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Automorphisms and connections on Higgs bundles over compact Kähler manifolds
Differential Geometry and its Applications 32 (2014) 139–152
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Automorphisms and connections on Higgs bundles over compact Kähler manifolds Indranil Biswas a,∗ , Steven B. Bradlow b , Adam Jacob c , Matthias Stemmler d a School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India b Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street (MC-382), Urbana, IL 61801, USA c Department of Mathematics, Harvard University, Cambridge, MA 02138, USA d Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße, Lahnberge, 35032 Marburg, Germany
a r t i c l e
i n f o
Article history: Received 17 September 2012 Available online 10 December 2013 Communicated by S. Merkulov MSC: 53C07 32L05 Keywords: Higgs bundle Automorphism Approximate Yang–Mills–Higgs connection
a b s t r a c t Let (E, ϕ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of E by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by G the direct sum of these stable quotients, and let the singular set of G be called S ⊂ X. We construct a 1-parameter family of filtration preserving C ∞ isomorphisms Φt : G|X\S → E|X\S , and a Hermitian metric h on G|X\S , such that as t → +∞, the Chern connection for the Hermitian Higgs bundle (Φ∗t E, Φ∗t ϕ, h)|X\S converges, in the C ∞ Fréchet topology over any relatively compact open subset of X\S, to the direct sum of the Yang–Mills–Higgs connections on the direct summands in G. We also prove an analogous result for principal Higgs G-bundles on X. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Constructing connections with optimal curvature properties has long been a fundamental problem within differential geometry. In the case of an indecomposable holomorphic vector bundle, one natural connection is a Hermitian–Einstein connection, defined by having curvature a constant multiple of the identity after tracing away the 2-form part. By the famous theorem of Donaldson, Uhlenbeck and Yau, such connections exist if and only if the holomorphic vector bundle is stable [8,19,9]. In the unstable case, rather than trying to find a specific canonical connection, one can instead look for a sequence of connections that approach a canonical connection on a related bundle (the holomorphic structure of the bundle may change in the limit). Kobayashi introduced the notion of an approximate Hermitian–Einstein structure in [15], and defined it to be a sequence of connections which approach in C 0 * Corresponding author. E-mail addresses: [email protected] (I. Biswas), [email protected] (S.B. Bradlow), [email protected] (A. Jacob), [email protected] (M. Stemmler). 0926-2245/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.difgeo.2013.11.008
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topology a Hermitian–Einstein connection. Kobayashi then proved the existence of such a structure to be equivalent to semistability of the bundle in the projective case. This result was extended to Kähler manifolds in [13] and to principal bundles in [3]. When the bundle is not semistable, the limiting connection cannot be a Hermitian–Einstein connection, yet there is a similar construction. In this case one looks for a sequence of connections for which the curvatures converge to an endomorphism with locally constant eigenvalues given by the slopes of the successive quotients of the Harder–Narasimhan filtration of the bundle. We call this sequence of connections an approximate Hermitian structure. The existence of these structures has been useful in the study of many geometric problems, most notably in [7]. In their paper Daskalopoulos and Wentworth used the existence of Lp approximate Hermitian structures on holomorphic bundles over Kähler surfaces to determine the limiting properties of the Yang–Mills flow. Their convergence results were extended to higher dimensions in [14,17]. We also note that Li and Zhang extended the result of Daskalopoulos and Wentworth to the context of Higgs bundles [12]. Hong and Tian proved that there is a subsequence of the Yang–Mills flow which converges smoothly on the complement of the singular set [10]. Despite these results, aside for the Riemann surfaces case it is not known whether the exponent p can be increased to infinity. The existence of an L∞ structure would give a full generalization of the equivalence between semistable bundles and approximate Hermitian–Einstein structures. In [6], the second author showed that the vector bundles on a Riemann surface admit C 0 approximate Hermitian–Einstein structures. This has been generalized to principal bundles over Riemann surfaces in [20]. The key difficulty in working with these structures is that for higher dimensions the Harder–Narasimhan filtration is given by subsheaves, not subbundles. Thus the limiting endomorphism in the approximate Hermitian structure is only defined away from an analytic subvariety of complex codimension at least two. In this paper, we show that by working away from this subvariety, we can produce a C ∞ approximate Hermitian structure which converges to a connection on the successive quotients of the Harder–Narasimhan filtration. Specifically, let (X, ω) be a compact connected Kähler manifold, and let (E, ϕ) be a Higgs vector bundle over X. Fix a filtration of E by coherent analytic subsheaves 0 = F0 F1 · · · F−1 F = E such that for each i ∈ [1, ], • the quotient Fi /Fi−1 is torsionfree, 1 • ϕ(Fi ) ⊂ ΩX ⊗ Fi , and • the Higgs sheaf (Fi /Fi−1 , ϕi ) is stable, where ϕi is the Higgs field on Fi /Fi−1 induced by ϕ. We note that such filtrations exist (see Proposition 2.6 below). Let S ⊂ X be the singular set for the graded object
G :=
Fi /Fi−1
i=1
associated to this filtration. This S is a closed complex analytic subset of X of complex codimension at least two. Denote by Fi (respectively, G) the C ∞ vector bundle underlying the holomorphic vector bundle Fi |X\S (respectively, G|X\S ). Write E = (E, ∂), where E is the underlying C ∞ vector bundle of E, and ∂ is the Dolbeault operator defining the holomorphic structure of E. Over X \ S, the two C ∞ vector bundles G and E are isomorphic. Take any filtration preserving C ∞ isomorphism over X \ S
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f : G −→ E that induces the identity map on G. Let ∂ i be the Dolbeault operator on Fi /Fi−1 induced by ∂, and let ϕi be the Higgs field on (Fi /Fi−1 , ∂ i ) induced by ϕ. Since each of the Higgs sheaves (Fi /Fi−1 , ϕi ) is stable, by [4, Theorem 3.1] (reproduced Theorem 2.5), there exists an admissible Yang–Mills–Higgs metric hi on (Fi /Fi−1 , ∂ i ) for each i ∈ [1, ]. Let D be the Chern connection on G defined by the holomorphic structure the Hermitian metric h := i=1 hi . We prove the following (see Theorem 3.1):
(1.1) i=1
∂ i , the Higgs field
i=1
ϕi and
Theorem 1.1. There is a smooth 1-parameter family {Φt }t∈R of C ∞ isomorphisms from G to E inducing the identity map on G, such that if Dϕ,t is the Chern connection of (G, Φ∗t ∂, Φ∗t ϕ, h), then as t → +∞, the connections Dϕ,t converge in the C ∞ Fréchet topology on every relatively compact open subset of X \ S to (see (1.1)). In particular, as t → +∞, the curvature forms of Dϕ,t converge in the the Chern connection D ∞ C Fréchet topology on every relatively compact open subset of X \ S to the curvature form of D. Note that if X is a Riemann surface, then the singular set S is empty. Consequently, in this case Theorem 1.1 gives convergence in the C ∞ Fréchet topology over the entire X. In Section 5, we address the question of the existence of a good Hermitian structure in the more general situation of principal Higgs bundles over a compact connected Kähler manifold. We prove for principal Higgs bundles a result similar to Theorem 1.1 (see Theorem 5.1). 2. Preliminaries Let (X, ω) be a compact connected Kähler manifold. The degree of a torsionfree coherent analytic sheaf E on X with respect to ω will be denoted by degree(E). If rank(E) > 0, the slope μ(E) of E is defined to be μ(E) :=
degree(E) . rank(E)
Definition 2.1. Let E be a torsionfree coherent analytic sheaf on X. 1 2 (i) A Higgs field on E is defined to be a section of ΩX ⊗ End(E) such that the section ϕ ∧ ϕ of ΩX ⊗ End(E) vanishes identically. (ii) If ϕ is a Higgs field on E, then (E, ϕ) is called a Higgs sheaf. If E is locally free, then (E, ϕ) is called a Higgs vector bundle.
Definition 2.2. Let (E, ϕ) be a Higgs sheaf on X. (i) (E, ϕ) is called stable (respectively, semistable) if for every coherent analytic subsheaf F ⊂ E with 0 < rank(F) < rank(E) such that • the quotient E/F is torsionfree, and 1 ⊗ F, • ϕ(F) ⊂ ΩX we have μ(F) < μ(E)
respectively, μ(F) μ(E) .
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(ii) (E, ϕ) is called polystable if
(E, ϕ) =
N
E i , ϕi ,
i=1
where each (E i , ϕi ) is a stable Higgs sheaf of slope μ(E i ) = μ(E). Therefore, a polystable Higgs sheaf is semistable. Now, let (E, ϕ) be a Higgs vector bundle over X, and let h be a Hermitian metric on E. Denote by D the unitary Chern connection of E with respect to the metric h. To the Hermitian Higgs bundle (E, ϕ, h) one can associate a natural connection Dϕ := D + ϕ + ϕ∗ ,
(2.1)
where ϕ∗ denotes the adjoint of ϕ with respect to the Hermitian metric h. (See [18, Section 3] for details.) This connection Dϕ will be called the Chern connection of (E, ϕ, h). Denote by F (Dϕ ) the curvature form of the connection Dϕ . Definition 2.3. The Hermitian metric h is called a Yang–Mills–Higgs metric on (E, ϕ) if √
−1 · ΛF Dϕ = λ · IdE ,
where Λ denotes the adjoint of wedging with the Kähler form ω and λ is a constant real number. If h is a Yang–Mills–Higgs metric on (E, ϕ), then the connection Dϕ is called a Yang–Mills–Higgs connection. Note that √
√ −1 · ΛF Dϕ = −1Λ F (D) + ϕ, ϕ∗ ,
where F (D) is the curvature of the Chern connection D, and [·, ·] is defined using the exterior product on differential forms and the Lie algebra structure of the fibers of End(E). We recall the definition of Yang–Mills–Higgs metrics on (singular) Higgs sheaves. Let (E, ϕ) be a Higgs sheaf on X, and let S ⊂ X be the singular set for E, meaning the closed complex analytic subset of X outside which E is locally free. Since E is torsionfree, the complex codimension of S is at least two. Definition 2.4. A Hermitian metric h on the holomorphic vector bundle E|X\S is called admissible if • the curvature F (D) of the Chern connection D of (E|X\S , h) is square-integrable, and • the contracted Chern curvature ΛF (D) is bounded. In [4], the following Donaldson–Uhlenbeck–Yau type correspondence for Higgs sheaves was established. Theorem 2.5. (See [4, Theorem 3.1].) Let (E, ϕ) be a Higgs sheaf on a compact Kähler manifold (X, ω), and let S be the singular set for E. If E is stable, then there exists an admissible Yang–Mills–Higgs metric on E|X\S . Moreover, the admissible Yang–Mills–Higgs connection is unique. The proof of Theorem 2.5 is based on the proof of the Donaldson–Uhlenbeck–Yau type correspondence for reflexive sheaves established by Bando and Siu [2]. We end this section with an existence proof for the type of filtration used in Theorem 1.1.
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Proposition 2.6. Given a Higgs sheaf (E, ϕ) over X, there is a filtration of E by coherent analytic subsheaves 0 = F0 F1 · · · F−1 F = E such that for each i ∈ [1, ], • the quotient Fi /Fi−1 is torsionfree, 1 • ϕ(Fi ) ⊂ ΩX ⊗ Fi , and • the Higgs sheaf (Fi /Fi−1 , ϕi ) is stable, where ϕi is the Higgs field on Fi /Fi−1 induced by ϕ. Proof. The proposition will be proved using induction on the rank of E. If rank(E) = 1, then the statement is obvious. Assume that the proposition is proved for all E with rank(E) r −1. Now assume that rank(E) = r. If the Higgs sheaf (E, ϕ) is stable, then there is nothing to prove. Otherwise, there is a coherent analytic subsheaf F ⊂ E with 0 < rank(F) < rank(E) such that • the quotient E/F is torsionfree, 1 • ϕ(F) ⊂ ΩX ⊗ F, and • μ(F) μ(E). Let ϕF (respectively, ϕE/F ) be the Higgs field on F (respectively, E/F) induced by the Higgs field ϕ. Since rank(F), rank(E/F) r − 1, by the induction hypothesis, both the Higgs sheaves (F, ϕF ) and (E/F, ϕE/F ) have a filtration satisfying the conditions in the proposition. The filtration of E/F given by the induction hypothesis produces a filtration of E using the natural projection E → E/F. Each term of this filtration contains F. Hence this filtration of E, and a filtration of F given by the induction hypothesis, together produce a filtration of E satisfying the conditions in the proposition. 2 3. Flow under a 1-parameter group of automorphisms As before, let (X, ω) be a compact connected Kähler manifold, and let (E, ϕ) be a Higgs vector bundle over X. Fix a filtration 0 = F0 F1 · · · F−1 F = E
(3.1)
satisfying the conditions in Proposition 2.6. Denote by S ⊂ X the singular set for the graded object G := i=1 Fi /Fi−1 associated to the filtration (3.1). Therefore, each Fi /Fi−1 is locally free on X \ S, and S is a closed complex analytic subset. Since the sheaf G is torsionfree, the codimension of S is at least two. Denote by Fi the C ∞ vector bundle underlying the holomorphic vector bundle Fi |X\S . Write E = (E, ∂), where E is the underlying C ∞ vector bundle of E, and ∂ is the Dolbeault operator defining the holomorphic structure of E. The restriction of E to X \ S will also be denoted by E. Fix C ∞ subbundles Vi ⊂ E over X \ S, 0 i , such that (1) the natural homomorphism
f:
i=0
is an isomorphism, and
Vi −→ E
(3.2)
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(2) for every j ∈ [0, ], f
j
Vi
= Fj .
i=0
Note that V0 = 0, and the composition Vi → Fi −→ Fi /Fi−1
(3.3)
is an isomorphism for all i ∈ [1, ]. Let ∂ i be the Dolbeault operator on Vi induced by ∂ using the isomorphism of Vi with Fi /Fi−1 in (3.3) (note that Fi /Fi−1 is a coherent analytic sheaf). Let ϕi be the Higgs field on the holomorphic vector bundle (Vi , ∂ i ) induced by ϕ using the isomorphism in (3.3). Since each of the Higgs sheaves Fi /Fi−1 on X equipped with the Higgs field induced by ϕ is stable, by Theorem 2.5 there exists an admissible Yang–Mills–Higgs metric hi on the holomorphic vector bundle (Fi /Fi−1 )|X\S . Denote by hi the Hermitian metric on Vi induced by hi using the isomorphism in (3.3). We note that hi is an admissible Yang–Mills–Higgs metric on the holomorphic Higgs bundle (Vi , ∂ i , ϕi ). Let D
(3.4)
be the Chern connection of the Hermitian Higgs bundle i=1 (Vi , ∂ i , ϕi , hi ). Let f ∗ ∂ := f −1 ◦ ∂ ◦ f be the Dolbeault operator on i=1 Vi given by ∂ using the isomorphism f in (3.2). Let f ∗ ϕ := f −1 ◦ ϕ ◦ f be the Higgs field on ( i=1 Vi , f ∗ ∂) given by ϕ using the isomorphism f . Let h := i=1 hi be the Hermitian metric on i=1 Vi , where hi is the above Yang–Mills–Higgs metric on (Vi , ∂ i , ϕi ). For every t ∈ R, define the C ∞ isomorphism Φt : i=1 Vi → E as follows: Φt := f ◦ At , where At is the C ∞ automorphism of
i=1
(3.5)
Vi given by
At :=
exp(−jt) · IdVj .
(3.6)
j=1
Define Φ∗t ∂ := Φ−1 t ◦ ∂ ◦ Φt
and Φ∗t ϕ := Φ−1 t ◦ ϕ ◦ Φt .
In particular, the Higgs bundle ( i=1 Vi , Φ∗t ∂, Φ∗t ϕ) is isomorphic to the Higgs bundle ( i=1 Vi , f ∗ ∂, f ∗ ϕ), and hence it is isomorphic to (E, ∂, ϕ), for all t. Keep the Hermitian metric h on i=1 Vi fixed, meaning it is not changed as t changes. Let Dϕ,t
(3.7)
be the Chern connection of the Hermitian Higgs bundle ( i=1 Vi , Φ∗t ∂, Φ∗t ϕ, h) (see (2.1)). Theorem 3.1. As t → +∞, the Chern connection Dϕ,t in (3.7) converges in the C ∞ Fréchet topology on in (3.4). In particular, as t → +∞, every relatively compact open subset of X \ S to the Chern connection D
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the curvature form of Dϕ,t converges in the C ∞ Fréchet topology on every relatively compact open subset of X \ S to the curvature form of D. Proof. For notational convenience, we only give a proof for the case where = 2. The proof for filtrations of arbitrary length works analogously. Since the Hermitian metric h on V1 ⊕ V2 is kept fixed, to prove the theorem it suffices to show that as t → +∞, (1) the Dolbeault operator Φ∗t ∂ converges to ∂ 1 ⊕∂ 2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S, and (2) the Higgs field Φ∗t ϕ converges to ϕ1 ⊕ ϕ2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. In terms of the direct sum decomposition V1 ⊕ V2 , we have f ∗∂ =
∂1 0
α ∂2
,
where α is a smooth (0, 1)-form on X \ S with values in Hom(V2 , V1 ). From the definitions of Φt and At (see (3.5) and (3.6)), we conclude that Φ∗t ∂
=
A−1 t
∗
◦ f ∂ ◦ At =
exp(−t) · α ∂2
∂1 0
.
This implies that as t → +∞, the Dolbeault operator Φ∗t ∂ converges to ∂ 1 ⊕ ∂ 2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. Analogously, we have f ∗ϕ =
β ϕ2
ϕ1 0
,
where β is a smooth (1, 0)-form on X \ S with values in Hom(V2 , V1 ). It follows that ∗ Φ∗t ϕ = A−1 t ◦ f ϕ ◦ At =
ϕ1 0
exp(−t) · β ϕ2
.
Therefore, as t → +∞, the Higgs field Φ∗t ϕ converges to ϕ1 ⊕ ϕ2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. This completes the proof. 2 4. Principal Higgs G-sheaves Let G be a connected reductive affine algebraic group defined over C. The Lie algebra of G will be denoted by g. A parabolic subgroup of G is a Zariski closed connected subgroup P ⊂ G such that the quotient variety G/P is complete. A large open subset of X is a dense open subset U ⊂ X such that the complement X \ U is a complex analytic subset of X of complex codimension at least two. The inclusion map U → X will be denoted by ιU . For a holomorphic principal G-bundle EG , the holomorphic vector bundle EG ×G g, the one associated to EG for the adjoint action of G on g, will be denoted by ad(EG ). This ad(EG ) is known as the adjoint vector bundle for EG . We note that the fibers of ad(EG ) are Lie algebras identified with g up to a conjugation.
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A principal G-sheaf on X is a pair (U, EG ), where U is large open subset of X and EG is a holomorphic principal G-bundle over U , such that the direct image ιU ∗ ad(EG ) is a coherent analytic sheaf on X. A principal Higgs G-sheaf is a triple of the form (U, EG , θ), where (U, EG ) is a principal G-sheaf, and θ ∈ H 0 U, ΩU1 ⊗ ad(EG ) is such that the section θ ∧ θ of ΩU2 ⊗ ad(EG ) vanishes identically. (The product θ ∧ θ is defined using the Lie algebra structure of the fibers of ad(EG ) and the natural homomorphism ΩU1 ⊗ ΩU1 −→ ΩU2 .) A principal Higgs G-sheaf (U, EG , θ) is called stable if for every triple of the form (U , Q, EQ ), where • U ⊂ U is a large open subset of X, • Q ⊂ G is a maximal proper parabolic subgroup, and • EQ ⊂ EG |U
(4.1)
is a holomorphic reduction of structure group of EG |U to Q over U such that θ|U is a section of ΩU1 ⊗ ad(EQ ), the following inequality holds: degree ad(EQ ) < 0, where ad(EQ ) = EQ ×Q Lie(Q) is the adjoint vector bundle defined as before. The above condition that degree(ad(EQ )) < 0 is equivalent to the condition that degree(ad(EG )/ad(EQ )) > 0, because degree(ad(EG )) = 0. Since ad(EQ ) is a subbundle of ad(EG ) over a large open subset, and ad(EG ) extends to X as a coherent analytic sheaf, it follows that ιU ∗ ad(EQ ) is a coherent analytic sheaf. The unipotent radical of a parabolic subgroup P ⊂ G will be denoted by Ru (P ). We recall that Ru (P ) is the unique maximal connected unipotent normal subgroup of P . The quotient P/Ru (P ), which we will denote by L(P ), is a connected reductive complex affine algebraic group. For a principal P -bundle EP defined over U , the quotient EP /Ru (P ) is a principal L(P )-bundle over U . Note that ad(EP /Ru (P )) is a quotient bundle of ad(EP ). Hence any section of ΩU1 ⊗ ad(EP ) produces a section of ΩU1 ⊗ ad(EP /Ru (P )). Lemma 4.1. Take any principal Higgs G-sheaf (U, EG , θ). There is a large open subset U ⊂ X, a parabolic subgroup P ⊂ G, and a holomorphic reduction of structure group EP ⊂ EG |U of EG to P , such that (1) θ ∈ H 0 (U , ΩU1 ⊗ ad(EP )), and is stable, where θ is the Higgs field on the holomorphic principal (2) the Higgs L(P )-sheaf (U , EP /Ru (P ), θ) L(P )-bundle EP /Ru (P ) induced by θ. Proof. If (U, EG , θ) is stable, then the lemma is proved, because G itself is a parabolic subgroup of G. Assume that (U, EG , θ) is not stable. Take a triple (U1 , Q, EQ ), where • U1 ⊂ U is a large open subset of X, • Q ⊂ G is a maximal proper parabolic subgroup, • EQ ⊂ EG |U1 is a holomorphic reduction of structure group of EG |U1 to Q over U1 with the property that θ|U1 is a section of ΩU1 1 ⊗ ad(EQ ), and • degree(ad(EQ )) 0.
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Consider the holomorphic principal L(Q)-bundle EL(Q) := EQ /Ru (Q) on U1 . Let θ1 be the Higgs field on EL(Q) given by θ. If (U1 , EL(Q) , θ1 ) is stable, then we stop. Note that since ad(EQ ) extends to X as a coherent analytic sheaf, and ad(EL(Q) ) is a quotient of ad(EQ ), it follows that ad(EL(Q) ) extends to X as a coherent analytic sheaf. If (U1 , EL(Q) , θ1 ) is not stable, then we replace (U, EG , θ) in the above construction by (U1 , EL(Q) , θ1 ). If the principal Higgs sheaf resulting from the above construction with (U, EG , θ) replaced by (U1 , EL(Q) , θ1 ) is stable, then we stop. Otherwise, we iterate the above construction. Note that at each step of the construction, the semisimple rank of the reductive group drops by one. If H is abelian reductive, then any principal Higgs H-sheaf is stable (there is no proper parabolic subgroup of H). Consequently, the iterative construction stops in at most n steps, where n is the semisimple rank of G. Consider the quotient map q : Q → L(Q). If Q1 is a parabolic subgroup of L(Q), then q −1 (Q1 ) is a parabolic subgroup of G. We note that this follows from the fact that G/q −1 (Q1 ) is a fiber bundle over G/Q with fiber L(Q)/Q1 ; since both G/Q and L(Q)/Q1 are complete varieties, it follows immediately that G/q −1 (Q1 ) is also complete. Note that in this argument we did not use that Q is a maximal proper parabolic subgroup. Let EQ1 ⊂ EL(Q) be a holomorphic reduction of structure group of EL(Q) to the parabolic subgroup Q1 over a large open subset U2 . Let q : EQ −→ EL(Q) = EQ /Ru (Q) be the quotient map. Then q −1 (EQ1 ) ⊂ EG |U2 is a holomorphic reduction of structure group of EG to the parabolic subgroup q −1 (Q1 ) of G. If the above defined Higgs field θ1 on EL(Q) = EQ /Ru (Q) is a holomorphic section of ΩU1 2 ⊗ ad(EQ1 ), then it follows immediately that θ ∈ H 0 U2 , ΩU1 2 ⊗ ad q −1 (EQ1 ) . Therefore, at each step of the construction, we get a parabolic subgroup of G, and a holomorphic reduction of structure group of EG to this parabolic subgroup over a large open subset, such that the Higgs field θ is a Higgs field on the reduction. Hence, after finitely many iterations of the construction, we get a reduction as in the statement of the lemma. 2 Fix a maximal compact subgroup K ⊂ G.
(4.2)
See [4, p. 553] for the definition of a Yang–Mills–Higgs reduction to K of a principal Higgs G-sheaf. The following is proved in [4]: Theorem 4.2. (See [4, p. 554, Theorem 4.6].) Any stable principal Higgs G-sheaf admits a Yang–Mills–Higgs reduction to K. The Yang–Mills–Higgs connection is unique. 5. Approximate Yang–Mills–Higgs connections on principal Higgs G-sheaves Let (U, EG , θ) be a principal Higgs G-sheaf. Take a triple (U , P, EP ) for (U, EG , θ) as in Lemma 4.1. Let
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EL(P ) := EP /Ru (P ) be the corresponding holomorphic principal P/Ru (P )-bundle. Let p : EP −→ EL(P )
(5.1)
be the quotient map. Let L be the Zariski closure of P ∩ K in G, where K is the subgroup in (4.2). We note that L is also the complex analytic closure of P ∩ K. The composition L → P −→ L(P ) = P/Ru (P )
(5.2)
is an isomorphism, where P → P/Ru (P ) is the quotient map. Such a subgroup L is called a Levi factor of P (see [11, p. 184], [5]). We will identify L with L(P ) using the above isomorphism. The principal G-bundle EG admits a C ∞ reduction of structure group to the subgroup K in (4.2). To see this, note that G/K is contractible. Hence the fiber bundle EG /K → U with fiber G/K admits a C ∞ section. The inverse image, for the quotient map EG → EG /K, of the image of any C ∞ section X → EG /K is a C ∞ reduction of structure group of EG to K. Fix a C ∞ reduction of structure group of EG 0 EK ⊂ EG
to K. 0 We note that EK ∩ EP is a C ∞ reduction of structure group of EG to P ∩ K over U . Let EL be the 0 ∩ EP . This means that for each point smallest fiberwise complex analytic subset of EG |U containing EK 0 x ∈ U , the fiber (EL )x is the smallest complex analytic subset (EP )x containing (EK )x ∩ (EP )x . We note that EL need not be a complex analytic subset of EP . It is easy to see that EL is the principal L-bundle 0 0 (EK ∩ EP ) ×P ∩K L obtained by extending the structure group of the principal P ∩ K-bundle EK ∩ EP 0 P ∩K using the inclusion of P ∩ K in L. We note that (EK ∩ EP ) × L is contained in EG using the map 0 0 (EK ∩ EP ) × L → EG defined by (z, g) → zg. (The total space (EK ∩ EP ) ×P ∩K L is the quotient of 0 (EK ∩ EP ) × L where two points (z1 , g1 ) and (z2 , g2 ) are identified if there is an element g0 ∈ P ∩ K such that z2 = z1 g0 and g2 = g0−1 g1 .) Therefore, EL is a C ∞ reduction of structure group of EG to L over U . 0 That EL coincides with (EK ∩ EP ) ×P ∩K L follows from the fact that L is the complex analytic closure of P ∩ K ⊂ P. The composition p
EL → EP −→ EP /Ru (P ) = EL(P ) |U is a C ∞ isomorphism (the map p is defined in (5.1)), and it commutes with the actions of L = L(P ). In other words, this composition identifies the principal L(P )-bundle EL(P ) |U with the C ∞ principal L-bundle EL using the isomorphism of L(P ) with L in (5.2). Using this isomorphism of EL(P ) |U with EL , the C ∞ principal L-bundle EL gets a holomorphic structure from the holomorphic structure of EL(P ) |U . The holomorphic principal L-bundle obtained this way will be denoted by EL . 0 Let EG be the holomorphic principal G-bundle over U obtained by extending the structure group of the holomorphic principal L-bundle EL using the inclusion of L in G. We note that there is a natural C ∞ isomorphism of principal G-bundles 0 γ : EG −→ EG |U
(5.3)
0 that sends the equivalence class of any (z, g) ∈ EL × G to zg ∈ EG (recall that EG is the quotient of EL × G where two points (z, g) and (z , g ) are identified if there is an element g0 ∈ L such that z = zg0
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0 and g = g0−1 g). It should be emphasized that this isomorphism γ between EG and EG |U need not be holomorphic. Let θ be the Higgs field on the holomorphic principal L(P )-bundle EL(P ) given by θ. Let θL be the 0 is the Higgs field on EL given by θ using the identification of EL(P ) |U with EL constructed above. Since EG extension of structure group of EL for the inclusion of L in G, we have
0 . ad(EL ) ⊂ ad EG 0 So we have H 0 (U , ΩU1 ⊗ ad(EL )) ⊂ H 0 (U , ΩU1 ⊗ ad(EG )). Let
0 θG ∈ H 0 U , ΩU1 ⊗ ad EG
(5.4)
0 be the Higgs field on EG given by θL using this inclusion. The group K ∩ L = K ∩ P will be denoted by KL . This KL is a maximal compact subgroup of the Levi factor L. is stable, and (EL , θL ) is isomorphic to it, we conclude from Theorem 4.2 that EL has a Since (EL(P ) , θ) Yang–Mills–Higgs reduction
EKL ⊂ EL = EL
(5.5)
for the Higgs field θL . Let EK := EKL ×KL K be the C ∞ principal K-bundle obtained by extending the structure group of EKL using the inclusion of KL in K. We note that 0 EK ⊂ EL ×L G = EG .
(5.6)
The above inclusion map is given by the inclusion map ι : EKL × K → EKL × G; the total spaces of the 0 are quotients of EKL × K and EKL × G respectively, and ι descends to the principal bundles EK and EG map in (5.6). 1 0 be the holomorphic principal G-bundle defined by the C ∞ principal G-bundle underlying EG Let EG equipped with the holomorphic structure given by the holomorphic structure of EG using the isomorphism 1 defined by θ using γ. So we have a Higgs G-sheaf γ in (5.3). Let θ1 be the Higgs field on EG
1 U , EG , θ1
(5.7)
0 1 such that the underlying C ∞ principal G-bundle on U is that for EG . Note that (U , EG , θ1 ) is holomorphically isomorphic to (U , EG , θ). Let ZL be the connected component of the center of L containing the identity element. Fix a homomorphism
ρ : C∗ −→ ZL
(5.8)
such that for any nontrivial character χ : P −→ C∗ trivial on Z0 (G) which is a nonnegative integral combination of simple roots, the composition homomorphism
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χ ◦ ρ : C∗ −→ C∗ is of the form c → cn with n positive. We note that it is enough the check the above condition for those characters χ of P that are simple roots and trivial on Z0 (G). Let Rn (p) be the Lie algebra of the unipotent radical Ru (P ). So Rn (p) is the nilpotent radical of the Lie algebra p of P . From the defining property of ρ it follows that lim
t∈R;t→−∞
Ad exp ρ(t) (v) = 0 for all v ∈ Rn (p)
(5.9)
(see [1, p. 708, Theorem 5]). Given a 1-parameter subgroup of G, there is a parabolic subgroup of G natural to it (see [16, p. 55, Definition 2.3/Proposition 2.6]). The parabolic subgroup of G associated to the above 1-parameter subgroup ρ is P itself. t of EL defined by Take any t ∈ C. Since ρ(exp(−t)) ∈ ZL , we get a C ∞ automorphism A t (z) = zρ exp(−t) , A
z ∈ EL .
t is a holomorphic automorphism of the Higgs L-sheaf (EL , θL ). Let We note that A 1 0 1 0 At : EG = EG −→ EG = EG 1 0 t . In other words, the automorphism At is the descent of be the C ∞ automorphism of EG = EG given by A 0 t is the automorphism At × IdG of EL × G. Note that At is a holomorphic automorphism of EG (because A 1 a holomorphic automorphism of EL ), but it is not necessarily a holomorphic automorphism of EG . For any t ∈ C, let 0 Φt := γ ◦ At : EG −→ EG |U
(5.10)
t be the C ∞ isomorphism of principal G-bundles, where γ is constructed in (5.3). Let EG be the holomorphic 0 principal G-bundle whose underlying C ∞ principal G-bundle is EG , and the holomorphic structure is the t one given by the holomorphic structure of EG using Φt in (5.10). Let θ t be the Higgs field on EG given by 0 θ using Φt . So the Higgs G-sheaf (U , EG , θ0 ) coincides with the one in (5.7). We note that (U , EG , θ) is t holomorphically isomorphic to (U , EG , θt ) for all t. 0 Let ∇0 be the connection on EG for the Higgs field θG (defined in (5.4)) and the reduction EK in (5.6). 0 We note that ∇0 coincides with the connection on EG induced by the Yang–Mills–Higgs connection on EL 0 for the Higgs field θL (this connection is constructed using θL and EKL in (5.5)). Note that since EG is an 0 extension of structure group of EL a connection on EL induces a connection on EG . 0 t For any t ∈ R, let ∇t be the connection on EG for the holomorphic structure EG , the Higgs field θ t (both defined above) and the reduction EK in (5.6).
Theorem 5.1. As t → ∞, the connection ∇t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the connection ∇0 . In particular, as t → ∞, the curvature of ∇t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the curvature of ∇0 . Proof. Since the reduction EK does not change with t, it suffices to prove the following two statements: t converges, in the C ∞ Fréchet topology over relatively (1) as t → ∞, the holomorphic structure of EG 0 compact open subsets of U , to the holomorphic structure of EG , and
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(2) as t → ∞, the Higgs field θ t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the Higgs field θG (defined in (5.4)). 0 t Let J0 (respectively, J t ) be the almost complex structure of EG (respectively, EG ). We have
0 αt := J t − J0 ∈ C ∞ U , Ω 0,1 ⊗ ad EG .
(5.11)
Let EP0 = EL ×L P be the holomorphic principal P -bundle over U obtained by extending the structure group of the holomorphic principal L-bundle EL using the inclusion of L in P . There is a canonical C ∞ isomorphism between the principal P -bundles EP0 and EP ; this isomorphism is the descent of the map EL × P → EP defined by (z, g) → zg. Let EP1 be the holomorphic principal P -bundle whose underlying C ∞ principal P -bundle is that of EP0 , and the holomorphic structure of EP1 is given by the holomorphic structure of EP using the above isomorphism of E P with EP0 . Let J 0 (respectively, J 1 ) be the almost complex structure of EP0 (respectively, EP1 ). We have α := J 1 − J 0 ∈ C ∞ U , Ω 0,1 ⊗ ad EP0 .
(5.12)
We note that α0 in (5.11) coincides with the image of the above element α by the injective homomorphism 0 C ∞ U , Ω 0,1 ⊗ ad EP0 −→ C ∞ U , Ω 0,1 ⊗ ad EG 0 defined by the inclusion of ad(EP0 ) in ad(EG ). As before, the nilpotent radical of the Lie algebra p of P will be denoted by Rn (p). Let
EL Rn (p) := EL ×L Rn (p) −→ U
(5.13)
be the vector bundle associated to EL for the adjoint action of L on Rn (p). Both the principal L(P )-bundles EL = EP0 /Ru (P ) and EP1 /Ru (P ) are canonically identified with EL(P ) := EP /Ru (P ). Both these identifications are holomorphic. Therefore, we conclude that α (in (5.12)) is in the image of C ∞ (U , Ω 0,1 ⊗ EL (Rn (p))) for the homomorphism C ∞ U , Ω 0,1 ⊗ EL Rn (p) → C ∞ U , Ω 0,1 ⊗ ad EP0
(5.14)
defined by the inclusion of EL (Rn (p)) in ad(EP0 ) (the inclusion EL (Rn (p)) in ad(EP0 ) is given by the inclusion of Rn (p) in p). Let α ∈ C ∞ U , Ω 0,1 ⊗ EL Rn (p)
(5.15)
be the element whose image by the homomorphism in (5.14) is α. t From the construction of the holomorphic structure on EG it follows that αt = Ad exp −ρ(t) ( α), where αt is constructed in (5.11). Now from (5.9) it follows that αt → 0 in the C ∞ Fréchet topology over any relatively compact open subset of U , as t → ∞. Therefore, we conclude that as t → ∞, the holomorphic t converges, in the C ∞ Fréchet topology over any relatively compact open subset of U , to structure of EG 0 . the holomorphic structure of EG To prove that θ t converges to θG , we note that βt := θ t − θG ∈ C ∞ U , Ω 1,0 ⊗ EL Rn (p) .
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This is because the Higgs fields on EL = EP0 /Ru (P ) and EP1 /Ru (P ) coincide with the Higgs field on EL(P ) using the holomorphic identifications of EL and EP1 /Ru (P ) with EL(P ) . In fact, βt = Ad(exp(−ρ(t)))(β0 ). Therefore, from (5.9) it follows that βt → 0 in the C ∞ Fréchet topology over any relatively compact open subset of U , as t → ∞. In other words, as t → ∞, the Higgs field θ t converges, in the C ∞ Fréchet topology over any relatively compact open subset of U , to the Higgs field θG . 2 References [1] B. Anchouche, H. Azad, I. Biswas, Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323 (2002) 693–712. [2] S. Bando, Y.-T. Siu, Stable sheaves and Einstein–Hermitian metrics, in: T. Mabuchi, et al. (Eds.), Geometry and Analysis on Complex Manifolds. Festschrift for Professor S. Kobayashi’s 60th Birthday, World Sci. Publishing, River Edge, NJ, 1994, pp. 39–50. [3] I. Biswas, A. Jacob, M. Stemmler, Existence of approximate Hermitian–Einstein structures on semistable principal bundles, Bull. Sci. Math. (2012), http://dx.doi.org/10.1016/j.bulsci.2012.02.005. [4] I. Biswas, G. Schumacher, Yang–Mills equation for stable Higgs sheaves, Int. J. Math. 20 (2009) 541–556. [5] A. Borel, Linear Algebraic Groups, second edition, Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. [6] S.B. Bradlow, Hermitian–Einstein inequalities and Harder–Narasimhan filtrations, Int. J. Math. 6 (1995) 645–656. [7] G.D. Daskalopoulos, R.A. Wentworth, Convergence properties of the Yang–Mills flow on Kähler surfaces, J. Reine Angew. Math. 575 (2004) 69–99. [8] S.K. Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3) 50 (1985) 1–26. [9] S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231–247. [10] M.-C. Hong, G. Tian, Asymptotical behaviour of the Yang–Mills flow and singular Yang–Mills connections, Math. Ann. 330 (2004) 441–472. [11] J.E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Springer-Verlag, New York, Heidelberg, Berlin, 1987. [12] J. Li, X. Zhang, The gradient flow of Higgs pairs, J. Eur. Math. Soc. 13 (2011) 1373–1422. [13] A. Jacob, Existence of approximate Hermitian–Einstein structures on semi-stable bundles, arXiv:1012.1888v2 [math.DG], 2011; Asian J. Math., submitted for publication. [14] A. Jacob, The Yang–Mills flow and the Atiyah–Bott formula on compact Kähler manifolds, arXiv:1109.1550v2 [math.DG], 2012. [15] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, vol. 15. Kanô Memorial Lectures 5, Princeton University Press, Princeton, NJ, 1987. [16] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer-Verlag, Berlin, 1994. [17] B. Sibley, Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit, preprint, arXiv:1206.5491 [math.DG]. [18] C.T. Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Am. Math. Soc. 1 (1988) 867–918. [19] K. Uhlenbeck, S.-T. Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) 257–293. [20] I. Biswas, S.B. Bradlow, A. Jacob, M. Stemmler, Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface, Ann. Global Anal. Geom. 44 (2013) 257–268.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Automorphisms and connections on Higgs bundles over compact Kähler manifolds Indranil Biswas a,∗ , Steven B. Bradlow b , Adam Jacob c , Matthias Stemmler d a School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India b Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street (MC-382), Urbana, IL 61801, USA c Department of Mathematics, Harvard University, Cambridge, MA 02138, USA d Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße, Lahnberge, 35032 Marburg, Germany
a r t i c l e
i n f o
Article history: Received 17 September 2012 Available online 10 December 2013 Communicated by S. Merkulov MSC: 53C07 32L05 Keywords: Higgs bundle Automorphism Approximate Yang–Mills–Higgs connection
a b s t r a c t Let (E, ϕ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of E by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by G the direct sum of these stable quotients, and let the singular set of G be called S ⊂ X. We construct a 1-parameter family of filtration preserving C ∞ isomorphisms Φt : G|X\S → E|X\S , and a Hermitian metric h on G|X\S , such that as t → +∞, the Chern connection for the Hermitian Higgs bundle (Φ∗t E, Φ∗t ϕ, h)|X\S converges, in the C ∞ Fréchet topology over any relatively compact open subset of X\S, to the direct sum of the Yang–Mills–Higgs connections on the direct summands in G. We also prove an analogous result for principal Higgs G-bundles on X. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Constructing connections with optimal curvature properties has long been a fundamental problem within differential geometry. In the case of an indecomposable holomorphic vector bundle, one natural connection is a Hermitian–Einstein connection, defined by having curvature a constant multiple of the identity after tracing away the 2-form part. By the famous theorem of Donaldson, Uhlenbeck and Yau, such connections exist if and only if the holomorphic vector bundle is stable [8,19,9]. In the unstable case, rather than trying to find a specific canonical connection, one can instead look for a sequence of connections that approach a canonical connection on a related bundle (the holomorphic structure of the bundle may change in the limit). Kobayashi introduced the notion of an approximate Hermitian–Einstein structure in [15], and defined it to be a sequence of connections which approach in C 0 * Corresponding author. E-mail addresses: [email protected] (I. Biswas), [email protected] (S.B. Bradlow), [email protected] (A. Jacob), [email protected] (M. Stemmler). 0926-2245/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.difgeo.2013.11.008
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topology a Hermitian–Einstein connection. Kobayashi then proved the existence of such a structure to be equivalent to semistability of the bundle in the projective case. This result was extended to Kähler manifolds in [13] and to principal bundles in [3]. When the bundle is not semistable, the limiting connection cannot be a Hermitian–Einstein connection, yet there is a similar construction. In this case one looks for a sequence of connections for which the curvatures converge to an endomorphism with locally constant eigenvalues given by the slopes of the successive quotients of the Harder–Narasimhan filtration of the bundle. We call this sequence of connections an approximate Hermitian structure. The existence of these structures has been useful in the study of many geometric problems, most notably in [7]. In their paper Daskalopoulos and Wentworth used the existence of Lp approximate Hermitian structures on holomorphic bundles over Kähler surfaces to determine the limiting properties of the Yang–Mills flow. Their convergence results were extended to higher dimensions in [14,17]. We also note that Li and Zhang extended the result of Daskalopoulos and Wentworth to the context of Higgs bundles [12]. Hong and Tian proved that there is a subsequence of the Yang–Mills flow which converges smoothly on the complement of the singular set [10]. Despite these results, aside for the Riemann surfaces case it is not known whether the exponent p can be increased to infinity. The existence of an L∞ structure would give a full generalization of the equivalence between semistable bundles and approximate Hermitian–Einstein structures. In [6], the second author showed that the vector bundles on a Riemann surface admit C 0 approximate Hermitian–Einstein structures. This has been generalized to principal bundles over Riemann surfaces in [20]. The key difficulty in working with these structures is that for higher dimensions the Harder–Narasimhan filtration is given by subsheaves, not subbundles. Thus the limiting endomorphism in the approximate Hermitian structure is only defined away from an analytic subvariety of complex codimension at least two. In this paper, we show that by working away from this subvariety, we can produce a C ∞ approximate Hermitian structure which converges to a connection on the successive quotients of the Harder–Narasimhan filtration. Specifically, let (X, ω) be a compact connected Kähler manifold, and let (E, ϕ) be a Higgs vector bundle over X. Fix a filtration of E by coherent analytic subsheaves 0 = F0 F1 · · · F−1 F = E such that for each i ∈ [1, ], • the quotient Fi /Fi−1 is torsionfree, 1 • ϕ(Fi ) ⊂ ΩX ⊗ Fi , and • the Higgs sheaf (Fi /Fi−1 , ϕi ) is stable, where ϕi is the Higgs field on Fi /Fi−1 induced by ϕ. We note that such filtrations exist (see Proposition 2.6 below). Let S ⊂ X be the singular set for the graded object
G :=
Fi /Fi−1
i=1
associated to this filtration. This S is a closed complex analytic subset of X of complex codimension at least two. Denote by Fi (respectively, G) the C ∞ vector bundle underlying the holomorphic vector bundle Fi |X\S (respectively, G|X\S ). Write E = (E, ∂), where E is the underlying C ∞ vector bundle of E, and ∂ is the Dolbeault operator defining the holomorphic structure of E. Over X \ S, the two C ∞ vector bundles G and E are isomorphic. Take any filtration preserving C ∞ isomorphism over X \ S
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f : G −→ E that induces the identity map on G. Let ∂ i be the Dolbeault operator on Fi /Fi−1 induced by ∂, and let ϕi be the Higgs field on (Fi /Fi−1 , ∂ i ) induced by ϕ. Since each of the Higgs sheaves (Fi /Fi−1 , ϕi ) is stable, by [4, Theorem 3.1] (reproduced Theorem 2.5), there exists an admissible Yang–Mills–Higgs metric hi on (Fi /Fi−1 , ∂ i ) for each i ∈ [1, ]. Let D be the Chern connection on G defined by the holomorphic structure the Hermitian metric h := i=1 hi . We prove the following (see Theorem 3.1):
(1.1) i=1
∂ i , the Higgs field
i=1
ϕi and
Theorem 1.1. There is a smooth 1-parameter family {Φt }t∈R of C ∞ isomorphisms from G to E inducing the identity map on G, such that if Dϕ,t is the Chern connection of (G, Φ∗t ∂, Φ∗t ϕ, h), then as t → +∞, the connections Dϕ,t converge in the C ∞ Fréchet topology on every relatively compact open subset of X \ S to (see (1.1)). In particular, as t → +∞, the curvature forms of Dϕ,t converge in the the Chern connection D ∞ C Fréchet topology on every relatively compact open subset of X \ S to the curvature form of D. Note that if X is a Riemann surface, then the singular set S is empty. Consequently, in this case Theorem 1.1 gives convergence in the C ∞ Fréchet topology over the entire X. In Section 5, we address the question of the existence of a good Hermitian structure in the more general situation of principal Higgs bundles over a compact connected Kähler manifold. We prove for principal Higgs bundles a result similar to Theorem 1.1 (see Theorem 5.1). 2. Preliminaries Let (X, ω) be a compact connected Kähler manifold. The degree of a torsionfree coherent analytic sheaf E on X with respect to ω will be denoted by degree(E). If rank(E) > 0, the slope μ(E) of E is defined to be μ(E) :=
degree(E) . rank(E)
Definition 2.1. Let E be a torsionfree coherent analytic sheaf on X. 1 2 (i) A Higgs field on E is defined to be a section of ΩX ⊗ End(E) such that the section ϕ ∧ ϕ of ΩX ⊗ End(E) vanishes identically. (ii) If ϕ is a Higgs field on E, then (E, ϕ) is called a Higgs sheaf. If E is locally free, then (E, ϕ) is called a Higgs vector bundle.
Definition 2.2. Let (E, ϕ) be a Higgs sheaf on X. (i) (E, ϕ) is called stable (respectively, semistable) if for every coherent analytic subsheaf F ⊂ E with 0 < rank(F) < rank(E) such that • the quotient E/F is torsionfree, and 1 ⊗ F, • ϕ(F) ⊂ ΩX we have μ(F) < μ(E)
respectively, μ(F) μ(E) .
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(ii) (E, ϕ) is called polystable if
(E, ϕ) =
N
E i , ϕi ,
i=1
where each (E i , ϕi ) is a stable Higgs sheaf of slope μ(E i ) = μ(E). Therefore, a polystable Higgs sheaf is semistable. Now, let (E, ϕ) be a Higgs vector bundle over X, and let h be a Hermitian metric on E. Denote by D the unitary Chern connection of E with respect to the metric h. To the Hermitian Higgs bundle (E, ϕ, h) one can associate a natural connection Dϕ := D + ϕ + ϕ∗ ,
(2.1)
where ϕ∗ denotes the adjoint of ϕ with respect to the Hermitian metric h. (See [18, Section 3] for details.) This connection Dϕ will be called the Chern connection of (E, ϕ, h). Denote by F (Dϕ ) the curvature form of the connection Dϕ . Definition 2.3. The Hermitian metric h is called a Yang–Mills–Higgs metric on (E, ϕ) if √
−1 · ΛF Dϕ = λ · IdE ,
where Λ denotes the adjoint of wedging with the Kähler form ω and λ is a constant real number. If h is a Yang–Mills–Higgs metric on (E, ϕ), then the connection Dϕ is called a Yang–Mills–Higgs connection. Note that √
√ −1 · ΛF Dϕ = −1Λ F (D) + ϕ, ϕ∗ ,
where F (D) is the curvature of the Chern connection D, and [·, ·] is defined using the exterior product on differential forms and the Lie algebra structure of the fibers of End(E). We recall the definition of Yang–Mills–Higgs metrics on (singular) Higgs sheaves. Let (E, ϕ) be a Higgs sheaf on X, and let S ⊂ X be the singular set for E, meaning the closed complex analytic subset of X outside which E is locally free. Since E is torsionfree, the complex codimension of S is at least two. Definition 2.4. A Hermitian metric h on the holomorphic vector bundle E|X\S is called admissible if • the curvature F (D) of the Chern connection D of (E|X\S , h) is square-integrable, and • the contracted Chern curvature ΛF (D) is bounded. In [4], the following Donaldson–Uhlenbeck–Yau type correspondence for Higgs sheaves was established. Theorem 2.5. (See [4, Theorem 3.1].) Let (E, ϕ) be a Higgs sheaf on a compact Kähler manifold (X, ω), and let S be the singular set for E. If E is stable, then there exists an admissible Yang–Mills–Higgs metric on E|X\S . Moreover, the admissible Yang–Mills–Higgs connection is unique. The proof of Theorem 2.5 is based on the proof of the Donaldson–Uhlenbeck–Yau type correspondence for reflexive sheaves established by Bando and Siu [2]. We end this section with an existence proof for the type of filtration used in Theorem 1.1.
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Proposition 2.6. Given a Higgs sheaf (E, ϕ) over X, there is a filtration of E by coherent analytic subsheaves 0 = F0 F1 · · · F−1 F = E such that for each i ∈ [1, ], • the quotient Fi /Fi−1 is torsionfree, 1 • ϕ(Fi ) ⊂ ΩX ⊗ Fi , and • the Higgs sheaf (Fi /Fi−1 , ϕi ) is stable, where ϕi is the Higgs field on Fi /Fi−1 induced by ϕ. Proof. The proposition will be proved using induction on the rank of E. If rank(E) = 1, then the statement is obvious. Assume that the proposition is proved for all E with rank(E) r −1. Now assume that rank(E) = r. If the Higgs sheaf (E, ϕ) is stable, then there is nothing to prove. Otherwise, there is a coherent analytic subsheaf F ⊂ E with 0 < rank(F) < rank(E) such that • the quotient E/F is torsionfree, 1 • ϕ(F) ⊂ ΩX ⊗ F, and • μ(F) μ(E). Let ϕF (respectively, ϕE/F ) be the Higgs field on F (respectively, E/F) induced by the Higgs field ϕ. Since rank(F), rank(E/F) r − 1, by the induction hypothesis, both the Higgs sheaves (F, ϕF ) and (E/F, ϕE/F ) have a filtration satisfying the conditions in the proposition. The filtration of E/F given by the induction hypothesis produces a filtration of E using the natural projection E → E/F. Each term of this filtration contains F. Hence this filtration of E, and a filtration of F given by the induction hypothesis, together produce a filtration of E satisfying the conditions in the proposition. 2 3. Flow under a 1-parameter group of automorphisms As before, let (X, ω) be a compact connected Kähler manifold, and let (E, ϕ) be a Higgs vector bundle over X. Fix a filtration 0 = F0 F1 · · · F−1 F = E
(3.1)
satisfying the conditions in Proposition 2.6. Denote by S ⊂ X the singular set for the graded object G := i=1 Fi /Fi−1 associated to the filtration (3.1). Therefore, each Fi /Fi−1 is locally free on X \ S, and S is a closed complex analytic subset. Since the sheaf G is torsionfree, the codimension of S is at least two. Denote by Fi the C ∞ vector bundle underlying the holomorphic vector bundle Fi |X\S . Write E = (E, ∂), where E is the underlying C ∞ vector bundle of E, and ∂ is the Dolbeault operator defining the holomorphic structure of E. The restriction of E to X \ S will also be denoted by E. Fix C ∞ subbundles Vi ⊂ E over X \ S, 0 i , such that (1) the natural homomorphism
f:
i=0
is an isomorphism, and
Vi −→ E
(3.2)
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(2) for every j ∈ [0, ], f
j
Vi
= Fj .
i=0
Note that V0 = 0, and the composition Vi → Fi −→ Fi /Fi−1
(3.3)
is an isomorphism for all i ∈ [1, ]. Let ∂ i be the Dolbeault operator on Vi induced by ∂ using the isomorphism of Vi with Fi /Fi−1 in (3.3) (note that Fi /Fi−1 is a coherent analytic sheaf). Let ϕi be the Higgs field on the holomorphic vector bundle (Vi , ∂ i ) induced by ϕ using the isomorphism in (3.3). Since each of the Higgs sheaves Fi /Fi−1 on X equipped with the Higgs field induced by ϕ is stable, by Theorem 2.5 there exists an admissible Yang–Mills–Higgs metric hi on the holomorphic vector bundle (Fi /Fi−1 )|X\S . Denote by hi the Hermitian metric on Vi induced by hi using the isomorphism in (3.3). We note that hi is an admissible Yang–Mills–Higgs metric on the holomorphic Higgs bundle (Vi , ∂ i , ϕi ). Let D
(3.4)
be the Chern connection of the Hermitian Higgs bundle i=1 (Vi , ∂ i , ϕi , hi ). Let f ∗ ∂ := f −1 ◦ ∂ ◦ f be the Dolbeault operator on i=1 Vi given by ∂ using the isomorphism f in (3.2). Let f ∗ ϕ := f −1 ◦ ϕ ◦ f be the Higgs field on ( i=1 Vi , f ∗ ∂) given by ϕ using the isomorphism f . Let h := i=1 hi be the Hermitian metric on i=1 Vi , where hi is the above Yang–Mills–Higgs metric on (Vi , ∂ i , ϕi ). For every t ∈ R, define the C ∞ isomorphism Φt : i=1 Vi → E as follows: Φt := f ◦ At , where At is the C ∞ automorphism of
i=1
(3.5)
Vi given by
At :=
exp(−jt) · IdVj .
(3.6)
j=1
Define Φ∗t ∂ := Φ−1 t ◦ ∂ ◦ Φt
and Φ∗t ϕ := Φ−1 t ◦ ϕ ◦ Φt .
In particular, the Higgs bundle ( i=1 Vi , Φ∗t ∂, Φ∗t ϕ) is isomorphic to the Higgs bundle ( i=1 Vi , f ∗ ∂, f ∗ ϕ), and hence it is isomorphic to (E, ∂, ϕ), for all t. Keep the Hermitian metric h on i=1 Vi fixed, meaning it is not changed as t changes. Let Dϕ,t
(3.7)
be the Chern connection of the Hermitian Higgs bundle ( i=1 Vi , Φ∗t ∂, Φ∗t ϕ, h) (see (2.1)). Theorem 3.1. As t → +∞, the Chern connection Dϕ,t in (3.7) converges in the C ∞ Fréchet topology on in (3.4). In particular, as t → +∞, every relatively compact open subset of X \ S to the Chern connection D
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the curvature form of Dϕ,t converges in the C ∞ Fréchet topology on every relatively compact open subset of X \ S to the curvature form of D. Proof. For notational convenience, we only give a proof for the case where = 2. The proof for filtrations of arbitrary length works analogously. Since the Hermitian metric h on V1 ⊕ V2 is kept fixed, to prove the theorem it suffices to show that as t → +∞, (1) the Dolbeault operator Φ∗t ∂ converges to ∂ 1 ⊕∂ 2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S, and (2) the Higgs field Φ∗t ϕ converges to ϕ1 ⊕ ϕ2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. In terms of the direct sum decomposition V1 ⊕ V2 , we have f ∗∂ =
∂1 0
α ∂2
,
where α is a smooth (0, 1)-form on X \ S with values in Hom(V2 , V1 ). From the definitions of Φt and At (see (3.5) and (3.6)), we conclude that Φ∗t ∂
=
A−1 t
∗
◦ f ∂ ◦ At =
exp(−t) · α ∂2
∂1 0
.
This implies that as t → +∞, the Dolbeault operator Φ∗t ∂ converges to ∂ 1 ⊕ ∂ 2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. Analogously, we have f ∗ϕ =
β ϕ2
ϕ1 0
,
where β is a smooth (1, 0)-form on X \ S with values in Hom(V2 , V1 ). It follows that ∗ Φ∗t ϕ = A−1 t ◦ f ϕ ◦ At =
ϕ1 0
exp(−t) · β ϕ2
.
Therefore, as t → +∞, the Higgs field Φ∗t ϕ converges to ϕ1 ⊕ ϕ2 in the C ∞ Fréchet topology on every relatively compact open subset of X \ S. This completes the proof. 2 4. Principal Higgs G-sheaves Let G be a connected reductive affine algebraic group defined over C. The Lie algebra of G will be denoted by g. A parabolic subgroup of G is a Zariski closed connected subgroup P ⊂ G such that the quotient variety G/P is complete. A large open subset of X is a dense open subset U ⊂ X such that the complement X \ U is a complex analytic subset of X of complex codimension at least two. The inclusion map U → X will be denoted by ιU . For a holomorphic principal G-bundle EG , the holomorphic vector bundle EG ×G g, the one associated to EG for the adjoint action of G on g, will be denoted by ad(EG ). This ad(EG ) is known as the adjoint vector bundle for EG . We note that the fibers of ad(EG ) are Lie algebras identified with g up to a conjugation.
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A principal G-sheaf on X is a pair (U, EG ), where U is large open subset of X and EG is a holomorphic principal G-bundle over U , such that the direct image ιU ∗ ad(EG ) is a coherent analytic sheaf on X. A principal Higgs G-sheaf is a triple of the form (U, EG , θ), where (U, EG ) is a principal G-sheaf, and θ ∈ H 0 U, ΩU1 ⊗ ad(EG ) is such that the section θ ∧ θ of ΩU2 ⊗ ad(EG ) vanishes identically. (The product θ ∧ θ is defined using the Lie algebra structure of the fibers of ad(EG ) and the natural homomorphism ΩU1 ⊗ ΩU1 −→ ΩU2 .) A principal Higgs G-sheaf (U, EG , θ) is called stable if for every triple of the form (U , Q, EQ ), where • U ⊂ U is a large open subset of X, • Q ⊂ G is a maximal proper parabolic subgroup, and • EQ ⊂ EG |U
(4.1)
is a holomorphic reduction of structure group of EG |U to Q over U such that θ|U is a section of ΩU1 ⊗ ad(EQ ), the following inequality holds: degree ad(EQ ) < 0, where ad(EQ ) = EQ ×Q Lie(Q) is the adjoint vector bundle defined as before. The above condition that degree(ad(EQ )) < 0 is equivalent to the condition that degree(ad(EG )/ad(EQ )) > 0, because degree(ad(EG )) = 0. Since ad(EQ ) is a subbundle of ad(EG ) over a large open subset, and ad(EG ) extends to X as a coherent analytic sheaf, it follows that ιU ∗ ad(EQ ) is a coherent analytic sheaf. The unipotent radical of a parabolic subgroup P ⊂ G will be denoted by Ru (P ). We recall that Ru (P ) is the unique maximal connected unipotent normal subgroup of P . The quotient P/Ru (P ), which we will denote by L(P ), is a connected reductive complex affine algebraic group. For a principal P -bundle EP defined over U , the quotient EP /Ru (P ) is a principal L(P )-bundle over U . Note that ad(EP /Ru (P )) is a quotient bundle of ad(EP ). Hence any section of ΩU1 ⊗ ad(EP ) produces a section of ΩU1 ⊗ ad(EP /Ru (P )). Lemma 4.1. Take any principal Higgs G-sheaf (U, EG , θ). There is a large open subset U ⊂ X, a parabolic subgroup P ⊂ G, and a holomorphic reduction of structure group EP ⊂ EG |U of EG to P , such that (1) θ ∈ H 0 (U , ΩU1 ⊗ ad(EP )), and is stable, where θ is the Higgs field on the holomorphic principal (2) the Higgs L(P )-sheaf (U , EP /Ru (P ), θ) L(P )-bundle EP /Ru (P ) induced by θ. Proof. If (U, EG , θ) is stable, then the lemma is proved, because G itself is a parabolic subgroup of G. Assume that (U, EG , θ) is not stable. Take a triple (U1 , Q, EQ ), where • U1 ⊂ U is a large open subset of X, • Q ⊂ G is a maximal proper parabolic subgroup, • EQ ⊂ EG |U1 is a holomorphic reduction of structure group of EG |U1 to Q over U1 with the property that θ|U1 is a section of ΩU1 1 ⊗ ad(EQ ), and • degree(ad(EQ )) 0.
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Consider the holomorphic principal L(Q)-bundle EL(Q) := EQ /Ru (Q) on U1 . Let θ1 be the Higgs field on EL(Q) given by θ. If (U1 , EL(Q) , θ1 ) is stable, then we stop. Note that since ad(EQ ) extends to X as a coherent analytic sheaf, and ad(EL(Q) ) is a quotient of ad(EQ ), it follows that ad(EL(Q) ) extends to X as a coherent analytic sheaf. If (U1 , EL(Q) , θ1 ) is not stable, then we replace (U, EG , θ) in the above construction by (U1 , EL(Q) , θ1 ). If the principal Higgs sheaf resulting from the above construction with (U, EG , θ) replaced by (U1 , EL(Q) , θ1 ) is stable, then we stop. Otherwise, we iterate the above construction. Note that at each step of the construction, the semisimple rank of the reductive group drops by one. If H is abelian reductive, then any principal Higgs H-sheaf is stable (there is no proper parabolic subgroup of H). Consequently, the iterative construction stops in at most n steps, where n is the semisimple rank of G. Consider the quotient map q : Q → L(Q). If Q1 is a parabolic subgroup of L(Q), then q −1 (Q1 ) is a parabolic subgroup of G. We note that this follows from the fact that G/q −1 (Q1 ) is a fiber bundle over G/Q with fiber L(Q)/Q1 ; since both G/Q and L(Q)/Q1 are complete varieties, it follows immediately that G/q −1 (Q1 ) is also complete. Note that in this argument we did not use that Q is a maximal proper parabolic subgroup. Let EQ1 ⊂ EL(Q) be a holomorphic reduction of structure group of EL(Q) to the parabolic subgroup Q1 over a large open subset U2 . Let q : EQ −→ EL(Q) = EQ /Ru (Q) be the quotient map. Then q −1 (EQ1 ) ⊂ EG |U2 is a holomorphic reduction of structure group of EG to the parabolic subgroup q −1 (Q1 ) of G. If the above defined Higgs field θ1 on EL(Q) = EQ /Ru (Q) is a holomorphic section of ΩU1 2 ⊗ ad(EQ1 ), then it follows immediately that θ ∈ H 0 U2 , ΩU1 2 ⊗ ad q −1 (EQ1 ) . Therefore, at each step of the construction, we get a parabolic subgroup of G, and a holomorphic reduction of structure group of EG to this parabolic subgroup over a large open subset, such that the Higgs field θ is a Higgs field on the reduction. Hence, after finitely many iterations of the construction, we get a reduction as in the statement of the lemma. 2 Fix a maximal compact subgroup K ⊂ G.
(4.2)
See [4, p. 553] for the definition of a Yang–Mills–Higgs reduction to K of a principal Higgs G-sheaf. The following is proved in [4]: Theorem 4.2. (See [4, p. 554, Theorem 4.6].) Any stable principal Higgs G-sheaf admits a Yang–Mills–Higgs reduction to K. The Yang–Mills–Higgs connection is unique. 5. Approximate Yang–Mills–Higgs connections on principal Higgs G-sheaves Let (U, EG , θ) be a principal Higgs G-sheaf. Take a triple (U , P, EP ) for (U, EG , θ) as in Lemma 4.1. Let
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EL(P ) := EP /Ru (P ) be the corresponding holomorphic principal P/Ru (P )-bundle. Let p : EP −→ EL(P )
(5.1)
be the quotient map. Let L be the Zariski closure of P ∩ K in G, where K is the subgroup in (4.2). We note that L is also the complex analytic closure of P ∩ K. The composition L → P −→ L(P ) = P/Ru (P )
(5.2)
is an isomorphism, where P → P/Ru (P ) is the quotient map. Such a subgroup L is called a Levi factor of P (see [11, p. 184], [5]). We will identify L with L(P ) using the above isomorphism. The principal G-bundle EG admits a C ∞ reduction of structure group to the subgroup K in (4.2). To see this, note that G/K is contractible. Hence the fiber bundle EG /K → U with fiber G/K admits a C ∞ section. The inverse image, for the quotient map EG → EG /K, of the image of any C ∞ section X → EG /K is a C ∞ reduction of structure group of EG to K. Fix a C ∞ reduction of structure group of EG 0 EK ⊂ EG
to K. 0 We note that EK ∩ EP is a C ∞ reduction of structure group of EG to P ∩ K over U . Let EL be the 0 ∩ EP . This means that for each point smallest fiberwise complex analytic subset of EG |U containing EK 0 x ∈ U , the fiber (EL )x is the smallest complex analytic subset (EP )x containing (EK )x ∩ (EP )x . We note that EL need not be a complex analytic subset of EP . It is easy to see that EL is the principal L-bundle 0 0 (EK ∩ EP ) ×P ∩K L obtained by extending the structure group of the principal P ∩ K-bundle EK ∩ EP 0 P ∩K using the inclusion of P ∩ K in L. We note that (EK ∩ EP ) × L is contained in EG using the map 0 0 (EK ∩ EP ) × L → EG defined by (z, g) → zg. (The total space (EK ∩ EP ) ×P ∩K L is the quotient of 0 (EK ∩ EP ) × L where two points (z1 , g1 ) and (z2 , g2 ) are identified if there is an element g0 ∈ P ∩ K such that z2 = z1 g0 and g2 = g0−1 g1 .) Therefore, EL is a C ∞ reduction of structure group of EG to L over U . 0 That EL coincides with (EK ∩ EP ) ×P ∩K L follows from the fact that L is the complex analytic closure of P ∩ K ⊂ P. The composition p
EL → EP −→ EP /Ru (P ) = EL(P ) |U is a C ∞ isomorphism (the map p is defined in (5.1)), and it commutes with the actions of L = L(P ). In other words, this composition identifies the principal L(P )-bundle EL(P ) |U with the C ∞ principal L-bundle EL using the isomorphism of L(P ) with L in (5.2). Using this isomorphism of EL(P ) |U with EL , the C ∞ principal L-bundle EL gets a holomorphic structure from the holomorphic structure of EL(P ) |U . The holomorphic principal L-bundle obtained this way will be denoted by EL . 0 Let EG be the holomorphic principal G-bundle over U obtained by extending the structure group of the holomorphic principal L-bundle EL using the inclusion of L in G. We note that there is a natural C ∞ isomorphism of principal G-bundles 0 γ : EG −→ EG |U
(5.3)
0 that sends the equivalence class of any (z, g) ∈ EL × G to zg ∈ EG (recall that EG is the quotient of EL × G where two points (z, g) and (z , g ) are identified if there is an element g0 ∈ L such that z = zg0
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0 and g = g0−1 g). It should be emphasized that this isomorphism γ between EG and EG |U need not be holomorphic. Let θ be the Higgs field on the holomorphic principal L(P )-bundle EL(P ) given by θ. Let θL be the 0 is the Higgs field on EL given by θ using the identification of EL(P ) |U with EL constructed above. Since EG extension of structure group of EL for the inclusion of L in G, we have
0 . ad(EL ) ⊂ ad EG 0 So we have H 0 (U , ΩU1 ⊗ ad(EL )) ⊂ H 0 (U , ΩU1 ⊗ ad(EG )). Let
0 θG ∈ H 0 U , ΩU1 ⊗ ad EG
(5.4)
0 be the Higgs field on EG given by θL using this inclusion. The group K ∩ L = K ∩ P will be denoted by KL . This KL is a maximal compact subgroup of the Levi factor L. is stable, and (EL , θL ) is isomorphic to it, we conclude from Theorem 4.2 that EL has a Since (EL(P ) , θ) Yang–Mills–Higgs reduction
EKL ⊂ EL = EL
(5.5)
for the Higgs field θL . Let EK := EKL ×KL K be the C ∞ principal K-bundle obtained by extending the structure group of EKL using the inclusion of KL in K. We note that 0 EK ⊂ EL ×L G = EG .
(5.6)
The above inclusion map is given by the inclusion map ι : EKL × K → EKL × G; the total spaces of the 0 are quotients of EKL × K and EKL × G respectively, and ι descends to the principal bundles EK and EG map in (5.6). 1 0 be the holomorphic principal G-bundle defined by the C ∞ principal G-bundle underlying EG Let EG equipped with the holomorphic structure given by the holomorphic structure of EG using the isomorphism 1 defined by θ using γ. So we have a Higgs G-sheaf γ in (5.3). Let θ1 be the Higgs field on EG
1 U , EG , θ1
(5.7)
0 1 such that the underlying C ∞ principal G-bundle on U is that for EG . Note that (U , EG , θ1 ) is holomorphically isomorphic to (U , EG , θ). Let ZL be the connected component of the center of L containing the identity element. Fix a homomorphism
ρ : C∗ −→ ZL
(5.8)
such that for any nontrivial character χ : P −→ C∗ trivial on Z0 (G) which is a nonnegative integral combination of simple roots, the composition homomorphism
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χ ◦ ρ : C∗ −→ C∗ is of the form c → cn with n positive. We note that it is enough the check the above condition for those characters χ of P that are simple roots and trivial on Z0 (G). Let Rn (p) be the Lie algebra of the unipotent radical Ru (P ). So Rn (p) is the nilpotent radical of the Lie algebra p of P . From the defining property of ρ it follows that lim
t∈R;t→−∞
Ad exp ρ(t) (v) = 0 for all v ∈ Rn (p)
(5.9)
(see [1, p. 708, Theorem 5]). Given a 1-parameter subgroup of G, there is a parabolic subgroup of G natural to it (see [16, p. 55, Definition 2.3/Proposition 2.6]). The parabolic subgroup of G associated to the above 1-parameter subgroup ρ is P itself. t of EL defined by Take any t ∈ C. Since ρ(exp(−t)) ∈ ZL , we get a C ∞ automorphism A t (z) = zρ exp(−t) , A
z ∈ EL .
t is a holomorphic automorphism of the Higgs L-sheaf (EL , θL ). Let We note that A 1 0 1 0 At : EG = EG −→ EG = EG 1 0 t . In other words, the automorphism At is the descent of be the C ∞ automorphism of EG = EG given by A 0 t is the automorphism At × IdG of EL × G. Note that At is a holomorphic automorphism of EG (because A 1 a holomorphic automorphism of EL ), but it is not necessarily a holomorphic automorphism of EG . For any t ∈ C, let 0 Φt := γ ◦ At : EG −→ EG |U
(5.10)
t be the C ∞ isomorphism of principal G-bundles, where γ is constructed in (5.3). Let EG be the holomorphic 0 principal G-bundle whose underlying C ∞ principal G-bundle is EG , and the holomorphic structure is the t one given by the holomorphic structure of EG using Φt in (5.10). Let θ t be the Higgs field on EG given by 0 θ using Φt . So the Higgs G-sheaf (U , EG , θ0 ) coincides with the one in (5.7). We note that (U , EG , θ) is t holomorphically isomorphic to (U , EG , θt ) for all t. 0 Let ∇0 be the connection on EG for the Higgs field θG (defined in (5.4)) and the reduction EK in (5.6). 0 We note that ∇0 coincides with the connection on EG induced by the Yang–Mills–Higgs connection on EL 0 for the Higgs field θL (this connection is constructed using θL and EKL in (5.5)). Note that since EG is an 0 extension of structure group of EL a connection on EL induces a connection on EG . 0 t For any t ∈ R, let ∇t be the connection on EG for the holomorphic structure EG , the Higgs field θ t (both defined above) and the reduction EK in (5.6).
Theorem 5.1. As t → ∞, the connection ∇t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the connection ∇0 . In particular, as t → ∞, the curvature of ∇t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the curvature of ∇0 . Proof. Since the reduction EK does not change with t, it suffices to prove the following two statements: t converges, in the C ∞ Fréchet topology over relatively (1) as t → ∞, the holomorphic structure of EG 0 compact open subsets of U , to the holomorphic structure of EG , and
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(2) as t → ∞, the Higgs field θ t converges, in the C ∞ Fréchet topology over relatively compact open subsets of U , to the Higgs field θG (defined in (5.4)). 0 t Let J0 (respectively, J t ) be the almost complex structure of EG (respectively, EG ). We have
0 αt := J t − J0 ∈ C ∞ U , Ω 0,1 ⊗ ad EG .
(5.11)
Let EP0 = EL ×L P be the holomorphic principal P -bundle over U obtained by extending the structure group of the holomorphic principal L-bundle EL using the inclusion of L in P . There is a canonical C ∞ isomorphism between the principal P -bundles EP0 and EP ; this isomorphism is the descent of the map EL × P → EP defined by (z, g) → zg. Let EP1 be the holomorphic principal P -bundle whose underlying C ∞ principal P -bundle is that of EP0 , and the holomorphic structure of EP1 is given by the holomorphic structure of EP using the above isomorphism of E P with EP0 . Let J 0 (respectively, J 1 ) be the almost complex structure of EP0 (respectively, EP1 ). We have α := J 1 − J 0 ∈ C ∞ U , Ω 0,1 ⊗ ad EP0 .
(5.12)
We note that α0 in (5.11) coincides with the image of the above element α by the injective homomorphism 0 C ∞ U , Ω 0,1 ⊗ ad EP0 −→ C ∞ U , Ω 0,1 ⊗ ad EG 0 defined by the inclusion of ad(EP0 ) in ad(EG ). As before, the nilpotent radical of the Lie algebra p of P will be denoted by Rn (p). Let
EL Rn (p) := EL ×L Rn (p) −→ U
(5.13)
be the vector bundle associated to EL for the adjoint action of L on Rn (p). Both the principal L(P )-bundles EL = EP0 /Ru (P ) and EP1 /Ru (P ) are canonically identified with EL(P ) := EP /Ru (P ). Both these identifications are holomorphic. Therefore, we conclude that α (in (5.12)) is in the image of C ∞ (U , Ω 0,1 ⊗ EL (Rn (p))) for the homomorphism C ∞ U , Ω 0,1 ⊗ EL Rn (p) → C ∞ U , Ω 0,1 ⊗ ad EP0
(5.14)
defined by the inclusion of EL (Rn (p)) in ad(EP0 ) (the inclusion EL (Rn (p)) in ad(EP0 ) is given by the inclusion of Rn (p) in p). Let α ∈ C ∞ U , Ω 0,1 ⊗ EL Rn (p)
(5.15)
be the element whose image by the homomorphism in (5.14) is α. t From the construction of the holomorphic structure on EG it follows that αt = Ad exp −ρ(t) ( α), where αt is constructed in (5.11). Now from (5.9) it follows that αt → 0 in the C ∞ Fréchet topology over any relatively compact open subset of U , as t → ∞. Therefore, we conclude that as t → ∞, the holomorphic t converges, in the C ∞ Fréchet topology over any relatively compact open subset of U , to structure of EG 0 . the holomorphic structure of EG To prove that θ t converges to θG , we note that βt := θ t − θG ∈ C ∞ U , Ω 1,0 ⊗ EL Rn (p) .
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This is because the Higgs fields on EL = EP0 /Ru (P ) and EP1 /Ru (P ) coincide with the Higgs field on EL(P ) using the holomorphic identifications of EL and EP1 /Ru (P ) with EL(P ) . In fact, βt = Ad(exp(−ρ(t)))(β0 ). Therefore, from (5.9) it follows that βt → 0 in the C ∞ Fréchet topology over any relatively compact open subset of U , as t → ∞. In other words, as t → ∞, the Higgs field θ t converges, in the C ∞ Fréchet topology over any relatively compact open subset of U , to the Higgs field θG . 2 References [1] B. Anchouche, H. Azad, I. Biswas, Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323 (2002) 693–712. [2] S. Bando, Y.-T. Siu, Stable sheaves and Einstein–Hermitian metrics, in: T. Mabuchi, et al. (Eds.), Geometry and Analysis on Complex Manifolds. Festschrift for Professor S. Kobayashi’s 60th Birthday, World Sci. Publishing, River Edge, NJ, 1994, pp. 39–50. [3] I. Biswas, A. Jacob, M. Stemmler, Existence of approximate Hermitian–Einstein structures on semistable principal bundles, Bull. Sci. Math. (2012), http://dx.doi.org/10.1016/j.bulsci.2012.02.005. [4] I. Biswas, G. Schumacher, Yang–Mills equation for stable Higgs sheaves, Int. J. Math. 20 (2009) 541–556. [5] A. Borel, Linear Algebraic Groups, second edition, Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. [6] S.B. Bradlow, Hermitian–Einstein inequalities and Harder–Narasimhan filtrations, Int. J. Math. 6 (1995) 645–656. [7] G.D. Daskalopoulos, R.A. Wentworth, Convergence properties of the Yang–Mills flow on Kähler surfaces, J. Reine Angew. Math. 575 (2004) 69–99. [8] S.K. Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3) 50 (1985) 1–26. [9] S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231–247. [10] M.-C. Hong, G. Tian, Asymptotical behaviour of the Yang–Mills flow and singular Yang–Mills connections, Math. Ann. 330 (2004) 441–472. [11] J.E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Springer-Verlag, New York, Heidelberg, Berlin, 1987. [12] J. Li, X. Zhang, The gradient flow of Higgs pairs, J. Eur. Math. Soc. 13 (2011) 1373–1422. [13] A. Jacob, Existence of approximate Hermitian–Einstein structures on semi-stable bundles, arXiv:1012.1888v2 [math.DG], 2011; Asian J. Math., submitted for publication. [14] A. Jacob, The Yang–Mills flow and the Atiyah–Bott formula on compact Kähler manifolds, arXiv:1109.1550v2 [math.DG], 2012. [15] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, vol. 15. Kanô Memorial Lectures 5, Princeton University Press, Princeton, NJ, 1987. [16] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer-Verlag, Berlin, 1994. [17] B. Sibley, Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit, preprint, arXiv:1206.5491 [math.DG]. [18] C.T. Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Am. Math. Soc. 1 (1988) 867–918. [19] K. Uhlenbeck, S.-T. Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) 257–293. [20] I. Biswas, S.B. Bradlow, A. Jacob, M. Stemmler, Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface, Ann. Global Anal. Geom. 44 (2013) 257–268.