Remarks on Bott residue formula and Futaki–Morita integral invariants

Topology and its Applications 160 (2013) 488–497

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Topology and its Applications www.elsevier.com/locate/topol

Remarks on Bott residue formula and Futaki–Morita integral invariants Ping Li a,b,∗,1 a b

Department of Mathematics, Tongji University, Shanghai 200092, China Department of Mathematics, Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan

a r t i c l e

i n f o

Article history: Received 31 March 2012 Received in revised form 8 December 2012 Accepted 26 December 2012 MSC: primary 58J20 secondary 32Q55 Keywords: Bott residue formula Futaki invariant Futaki–Morita integral invariant Hirzebruch χ y -genus

a b s t r a c t When a compact complex manifold admits a non-degenerate holomorphic vector field, the famous Bott residue formula reduces the calculations of Chern numbers to the zero set of this vector field. The Futaki invariant obstructs the existence of Kähler–Einstein metric with positive scalar curvature. Inspired by the proof of Bott residue formula, Futaki and Morita defined a family of integral invariants, which include Futaki’s original invariant as a special case, and gave them corresponding residue formulae which have the same feature as that of Bott. They also proved some properties of these integral invariants when the underlying manifolds are Kähler. We remark that some considerations of Futaki and Morita on these integral invariants are closely related to some much earlier literatures and recent work of the author. The purpose of this paper is to generalize some considerations of them and give some new properties of these integral invariants. Some related remarks and articles are also discussed in this note. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In mid-1980s, inspired by Bott’s proof of his well-known residue formula [4] and Futaki’s work on Kähler–Einstein metric [5], Futaki and Morita constructed a family of integral invariants, now called Futaki–Morita integral invariants, and proved some properties of them [7,8], [6, Chapter 5]. The purpose of this paper is two-fold. On the one hand, we shall explain that Futaki and Morita’s methods are closely related to two features of the famous Hirzebruch χ y -genus [9]: rigidity property and “−1” phenomenon (details can be found in Section 4). Rigidity means that the equivariant χ y -genus is invariant under one-parameter group action, which was observed by Lusztig [16] and Kosniowski [11]. The so-called “−1” phenomenon, which appears in several independent articles [18,6,15,19], tells us that the coefficients of Taylor expansion of χ y -genus at y = −1 have explicit expressions. On the other hand, these properties of χ y -genus, together with some recent work of the author and Liu [12–14], can be applied to both improving Futaki–Morita’s original ideas and obtaining some new results. So in this paper my sole contribution is to synthesize various ideas found in various articles, clarify their relationships and, as by-products, obtain some new results. This paper is organized as follows. Section 2 contains some preliminaries. In Section 2.1, we review some basic facts about Bott residue formula and Futaki–Morita integral invariants. In Section 2.2 we introduce some notation and prove a technical lemma (Lemma 2.5), which will play a crucial role in the proofs of our main results. Section 3 is devoted to explaining Futaki–Morita’s methods and results and stating our new results (Theorems 3.1, 3.5 and 3.7). In Sections 4.1 and 4.2 we shall clarify the relations between Futaki–Morita’s methods and the two above-mentioned features of χ y -genus,

* 1

Correspondence to: Department of Mathematics, Tongji University, Shanghai 200092, China. E-mail address: [email protected]. The author is supported by National Natural Science Foundation of China (Grant No. 11101308) and JSPS Postdoctoral Fellowship for Foreign Researchers.

0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.12.007

P. Li / Topology and its Applications 160 (2013) 488–497

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from which the proofs of Theorems 3.1 and 3.5 will follow. The proof of Theorem 3.7 is given in Section 4.3. In the last section, Section 5, we will propose two questions, which are closely related to the discussions in this article and whose answers are unknown to the author up to now. 2. Preliminaries 2.1. Bott residue formula and Futaki–Morita integral invariants Let M be an n-dimensional compact complex manifold and X be a holomorphic vector field on M. Given a Hermitian metric on M, we have the Hermitian connection ∇ . Put

L ( X ) := ∇ X − L X , where L X is the Lie derivative with respect to X . It is easy to check that, for any smooth function f and any (1, 0)-type vector field Y on M, we have

L ( X )( f Y ) = f L ( X )Y . Thus L ( X ) can be viewed as a section of the endomorphism bundle of the (1, 0)-type tangent bundle of M. Suppose P ∈ zero( X ) is an isolated zero point of X . Then L ( X ) induces an endomorphism on the holomorphic tangent space to P ,

L P ( X ) : T P M → T P M. We say X is non-degenerate if zero( X ) consists of isolated points and, at each P ∈ zero( X ), the endomorphism L P ( X ) is non-degenerate. Before stating Bott residue formula, let us introduce a notation, which will be used throughout this paper. Let I p (G L (n, C)) be the set of all holomorphic G L (n, C)-invariant polynomials of degree p. That is, any element ϕ in I p (G L (n, C)) is a map ϕ : gl(n, C) → C such that, for any A ∈ gl(n, C) and P ∈ G L (n, C), ϕ ( A ) is a polynomial of degree p whose elements are entries of A and ϕ ( P −1 A P ) = ϕ ( A ). Here gl(n, C) (resp. G L (n,pC)) is the set of all n × n matrices (resp. nonsingular n × n matrices) over C. It is well known that I ∗ (G L (n, C)) = p 0 I ( G L (n, C)) is multiplicatively generated by n elements c 1 , . . . , cn , which are characterized by

det(1 + t A ) = 1 + tc 1 ( A ) + t 2 c 2 ( A ) + · · · + t n cn ( A ),

A ∈ gl(n, C),

deg(c i ) = 1.

In other words, if λ1 , . . . , λn are the eigenvalues of A, then

ci ( A) =



λk1 · · · λki .

1k1 <···
Let c i ( M ) ∈ H 2i ( M ; Z) (1  i  n) be the i-th Chern class of M. For any ϕ = ϕ (c 1 , . . . , cn ) ∈ I ∗ (G L (n, C)), by replacing c i with c i ( M ), we obtain a cohomology class ϕ ( M ) := ϕ (c 1 ( M ), . . . , cn ( M )) ∈ H ∗ ( M ; Z). We denote by ϕ [ M ] the evaluation of ϕ ( M ) on the fundamental class of M. In particular, when ϕ is a monomial whose degree is exactly n, ϕ [ M ] is the corresponding Chern number of M. The following theorem is a famous result of Raoul Bott [4, p. 232], which reduces the calculations of Chern numbers of M to the local information around zero( X ). Theorem 2.1 (Bott residue formula). Let the notation be as above. Then for each ϕ ∈ I ∗ (G L (n, C)) whose degree is no more than n, we have

ϕ [M ] =

 P ∈zero( X )

ϕ ( L P ( X )) cn ( L P ( X ))

.

(2.1)

Remark 2.2. When the degree of ϕ is less than n, the left-hand side of (2.1) vanishes and thus its right-hand side also vanishes. So this residue formula tells us more information about the restrictions to the eigenvalues of L P ( X ) rather than just how to compute the Chern numbers of M. Note that Bott residue formula says nothing for those ϕ whose degrees are bigger than n. For such ϕ the left-hand side of (2.1) clearly vanish by the dimensional reason. While this is not always the case for the right-hand side of (2.1) (see Proposition 3.3). Therefore, a natural question is whether or not we can give an interpretation of the right-hand side of (2.1) for those ϕ whose degree are bigger than n. Futaki and Morita found that [7,8] these are also residue formulae for a family of integral invariants, which include the famous Futaki invariant as a special case and we shall recall in what follows.

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In complex geometry it is an important topic to search for canonical metrics on M, for example, Kähler–Einstein metrics. If a compact complex manifold M admits a Kähler–Einstein metric, the first Chern class of M, c 1 ( M ), is necessarily positive, zero or negative, according to the sign of the Ricci curvature of this metric. Conversely, if c 1 ( M ) is zero or negative, then by the work of Yau [21] and Yau, Aubin [21,3] in the 1970s we know that M has a Kähler–Einstein metric. However, if c 1 ( M ) is positive, there are obstructions to the existence of Kähler–Einstein metrics. The first obstruction was found by Matsushima [17], who showed that if M admits a Kähler–Einstein metric, then h( M ), the Lie algebra of holomorphic vector fields of M, must be reductive. In [5], Futaki defined an important Lie algebra homomorphism F : h( M ) → C, now called the Futaki invariant, and proved that it depends only on the complex structure of M and vanishes identically if M admits a Kähler–Einstein metric with positive scalar curvature. Soon after the discovery of Futaki invariant, inspired by Bott’s proof of his residue formula (2.1), Futaki and Morita generalized the Futaki invariant by introducing a family of integral invariants [7,8], [6, Chapter 5], which are defined on any compact complex manifold, also depend only on the complex structure, and include the Futaki invariant F as a special case. To be more precise, for each ϕ ∈ I n+k (G L (n, C)) (k is any positive integer), Futaki and Morita defined the following integral invariants f ϕ : h( M ) → C [8, p. 136] by

 f ϕ ( X ) :=



ϕ L( X ) +

√ −1

 ∇ . 2

2π

M

In particular, Futaki’s original invariant F coincides with holomorphic vector field, then [8, p. 136]



fϕ(X) =

P ∈zero( X )

ϕ ( L P ( X )) cn ( L P ( X ))

1 f n+1 cn1+1

[8, Proposition 2.3]. Moreover, if X is a non-degenerate

.

Remark 2.3. [7] is an announcement of the properties of their integral invariants, whose proofs are contained in [8] and Chapter 5 of Futaki’s book [6]. Note that our L ( X ) differs from that in [8] by a sign (three lines below Proposition 2.1 on page 136 of [8]). We summarize the above discussions into the following theorem. Theorem 2.4 (Bott, Futaki–Morita). Suppose M is an n-dimensional compact complex manifold and X is a non-degenerate holomorphic vector field. With the above notation understood, we have

⎧

ϕ ( L P ( X )) ⎨ 0, ϕ [ M ],

 P ∈zero( X )

cn ( L P ( X )) ⎩ f ( X ), ϕ

deg(ϕ ) < n, deg(ϕ ) = n, deg(ϕ ) > n.

2.2. Technical preliminaries From now on we use σi (·, . . . , ·) to denote the i-th elementary symmetric polynomial of the variables in the bracket. Of course, i is no more than the number of the variables in the bracket. Given m variables x1 , . . . , xm and an indeterminate y, we consider the following expression, which stems from the Hirzebruch–Riemann–Roch formula m

xi (1 + ye −xi ) i =1

1 − e − xi

=

 m 

σi e

−x1

,...,e

−xm

m



·

i =0

j =1

xj 1 − e −x j

 · yi .

(2.2)

Set c i := σi (x1 , . . . , xm ) and deg(xi ) = 1, and let T si (c 1 , . . . , cm ) be the homogeneous part of degree s in the expression m





σi e−x1 , . . . , e−xm ·

j =1

Then we have m

xi (1 + ye −xi ) i =1

1 − e − xi

=

xj 1 − e −x j

∞ m   i =0

.

 T si (c 1 , . . . , cm )

· yi .

s =0

The readers familiar with H-R-R may notice that these T si (c 1 , . . . , cm ) are nothing else but the Todd polynomials. For the sake of later convenience, we also introduce A is (c 1 , . . . , cm ) as follows:

P. Li / Topology and its Applications 160 (2013) 488–497

m

xi (1 + ye −xi ) i =1

1 − e − xi

=:

∞ m   i =0

491

 A is (c 1 , . . . , cm ) · (1 + y )i ,

s =0

which means that A is (c 1 , . . . , cm ) is the homogeneous part of degree s in the coefficient of (1 + y )i when expanding at y = −1. The following lemma is crucial to the proofs of our main results Theorems 3.1 and 3.5. Lemma 2.5. A is (c 1 , . . . , cm ) is a linear combination of T si (c 1 , . . . , cm ), T si +1 (c 1 , . . . , cm ), . . . , T sm (c 1 , . . . , cm ). For any two positive integers n and k, we have

A is+k (c 1 , . . . , cn , cn+1 , . . . , cn+k )|cn+1 =···=cn+k =0 = A is+k (c 1 , . . . , cn , 0, . . . , 0) = A is (c 1 , . . . , cn ) (0  i  n).

(2.3)

Here c i in the first (resp. third) line of (2.3) should be understood as c i = σi (x1 , . . . , xn+k ) (resp. c i = σi (x1 , . . . , xn )). Proof. The first statement is direct from the definitions of A is and T si . For the second statement, we have

∞ n+k   i =0



A is (c 1 , . . . , cn+k )|cn+1 =···=cn+k =0

· (1 + y )i

s =0

n+k 

xi (1 + ye −xi )  =   1 − e − xi i =1

 = xn+1 =···=xn+k =0

n

xi (1 + ye −xi ) i =1

1 − e − xi

Comparing the corresponding coefficients will lead to (2.3).

 k

(1 + y ) =

∞ n   i =0

 A is (c 1 , . . . , cn )

· (1 + y )i +k .

s =0

2

3. Main observations Combining Atiyah–Bott fixed point formula and the fact that the Dolbeault cohomology groups of compact Kähler manifolds have the property of homotopy-invariance, Futaki and Morita gave the following result for Kähler manifolds [7, Theorem 6.2], [6, Theorem 5.3.11]. Here we remark that, by applying a result of Lusztig [16], the following result is valid for any complex manifold. The details of the proof are given in Section 4.1. Theorem 3.1. Suppose M is an n-dimensional compact complex manifold and X is a non-degenerate holomorphic vector field. Then

f T p (c ,...,cn ) ( X ) = 0, n+k 1

∀0  p  n, ∀k  1. p

Remark 3.2. For any 0  p  n and k  1, we know from Lemma 2.5 that A n+k (c 1 , . . . , cn ) is a linear combination of p p +1 T n+k (c 1 , . . . , cn ), T n+k (c 1 , . . . , cn ), . . . , T nn+k (c 1 , . . . , cn ). Thus Theorem 3.1 also implies that

f A p (c ,...,cn ) ( X ) = 0, n+k 1

∀0  p  n, ∀k  1,

(3.1)

which, together with the “−1” phenomenon introduced in Section 4.2, will be the main ingredients in the proof of our second main result, Theorem 3.5. As an application of this theorem, Futaki and Morita, through expanding the Hirzebruch χ y -genus at y = −1 and explicitly computing the coefficient of the term ( y + 1)2 (more details can be found in Section 4.2), showed that f c1 cn ( X ) = 0 holds for all non-degenerate holomorphic vector fields on Kähler manifolds [7, Corollary 6.3], [6, Corollary 5.3.12]. Moreover, they also indicated that [7, Proposition 5.2], c 1 cn is the unique monomial of degree n + 1 in I n+1 (G L (n, C)) satisfying the above-mentioned property. Thus, this property can be strengthened as follows. Proposition 3.3. c 1 cn is the unique monomial of degree n + 1 in I n+1 (G L (n, C)) such that, for any non-degenerate holomorphic vector field X on any compact complex manifold M,

f c 1 c n ( X ) = 0. (i )

(i )

Remark 3.4. If we write zero( X ) = { P 1 , . . . , P r } and let the eigenvalues of L P i ( X ) be λ1 , . . . , λn , then f c1 cn ( X ) = 0 is equivalent to r  n  i =1 j =1

λ(ji ) = 0.

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P. Li / Topology and its Applications 160 (2013) 488–497

(i )

In fact, there is a stronger statement, due to Kosniowski [11, Theorem 4], which states that the elements in {λ j | 1  i  r , 1  j  n} can be paired such that the sum of the elements in any pair is zero. As we have mentioned above, Futaki–Morita’s trick obtaining f c1 cn ( X ) = 0 is to expand χ y ( M ) at y = −1 and compute the coefficient of ( y + 1)2 . In fact, to the author’s knowledge, several articles [18,15,19,10], explicitly or implicitly, have devoted to the calculations of the coefficients of ( y + 1) p in χ y ( M ). The following result would be clear after we give some explanations in Section 4. Theorem 3.5. For any n-dimensional compact complex manifold M and any non-degenerate holomorphic vector field X on it, there p are explicit sequences {ak (c 1 , . . . , cn ) | 1  k  2p  n + k}, whose degrees are bigger than n and which stem from the coefficients of 2p ( y + 1) in χ y ( M ), such that

f a p (c ,...,cn ) ( X ) = 0, k 1

1  k  2p  n + k.

In particular,

a11 = c 1 cn ,

a12 = 0,









a21 = c 12 + 3c 2 cn−1 − c 13 − 3c 1 c 2 + 3c 3 cn−2 ,



a22 = c 12 + 3c 2 cn − c 13 − 3c 1 c 2 + 3c 3 cn−1 ,



a23 = c 13 − 3c 1 c 2 + 3c 3 cn ,

a24 = 0,



a31 = (7c 1 c 2 cn−2 − 4c 2 cn−1 + 14c 3 cn−2 ) − 20c 4 + c 1 c 3 − 10c 22 − c 12 c 2 + 2c 14 cn−3



+ 2 5c 5 − 5c 1 c 4 − 5c 2 c 3 + 5c 12 c 3 + 5c 1 c 22 − 5c 13 c 2 + c 15 cn−4 ,

···. From Proposition 3.3 we know that, for a monomial ϕ ∈ I n+1 (G L (n, C)) rather than c 1 cn , f ϕ ( X ) is not zero in general. Now we turn to giving a sufficient condition to the vanishing of f ϕ ( X ) for those ϕ , which is related to some recent work of the author and Liu [12–14]. n Let μ = (1m1 (μ) 2m2 (μ) · · · nmn (μ) ) be a partition of weight n + 1. That is, m j (μ) are all non-negative integers and j =1 j · m j (μ) = n + 1. For the sake of convenience we set

c μ :=

n

(c j )m j (μ) .

j =1

Before introducing our next result, we give a definition, which associates to each partition integer m(μ) as follows. Definition 3.6. Given a partition



m(μ) :=

μ of weight n + 1 a positive

μ = (1m1 (μ) 2m2 (μ) · · · nmn (μ) ), set

max{m1 (μ), m2 (μ), . . . , mn (μ)}, max{m1 (μ), m2 (μ), . . . , mn (μ)} + 1,

if max{m2 (μ), . . . , mn (μ)} < m1 (μ), if max{m2 (μ), . . . , mn (μ)}  m1 (μ).

Our next result gives a sufficient condition to the vanishing of f c μ ( X ). Theorem 3.7. Let μ be a partition of weight n + 1 and X ∈ h( M ) be a non-degenerate holomorphic vector field on a compact complex 1 f ( X ) = 0. manifold M. If |zero( X )| < m(μ), then f c μ ( X ) = 0. In particular, if |zero( X )| < n + 1, the Futaki invariant F ( X ) = n+ 1 c n +1 1

Remark 3.8. Futaki’s result in [5] says that the vanishing of F obstructs the existence of a Kähler–Einstein metric with positive scalar curvature, while our above result shows that, in some sense, the vanishing of F also obstructs the upper bound of |zero( X )|. Before ending this section, the author would like to point out one fact, which might be known or even well known to experts. But, to the author’s best knowledge, it did not appear explicitly in previous literatures. The existence of a Kähler–Einstein metric on a compact complex manifold M with nonzero scalar curvature implies that the Chern number cn1 is nonzero, the sign of which is related to that of the scalar curvature and the parity of n. While the non-vanishing of cn1 implies that any non-degenerate holomorphic vector field on M has at least n + 1 zero points [13, Corollary 1.5]. Thus, we have the following fact, which obstructs the existence of Kähler–Einstein metrics on compact complex manifolds with nonzero scalar curvature.

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Proposition 3.9. Suppose M is an n-dimensional compact complex manifold with nonzero Chern number cn1 . Then any non-degenerate holomorphic vector field on M has at least n + 1 zero points. In particular, if a compact complex manifold admits a Kähler–Einstein metric with nonzero scalar curvature, then any non-degenerate holomorphic vector field on it has at least n + 1 zero points. 4. Proofs 4.1. Proof of Theorem 3.1 We keep using the notation and symbols in Sections 1, 2 and 3. For an n-dimensional compact complex manifold M and p ,q each pair 0  p , q  n, we can define the Dolbeault cohomology group H ¯ ( M ). The Hirzebruch χ y -genus of M, χ y ( M ), is ∂ defined as follows: p ,q

h p ,q ( M ) := dimC H ¯ ( M ),

χ p ( M ) :=

∂

n  (−1)q h p ,q ( M ),

χ y ( M ) :=

q =0

n 

χ p (M ) y p .

(4.1)

p =0

If the total Chern class of M, c ( M ), has the following formal factorization:

c ( M ) = 1 + c 1 + · · · + cn =

n

(1 + xi ),

deg(xi ) = 1,

i =1

then Hirzebruch–Riemann–Roch formula [9], [2, §4], tells us that



χ y (M ) =

n

xi (1 + ye −xi )



1 − e − xi

i =1

· [ M ].

(4.2)

Using the notation introduced in Section 2.2, (4.2) is equivalent to

χ p ( M ) = T np (c1 , . . . , cn )[ M ], 0  p  n.

(4.3) p ,q

p ,q

For a holomorphic self-map g of M, it acts on H ¯ ( M ) and thus the corresponding trace, tr( g | H ¯ ( M )), is defined. Then ∂ ∂ we can define the equivariant Hirzebruch χ y -genus, χ y ( g , M ), as follows:

χ p ( g , M ) :=

n  q =0

 p ,q

(−1)q tr g  H ∂¯ ( M ) ,

χ y ( g , M ) :=

n 

χ p (g, M) y p .

p =0

Now let X be a holomorphic vector field of M. We consider the one-parameter group of X , exp(t X ). So χ y (exp(t X ), M ) can also be defined accordingly. When X is non-degenerate, the celebrated Atiyah–Bott fixed point formula [1] allows us to compute χ y (exp(t X ), M ) in terms of the eigenvalues of L P ( X ) for P ∈ zero( X ). More precisely,





χ y exp(t X ), M =

n

1 + ye λi t

P ∈zero( X ) i =1

1 − e λi t

,

where λ1 , . . . , λn are eigenvalues of L P ( X ). p ,q Futaki and Morita noticed that [6, Remark 5.3.9], when M admits a Kähler metric, these H ¯ ( M ) can be viewed as ∂ subspaces of the DeRham cohomology groups, and hence have the property of homotopy-invariance. Thus,





trace exp(t X ) H ¯ ( M ) ≡ h p ,q ( M ), p ,q

∂



χ y exp(t X ), M ≡ χ y ( M ), ∀t ,

which means that

χ y (M ) ≡



n

1 + ye λi t

P ∈zero( X ) i =1

1 − e λi t

,

∀t .

(4.4) p ,q

Lusztig noticed that [16], [11, p. 47], for a general compact complex manifold M, trace(exp(t X ), | H ¯ ( M )) may not be ∂ identically equal to h p ,q ( M ), but χ p (exp(t X ), M ) (0  p  n) are still identically equal to χ p ( M ). This means that, for any compact complex manifold M, (4.4) still holds.

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Now we can give a proof of Theorem 3.1 by expanding the right-hand side of (4.4) as a Laurent series in t as follows:



n

1 + ye λi t

P ∈zero( X ) i =1

1 − e λi t



=

λ1 · · · λn (−t )n

P ∈zero( X )



=

λ1 · · · λn (−t )n



1

λ1 · · · λn (−t )n

P ∈zero( X )

∞ n   = (−t ) j −n p =0

j =0

∞ n   = (−t ) j −n p =0

p =0

j =0

p =0

j =0

p T j 1 (−λ1 t , . . . , −λn t ), . . . ,

σ



σn (−λ1 t , . . . , −λn t ) · y p

∞  n  

j p (−t ) T j σ1 (λ1 , . . . , λn ), . . . , σn (λ1 , . . . , λn ) · y p p T j ( 1 (λ1 , . . . , λn ), . . . ,



σ

p



σn (λ1 , . . . , λn ))

λ1 · · · λn

P ∈zero( X )

T j ( L P ( X ))

P ∈zero( X )

j =0

1 − e λi t

i =1

∞ n  

1

P ∈zero( X )

=

n

(−λi t )(1 + ye λi t )

1

 · yp

 · yp.

cn ( L P ( X ))

(4.5) p

The second equality in (4.5) comes from the definition of T j given in Section 2.2. Comparing (4.4) with (4.5) leads to, for each 0  p  n, p

χ (M ) ≡

∞  

(−t )

which means that

P ∈zero( X )

T j ( L P ( X ))

P ∈zero( X )

j =0



p



j −n



p

T j ( L P ( X )) cn ( L P ( X ))

cn ( L P ( X ))

 ,

∀t ,

0  j < n,

0,

χ p ( M ), j = n,

=

0,

(4.6)

j > n.

When 0  j < n, (4.6) is a direct corollary of Bott residue formula (2.1). When j = n, this is given by Bott residue formula (2.1) and Hirzebruch–Riemann–Roch formula (4.3). Those corresponding to j > n are what we want in Theorem 3.1 and it can not be derived from residue formula and H-R-R themselves. 4.2. Proof of Theorem 3.5 In this section we will explain why we have Theorem 3.5, which stems from another interesting phenomenon of Hirzebruch χ y -genus. This phenomenon has been observed, explicitly or implicitly, in several articles, which we shall gather together in what follows. Given an m-dimensional compact complex manifold V , many mathematicians have noticed that the coefficients of the polynomial χ y ( V ) at y = −1 are more interesting than χ p ( V ) themselves. More precisely, if we write

χ y (V ) =

m 

χ p · yp =

p =0

m 

p

A m (c 1 , . . . , cm )[ M ] · ( y + 1) p ,

p =0

p A m (c 1 , . . . , cm )

p

then can be expressed explicitly from Hirzebruch–Riemann–Roch formula (4.2). Here we list A m (c 1 , . . . , cm ) for p  4 and then give the related literatures.

1 1 Am (c 1 , . . . , cm ) = − mcm , 2   1 m(3m − 5) 2 A m (c 1 , . . . , cm ) = c m + c 1 c m −1 , 12 2 0 Am (c 1 , . . . , cm ) = cm ,

3 Am (c 1 , . . . , cm ) = − 4 Am (c 1 , . . . , cm )

=

1



m(m − 2)(m − 3)

24 1

5760

2



3



cm + (m − 2)c 1 cm−1 ,



m 15m − 150m2 + 485m − 502 cm + 4 15m2 − 85m + 108 c 1 cm−1

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 + 8 c 12 + 3c 2 cm−2 − 8 c 13 − 3c 1 c 2 + 3c 3 cm−3 , ···. 0 1 = cm is quite well known as Am = χ y | y=−1 is the Euler characteristic of V [9, Theorem 15.8.1]. The computation of Am only needs one fact χ p = (−1)m χ m− p : 0 Am

1 Am

 m m  m p m  = =− p χ p (−1) p = − χ (−1) p = − cm .  d y y =−1 2 2 dχ y 

p =0

p =0

2 Am

The calculation of has been obtained implicitly by Narasimhan and Ramanan in [18, p. 18]. As we have mentioned 2 in Section 1, Futaki and Morita also derived A m in their proof of f c1 cn ( X ) = 0 [6, Corollary 5.3.12]. Libgober and Wood might be the first who wrote down the relation between χ p and c 1 cm−1 explicitly [15, pp. 141–143], and gave some 2k+1 applications [15, Theorems 1 and 2], by using this formula. They also noticed that [15, p. 144], for each odd 2k + 1, A m 0 2 4 2k is a linear combination of A m , Am , Am , . . . , Am . Inspired by [18], Salamon systematically investigated this phenomenon in 2 [19, Section 3] and applied A m [19, Corollary 3.4], to obtain a restriction on the Betti numbers of hyper-Kähler manifolds 3 4 5 6 and A m are included in [19, p. 145]. The expressions of A m and A m are also [19, Theorem 4.1]. The expressions of A m 1 2 3 included in [20, p. 300], whose calculations are due to R. Jung. Hirzebruch used A m , A m and A m to obtain a divisibility result on the Euler characteristic of those almost-complex manifolds whose c 1 cm−1 = 0 [10]. In particular, those almosti complex manifolds with c 1 = 0 satisfy this property. Hirzebruch [10, p. 805] states that R. Jung has generated the A m up to i = 10 by computer. Now we turn to the proof of Theorem 3.5. Proof. We suppose 1  k  2p  n + k, which, through Remark 3.1, implies that

f

2p −k

A n+k (c 1 ,...,cn )

( X ) = 0,

1  k  2p  n + k.

But from Lemma 2.5 we know that 2p −k

2p

A n+k (c 1 , . . . , cn ) = A n+k (c 1 , . . . , cn+k )|cn+1 =···=cn+k =0 .

(4.7)

2 Taking p = 1 in (4.7) and using the explicit expression of A m (c 1 , . . . , cm ) we have

A n2+1 (c 1 , . . . , cn , 0) =

1 12

c 1 cn ,

A n0+2 (c 1 , . . . , cn , 0, 0) = 0,

which give the expressions a11 and a12 in Theorem 3.5. 4 (c 1 , . . . , cm ), we have Taking p = 2 in (4.7) and using the explicit expression of A m







A n4+1 (c 1 , . . . , cn , 0) = (·)c 1 cn + (·) c 12 + 3c 2 cn−1 − c 13 − 3c 1 c 2 + 3c 3 cn−2 ,







A n4+2 (c 1 , . . . , cn , 0, 0) = (·) c 12 + 3c 2 cn − c 13 − 3c 1 c 2 + 3c 3 cn−1 ,



A n4+3 (c 1 , . . . , cn , 0, 0, 0) = (·) c 13 − 3c 1 c 2 + 3c 3 cn , A n4+4 (c 1 , . . . , cn , 0, 0, 0, 0) = 0, which give the desired expressions a21 , a22 , a23 and a24 in Theorem 3.5. Here (·) denotes some nonzero rational number. p When going on taking bigger p we can obtain corresponding ak (c 1 , . . . , cn ) for p  3. This gives the desired proof of Theorem 3.5. 2 Remark 4.1. Note that χ p ( M ) can be defined for any compact, almost-complex manifold ( M 2n , J ). In fact, the choice of an ¯ of the almost Hermitian metric on M enables us to define the Hodge star operator ∗ and the formal adjoint ∂¯ ∗ = −∗∂∗ ∂¯ -operator. Then for each 0  p  n, we have an elliptic differential operator

 q even

¯ ∂¯ ∗ ∂+

Ω p ,q ( M ) −→



Ω p ,q ( M ),

(4.8)

q odd

where Ω p ,q ( M ) := Γ (Λ p T ∗ M ⊗ Λq T ∗ M ). Here T ∗ M is the dual of holomorphic tangent bundle T M in the sense of J . The index of this operator is denoted by





χ p ( M ) := dimC ker ∂¯ + ∂¯ ∗ − dimC coker ∂¯ + ∂¯ ∗ . When J is integrable, i.e., M is an n-dimensional complex manifold, this definition of χ p ( M ) coincides with that of (4.1) and (4.2) also holds by Atiyah–Singer index theorem. So all the discussions in this subsection are also valid for compact, almost-complex manifolds.

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P. Li / Topology and its Applications 160 (2013) 488–497

4.3. Proof of Theorem 3.7 The idea of the proof of Theorem 3.7 is quite similar to that of [14, Proposition 1.1], and has been generalized in [12, Proposition 2.3]. Here, for the sake of completeness, we still give a detailed proof. (i ) (i ) We set zero( X ) = { P 1 , . . . , P r } and at each P i , the eigenvalues of L P i ( X ) are λ1 , . . . , λn . Given a partition μ = (1m1 (μ) 2m2 (μ) · · · nmn (μ) ) of weight n + 1, what we need to show is that, if r < m(μ) (see Definition 3.6), then f cμ ( X ) = 0. Case 1. m1 (μ) > max{m2 (μ), . . . , mn (μ)}. In this case, we have m1 (μ) = max{m1 (μ), m2 (μ), . . . , mn (μ)} = m(μ). Let



 n 

 (i )  c1 L P i ( X ) = λ j  1  i  r = {s1 , . . . , sl } ⊂ C.

j =1

Clearly l  r and s1 , . . . , sl are mutually distinct. We define



A t :=

1i r c 1 ( L P i ( X ))=st

[c 2 ( L P i ( X ))]m2 (μ) · · · [cn ( L P i ( X ))]mn (μ) , cn ( L P i ( X ))

1  t  l.

Now we consider the following m1 (μ) − 1 partitions



μ( j) := 1 j 2m2 (μ) · · · nmn (μ) , 0  j  m1 (μ) − 2. μ( j) are all less than n as the weight of μ is exactly n + 1. Then Bott residue formula tells us that ⎧ A 1 + A 2 + · · · + A l = 0, ⎪ ⎪ ⎨ s1 A 1 + s2 A 2 + · · · + sl Al = 0, (4.9) .. ⎪ ⎪ ⎩ . m (μ)−2 (s1 ) 1 A 1 + (s2 )m1 (μ)−2 A 2 + · · · + (sl )m1 (μ)−2 A l = 0.

The weights of these

If r < m(μ), which is the assumption of Theorem 3.7, then l  r  m(μ) − 1 = m1 (μ) − 1, which means that (4.9) has at least l rows. Thus the first l lines of (4.9) imply that A 1 = · · · = Al = 0 as s1 , . . . , sl are mutually distinct. Thus

f c μ ( X ) = (s1 )m1 (μ) A 1 + · · · + (sl )m1 (μ) A l = 0. Case 2. m1 (μ)  max{m2 (μ), . . . , mn (μ)}. In this case, we may assume, without loss of generality, that m2 (μ) = max{m1 (μ), m2 (μ), . . . , mn (μ)}. Then we consider the following m2 (μ) partitions



μ( j) := 1m1 (μ) 2 j 3m3 (μ) · · · nmn (μ) , 0  j  m2 (μ) − 1, the weights of which are all less than n as the weight of



 c 2 L P i ( X )  1  i  r = {s1 , . . . , sl } ⊂ C

μ is exactly n + 1. We can define



and A t accordingly. Using the same idea as in Case 1 we can show that, if r < m2 (μ) + 1, then A 1 = · · · = Al = 0 and thus f c μ ( X ) = 0. This completes the proof of Theorem 3.7. 5. Concluding remarks In this section we will propose two questions, which are closely related to the discussions in this paper and whose answers are unknown to the author up to now. Given any pair of positive integers (n, k), we define a set Φ(n; k) (⊂ I n+k (G L (n, C))) as follows. Definition 5.1. We define φ ∈ Φ(n; k), if, for any n-dimensional compact complex manifold M and any non-degenerate holomorphic vector field X on it, we have

 P ∈zero( X )

φ( L P ( X )) cn ( L P ( X ))

= 0.

It is very challenging (also very difficult, at least from the author’s viewpoint) to determine Φ(n; k) for any given positive integer pair (n, k). Clearly, Φ(n; k) are linear subspaces of I n+k (G L (n, C)). We end this note by raising the following three questions.

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Question 5.2. Given a pair of positive integers (n, k), dimC Φ(n; k) = ? Theorem 3.5 tells us that each Φ(n; k) is non-empty and thus dimC Φ(n; k)  1. Another related question is Question 5.3. For any (n, k), do the elements in Φ(n; k) essentially all come from Theorem 3.5? Acknowledgement I would like to express my sincere thanks to my host researcher in Waseda University, Professor Martin Guest, for his hospitality and help. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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