rel to the absolute perfect • obstruction theories F • (given in Proposition 2.5) and Frel (given by (36)): Proposition 4.5. Let α be a cohomology class in the total space of S[n1 ≥n2 ] , then, ⎛

 α=



⎜ ⎝

n=n +n

[S [n1 ≥n2 ] ]vir

⎞ ⎛

 [(S1

⎟ ⎜ α⎠ · ⎝

  /D)[n1 ≥n2 ] ]vir

⎞

 [(S2

⎟ α⎠ .



  /D)[n1 ≥n2 ] ]vir

Remark 4.6. In Proposition 4.5, if n = n1 = n2 then S[n1 ≥n2 ] ∼ = S[n] constructed by [34], and by the same argument as in proof of Theorem 3.1 part 1, one can recover the usual degeneration formula for the Hilbert schemes of points used in [45,33,12]: ⎛

 α= S [n]

 n=n +n

⎜ ⎝

⎞ ⎛

 (S1 /D)

⎟ ⎜ α⎠ · ⎝ [n ]

⎞

 [(S2 /D)

⎟ α⎠ . [n ]

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4.4. Double point relation Let (S, D) be a pair of nonsingular projective surface and a nonsingular effective divisor as in Section 4.3, and let M be line bundle on S. Also, recall the definitions of π : S ×A[n] (S/D)[n] → (S/D)[n] ,

p : S ×A[n] (S/D)[n] → S.





Let D ⊂ S be the proper transform of D ⊂ S via p. Note that we use π and p for the similar natural morphisms from S ×C[n] S[n1 ≥n2 ] as well. Definition 4.7. Define the following element in K((S/D)[n1 ≥n2 ] ) of rank n1 + n2 : [n ≥n2 ]

KM1

  := [Rπ∗ p∗ M ] − RHomπ (I [n1 ] , I [n2 ] ⊗ p∗ M ) .

Define the following generating series: 



Znest (S/D, M ) :=

[n ≥n2 ]

q1n1 q2n2

n1 ≥n2 ≥0

c(KM1

).

[(S/D)[n1 ≥n2 ] ]vir

If D = 0 we drop it from the notation. Lemma 4.8. Given a good degeneration S  S0 := S1 ∪D S2 , and a choice of degeneration of line bundles Pic(S)  M  Mi ∈ Pic(Si ) [n ≥n2 ]

we get the degeneration of the class c(KM1 

i = 1, 2,

) whose restriction to the component







(S1 /D)[n1 ≥n2 ] × (S2 /D)[n1 ≥n2 ] [n ≥n2 ]

of the central fiber of S[n1 ≥n2 ] is c(KM11

[n ≥n 2]

)  c(KM12

).

Proof. Let M be the line bundle over the total space of the good degeneration of S that gives the degeneration of M as in the lemma. The derived pullbacks of the perfect   complexes RHomπ (I[n1 ] , I[n2 ] ⊗ p∗ M) and Rπ∗ p∗ M to the component (S1 /D)[n1 ≥n2 ] ×   (S2 /D)[n1 ≥n2 ] fits in the exact triangles RHomπ (I[n1 ] , I[n2 ] ⊗ p∗ M) 







→ RHomπ (I [n1 ] , I [n2 ] ⊗ p∗ M1 ) ⊕ RHomπ (I [n1 ] , I [n2 ] ⊗ p∗ M2 ) → RHomπ (OD , OD ⊗ p∗ M |D ) ∼ = Rπ∗ p∗ M |D ,

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and Rπ∗ p∗ M → ⊕2i=1 Rπ∗ p∗ Mi → Rπ∗ p∗ M |D . Now taking the difference of the Kgroup classes from the exact triangles above, and applying the total Chern class, we [n ≥n ] conclude that c(KM1 2 ) degenerates to a class whose restriction to the component 







(S1 /D)[n1 ≥n2 ] × (S2 /D)[n1 ≥n2 ] [n ≥n2 ]

is c(KM11

[n ≥n 2]

 KM12

). 

A direct corollary of Proposition 4.5 and Lemma 4.8 is Proposition 4.9. Given a good degeneration S  S0 := S1 ∪D S2 , and a choice of degeneration of line bundles Pic(S)  M  Mi ∈ Pic(Si )

i = 1, 2,

we have Znest (S, M ) = Znest (S1 /D, M1 ) · Znest (S2 /D, M2 ).



(41)

In the situation of Proposition 4.9, Let P be either of the projective bundles P (OD + NS1 /D ) ∼ = P (OD + NS2 /D ), and let MP be the pullback of M |D to P . Applying Proposition 4.9 to the degeneration to the normal cone of D ⊂ Si gives Znest (Si , Mi ) = Znest (Si /D, Mi ) · Znest (P /D, MP ).

(42)

Similarly, the degeneration to the normal cone of D ⊂ P gives Znest (P , MP ) = Znest (P /D, MP ) · Znest (P /D, MP ).

(43)

Let M2,1 (C)+ be the group completion of the set of isomorphism classes of the pairs (S, M ), where S is a smooth projective surface over C and M is a line bundle on S (see [28, Definition 3]). Corollary 4.10. Znest (−, −) satisfies the relation13 Znest (S, M ) · Znest (S1 , M1 )−1 · Znest (S2 , M2 )−1 · Znest (P , MP ) = 1

(44)

and hence it respects the double point relations in M2,1 (C)+ . In other words, Znest (−, −) descends to a homomorphism Znest (−, −) : ω2,1 (C) ⊗Z Q → Q[[q1 , q2 ]]∗ , 13

This relation is the analog of the relation (0.10) in [30].

44

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where ω2,1 (C) is the double point cobordism theory for line bundles on surfaces obtained by taking the quotient of M2,1 (C)+ by all the double point relations. Proof. Relation (44) follows immediately from relations (41)-(43).  It is known that ω2,1 (C) is generated by the following classes (see [45,28]) [P 2 , O],

[P 2 , O(1)],

[P 1 × P 1 , O],

[P 1 × P 1 , O(1, 0)].

(45)

Let B1 , B2 , B3 , B4 be Znest (S, M ) where (S, M ) is one of the pairs above respectively from left to right. Define −3/2

A1 := B1−1 B2 B3 B4 3/2

1/2

−1/2

, A2 := B3 B4

−1/3

, A3 := B1

−1/4

B3

−2/3

, A4 := B1

3/4

B3 .

Proposition 4.11. Let S be a nonsingular projective surface and M be a line bundle on S. Let A1 , A2 , A3 , A4 ∈ Q[[q1 , q2 ]]∗ be defined as above (independent of S) then K2

M ·KS Znest (S, M ) = AM A3 S A42 1 A2 2

c (S)

.

Proof. By basechange, [n ≥n2 ]

c(EnM1 ,n2 )|S [n1 ≥n2 ] = c(KM1

),

and so by Proposition 4.4, we have Znest (S, M ) =





cn1 +n2 (En1 ,n2 ) ∪ c(EnM1 ,n2 ),

q1n1 q2n2

n1 ≥n2 ≥0

S [n1 ] ×S [n2 ]

if (S, M ) is one of the generators (45). For a general (S, M ) as in the proposition we can express the class [S, M ] ∈ ω2,1 (C) as a linear combination of the generators (45) [45, Proposition 4.1]. The result then follows by applying the homomorphism Znest (−, −) of Corollary 4.10 and then rearranging the factors as in the proof of [45, Proposition 4.1].  Corollary 4.12. Let S be a nonsingular projective surface and M be a line bundle on S. Then the integral 

[n ≥n2 ]

c(KM1

)

[S [n1 ≥n2 ] ]vir

can be written as a degree n1 + n2 universal polynomial in M 2 , M · KS , KS2 , c2 (S).

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45

Proof. The integral in the proposition is the coefficient of q1n1 q2n2 in Znest (S, M ). The result follows after expanding the right hand side of the formula in Proposition 4.11 and extracting the coefficient of q1n1 q2n2 .  Corollary 4.13. For any nonsingular projective surface S and M ∈ Pic(S) let P := [n ≥n ] c(KM1 2 ). Then  cn1 +n2 (En1 ,n2 ) ∪ c(EnM1 ,n2 ).

NS (n1 , n2 , 0; P) = S [n1 ] ×S [n2 ]

Proof. By Corollary 4.12 we know that the LHS of the statement above i.e. 

[n ≥n2 ]

NS (n1 , n2 , 0; P) =

c(KM1

)

[S [n1 ≥n2 ] ]vir

is a universal polynomial P1 in M 2 , M · KS , KS2 , c2 (S). On the other hand, using Grothendieck-Riemann-Roch formula and the induction scheme of Ellingsrud, Göttsche, and Lehn (see [8, Sections 3, 4] and [3, Section 3]), we can express the RHS of the corollary in terms of a universal polynomial P2 in M 2 , M · KS , KS2 , c2 (S). But since the equality in the corollary holds for any (S, M ) in which S is toric (by Proposition 4.4), we conclude that P1 = P2 , and the result follows.  This corollary allows us to write integrals against [S [n1 ≥n2 ] ]vir in terms of integrations over the product of Hilbert schemes of points. As said in the introduction the following generalization of this corollary is used in [15] in the context of reduced localized DT [n ≥n ] invariants. In fact the only property of the integrand c(KM1 2 ) that was needed in the argument leading to Corollary 4.13 was its decomposition property under good degenerations of S as stated in Lemma 4.8. We can therefore prove similar identities as in Corollary 4.13 for other integrands with such a decomposition property. For example, by a same proof as Lemma 4.8 one can see that the Chern class of the twisted tangent bundle (Definition 4.2), c(TM ), also has this property, and so does any product/quoS [ni ] tient of these Chern classes. In particular, we can extend Corollary 4.13 to the following more general statement: Proposition 4.14. Let L1 , . . . , Ls , L1 , . . . , Ls , M1 , . . . , Mt , be some line bundles on the nonsingular projective surface S, and l1 , . . . , ls , l1 , . . . , ls  , m1 , . . . , mt be finite sequences of ±1. Define P :=

s  i=1

Then,



i c(TL )li S [n1 ]

∪

s  i=1

L



i c(TS [n )li ∪ 2]

t  i=1

[n ≥n2 ] mi

c(KM1i

)

.

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46

NS (n1 , n2 , 0; P) =  cn1 +n2 (E

n1 ,n2

)∪

s 



i c(TL )li S [n1 ]

∪

i=1

S [n1 ] ×S [n2 ]

s 

 Li c(TS [n )li 2]

∪

i=1

t 

c(EnM1i,n2 )mi .



i=1

5. Vertex operator formulas and proof of Theorem 4 Let (S, M ) be a pair of a projective nonsingular surface S and a line bundles M on S. Let F = ⊕n H ∗ (S [n] , Q). Carlsson and Okounkov defined the operator W (M1 ) in End(F)[[z1 , z1−1 ]] by

W (M1 )η1 , η2  :=



z1n2 −n1

p∗1 η1 ∪ p∗2 η2 ∪ cn1 +n2 (EnM11,n2 ),

S [n1 ] ×S [n2 ]

where −, − is the Poincaré pairing, ηi ∈ H ∗ (S [ni ] , Q), pi is the projection to the i-th factor of S [n1 ] × S [n2 ] , and EnM11,n2 ∈ K(S [n1 ] × S [n2 ] ) is as in Definition 4.2. In other words, using [10, Definition 16.1.2], W (M1 ) is the operator associated to the family of correspondences cn1 +n2 (EnM11,n2 ) : S [n1 ]  S [n2 ]

n1 , n2 ≥ 0.

If M2 is another line bundle on S, we define W (M1 , M2 )(z1 , z2 ) := W (M2 )(z2 ) ◦ W (M1 )(z1 ). By [10, Proposition 16.1.2],

W (M1 , M2 )η1 , η3  =  z1n2 −n1 z2n3 −n2 n2



p∗1 η1 ∪ p∗3 η3 ∪ cn1 +n2 (EnM11,n2 ) ∪ cn2 +n3 (EnM22,n3 ),

S [n1 ] ×S [n2 ] ×S [n3 ]

where ηi ∈ H ∗ (S [ni ] , Q), and pi is the projection to the i-th factor of S [n1 ] × S [n2 ] × S [n3 ] . Carlsson and Okounkov found an explicit formula for W (−) in terms of vertex operators. Let α± (−) denote Nakajima’s annihilation/creation operators. Theorem 5.1. (Carlsson-Okounkov [3]) W (M1 ) = Γ− (−M1 , −z1 ) ◦ Γ+ (−M1D , z1 ), where 

  z ∓n 1 Γ± (M1 , z1 ) := exp α±n (M1 ) . n n>0



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Note that the operators Γ± satisfy the commutation relations [Γ± , Γ± ] = 0, and moreover, Γ+ (M2 , z2 ) ◦ Γ− (M1 , z1 ) = (1 + Let c := (1 −

z1 M1 ,M2D . z2 )

z1 M1 ,M2 ) Γ− (M1 , z1 ) ◦ Γ+ (M2 , z2 ). z2

Using these properties, we can write

W (M1 , M2 ) = Γ− (−M2 , −z2 ) ◦ Γ+ (−M2D , z2 ) ◦ Γ− (−M1 , −z1 ) ◦ Γ+ (−M1D , z1 ) = c Γ− (−M2 , −z2 ) ◦ Γ− (−M1 , −z1 ) ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ). Let N be the number-of-points operator: N|Fn = n id. It satisfies q N ◦ Γ− (Mi , zi ) = Γ− (Mi , qzi ) ◦ q N .   Starting with str q N ◦ W (M1 , M2 ) and using the commutation relation of the supertrace str(A ◦ B) = str(B ◦ A), we obtain   c str q N ◦ Γ− (−M2 , −z2 ) ◦ Γ− (−M1 , −z1 ) ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ) =   c str Γ− (−M2 , −z2 q) ◦ Γ− (−M1 , −z1 q) ◦ q N ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ) =   c str q N ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ) ◦ Γ− (−M2 , −z2 q) ◦ Γ− (−M1 , −z1 q) = D D z2 q M1D ,M2 z1 q M1 ,M2D ) (1 − ) (1 − q)M1 ,M1 +M2 ,M2 (1 − z2 z1  N  c str q ◦ Γ− (−M2 , −z2 q) ◦ Γ− (−M1 , −z1 q) ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ) . Iterating this process, we get   str q N ◦ W (M1 , M2 ) =  D D z1 q n M1 ,M2D z2 q n M1D ,M2 (1 − ) (1 − ) (1 − q n )M1 ,M1 +M2 ,M2 z2 z1 n>0  N  c str q ◦ Γ+ (−M2D , z2 ) ◦ Γ+ (−M1D , z1 ) =   D D z1 q n M1 ,M2D z2 q n M1D ,M2 (1 − ) (1 − q n )M1 ,M1 +M2 ,M2 −e(S) (1 − ) , z2 z1 n>0 n≥0

where for the last equality, we have used the fact that Γ+ is a lower triangular operator, and Göttsche’s formula  n≥0

Define

e(S [n] )q n =



(1 − q n )−e(S) .

n≥0

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48

q1 := qz1−2 ,

q2 := z12 ,

then we have shown   str q N ◦ W (M1 , M2 )(z1 , 1/z1 ) = (46)   D D D D (1 − q2n+1 q1n )M1 ,M2 (1 − (q1 q2 )n )M1 ,M1 +M2 ,M2 −e(S) (1 − q2n−1 q1n )M1 ,M2 . n>0

n≥0

On the other hand, by [10, Example 16.1.3], 

  2(n −n ) str q N ◦ W (M1 , M2 )(z1 , 1/z1 ) = q n1 z1 2 1

cn1 +n2 (EnM11,n2 )cn1 +n2 (EnM22,n1 )

S [n1 ] ×S [n2 ]

(47)



,n2 cn1 +n2 (EnM11,n2 )cn1 +n2 (EnM1D ),

= (−1)n1 +n2 q1n1 q2n2

2

S [n1 ] ×S [n2 ]

where the last equality is because of Grothendieck-Verdier duality. Notation. If Z =

r1 ,r2 ≥0

ar1 ,r2 q1r1 q2r2 is a formal series, we define Z [q1n1 q2n2 ] := an1 ,n2 .

The following proposition completes the proof of Theorem 4: Proposition 5.2. Let (S, M ) be a pair of a projective nonsingular surface S and a line bundles M on S, then, 

[n ≥n2 ]

cn1 +n2 (KM1 [S [n1 ≥n2 ] ]vir

(−1)n1 +n2



)=

(1 − q2n−1 q1n )KS ,M

D



(1 − (q1 q2 )n )M

D

,M −e(S)

n>0

Proof. By (47),  cn1 +n2 (En1 ,n2 ) ∪ cn1 +n2 (EnM1 ,n2 ) = S [n1 ] ×S [n2 ]

  (−1)n1 +n2 str q N ◦ W (OS , M D )(z1 , 1/z1 ) [q1n1 q2n2 ] .

The result now follows immediately from (46) and Corollary 4.13.



[q1n1 q2n2 ] .

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