HIGGS BUNDLES, INTEGRABILITY, AND HOLOMORPHIC FORMS

DONU ARAPURA

In this article, I want to consider how conditions on the fundamental group lead to the existence of holomorphic sections of the cotangent bundle and its tensor powers. Perhaps the earliest instance of this sort of thing is the the fundamental equality b_{1}(X) = 2q(X), where b_{1}(X) is the first Betti number and q(X) = dim H^{0}(X, Ω_{X}^{1}) is the irregularity. This implies that X has plenty of holomorphic 1-forms whenever π_{1}(X) has lots of rank one representations. A more recent example is Kollár’s work [K1, K2] on the plurigenera of varieties with “large” fundamental group. In fact, the present work grew out of my attempt to gain a better understanding of Kollár’s.

As I have indicated, the existence of nonzero 1-forms follows directly from topology. Nonzero forms of higher degree can be produced by taking wedge products provided that the dimension of the image of the Albanese map is sufficiently large. The main idea here is to play the same game with higher rank representations. In this case, the 1-forms get replaced with (locally) matrix valued forms, i.e. Higgs fields in the sense of Hitchin and Simpson [H2, S1]. Forms in the usual sense can then be produced by taking traces of products.

The analogue of b_{1}(X) = 2q(X) in the context of Higgs bundles is an interesting story in its own right. It will be deduced from an integrability result which had been establish earlier in the preprint [A2]. Since the “pre” is likely to remain in perpetuity, I am including the relevant details here.

1. Consequences of the abelian theory

The equation b_{1}(X) = 2q(X), which is a direct consequence of the Hodge decomposition on H^{1}(X, ℂ) = Hom(π_{1}(X), ℂ), implies that X carries nontrivial holomorphic 1-forms if and only if b_{1}(X) ≠ 0. Sections of Ω_{X}^{p} can in principle be obtained by taking wedge products. A simple criterion for this procedure to yield nonzero forms can be given in terms of the the Albanese a : X → Alb(X).

Lemma 1.1. The natural map

∧^{p}H^{0}(Ω_{X}^{1}) → H^{0}(Ω_{X}^{p})

is nonzero if and only if dim a(X) ≥ p.

Proof. This follows immediately from the observation that the image of ∧^{p}H^{0}(Ω_{X}^{1}) is just a^{∗}H^{0}(Ω_{Alb(X)}^{p}).

The dimension of a(X) has an infinitesimal interpretation, namely as the codimension of ker(da_{x}) for a general point x ∈ X. After making this replacement, the above lemma admits a generalization to the nonabelian setting. To prepare the ground, observe that a tangent vector v ∈ T_{x} lies in ker(da_{x}) if and only if it is annihilated by pullbacks of all the cotangent vectors in T_{a(x)}^{∗}. As the cotangent bundle of Alb(X) is trivial, any element of a^{∗}T_{a(x)}^{∗} automatically extends to a global 1-form on X. Thus ker(da_{x}) is the subspace of T_{x} annihilated by H^{0}(Ω_{X}^{1}).

Based on talk given at Irvine, June 1998. Support from the conference and NSF is gratefully acknowledged.

2. Higgs bundles and all that

Deeper information about π_{1}(X), can be obtained by considering representations into nonabelian groups. The set of conjugacy classes of semisimple representations of π_{1}(X) into GL_{n}(ℂ), which will be denoted by M_{B}(X, n), has the structure of an affine algebraic scheme of finite type over spec ℂ (or more accurately, it is the set of ℂ-rational points of a scheme). Let M_{U}(X, n) be the subset of conjugacy classes of unitary representations; this forms a real algebraic subvariety of M_{B}.

Given a representation ρ ∈ M_{U}(X, n), let E_{ρ} be the associated flat vector bundle. This carries a flat metric, and consequently has trivial rational Chern classes. Furthermore the curvature decreasing property implies that E is semistable with respect to any polarization of X. Conversely Donaldson, and Uhlenbeck-Yau have shown that any such semistable bundle arises this way. The extension to nonunitary representations involves the notion of Higgs bundle. Given a vector bundle F, an F -Higgs bundle consists of a second vector bundle E and map θ : E → F ⊗ E such that θ ∧ θ, the composite of

E ⟶^{θ} F ⊗ E ⟶^{id_{F} ⊗θ} F ⊗ F ⊗ E → ∧^{2}F ⊗ E,

vanishes; or equivalently, such that θ ⊗ θ lies in in the image of S^{2}(F) ⊗ End(E) under the product of the natural splitting of F ⊗ F → S^{2}(F) with id_{End(E)}. A Higgs bundle is just an Ω_{X}^{1}-Higgs bundle. The usual (semi)stability condition for vector bundles can be modified for a Higgs bundle (E, θ) by requiring that the normalized degree or slope of every θ-invariant coherent subsheaf be less than (or equal to) that of E. This reduces to the usual notion when θ = 0. A direct sum of stable Higgs bundles with the same slope is called polystable. Building on earlier work of Corlette, Donaldson, Eels-Sampson, Uhlenbeck-Yau and others, Simpson [S1, S2] has established the basic correspondence:

Theorem 2.1. There is an affine scheme of finite type over ℂ, M_{B}(X, n) whose (closed) points parameterize the isomorphism classes of semisimple representation of π_{1}(X) into Gl_{n}(ℂ). There is a quasiprojective scheme M_{Dol}(X, n), whose points parameterize polystable rank n Higgs bundles with vanishing first and second rational Chern classes. There are open subsets M_{Dol}^{s}(X, n) ⊆ M_{Dol}(X, n) and M_{B}^{irr}(X, n) ⊆ M_{B}(X, n) which parametrize stable bundles and irreducible representations respectively. The associated analytic spaces M_{Dol}(X, n)^{an} and M_{B}(X, n)^{an} are homeomorphic, and M_{Dol}^{s}(X, n) and M_{B}^{irr}(X, n) correspond under this homeomorphism. Under this correspondence, M_{U}(X, n) is identified with the subset M_{V} (X, n) of Higgs bundles with zero Higgs field.

Note that M_{V} (X, n) can be identified with moduli space of semistable vector bundle with c_{1} = c_{2} = 0. M_{B} and M_{Dol} are almost never isomorphic as varieties. For example the connected components of M_{B}(X, 1) are isomorphic to products of ℂ^{∗}’s whereas M_{Dol}(X, 1) ≅ H^{0}(X, Ω_{X}^{1}) × P ic^{τ} (X).

Given an F -Higgs bundle (E, θ), powers of θ can be viewed as sections of S^{∗}F ⊗ End(E). Define

h_{X}(E, θ) = h(E, θ) ∈ ⨁_{i=1}^{rk E} H^{0}(S^{i}F)

by

h(E, θ) = ∑ trace(θ^{i}).

h(E, θ) is equivalent, in some sense, to the characteristic polynomial of the transpose F^{∗} → End(E) (also denoted by θ) viewed as a function on the total space of F^{∗}. In particular, h(E, θ) evaluates to 0 at v ∈ F_{x}^{∗} if and only if θ_{x}(v) is nilpotent.

Theorem 2.2. (Hitchin, Nitsure, Simpson). The Hitchin map

h : M_{Dol}(X, n) → ⨁_{i=1}^{n} H^{0}(S^{i}Ω_{X}^{1})

is proper.

Remark 2.3. By construction, M_{Dol}(X, n) is quasiprojective, therefore h is in fact projective.

An easy application of these ideas yields the existence of symmetric forms:

Proposition 2.4. If π_{1}(X) has an infinite number of nonisomorphic semisimple representations into GL_{n}(ℂ), then H^{0}(S^{i}Ω_{X}^{1}) ≠ 0 for some 1 ≤ i ≤ n.

Proof. The hypothesis shows that some component of M_{B}(X, n) is a positive dimensional affine variety. Therefore this space is noncompact (for the classical topology), and the same goes for M_{Dol}(X, n). As

h_{X} : M_{Dol}(X, n) → ⨁_{i=1}^{n} H^{0}(S^{i}Ω_{X}^{1})

is proper, it must be nonconstant.

Note that there exists examples of simply connected varieties where H^{0}(X, S^{i}Ω_{X}) ≠ 0 for some i > 1. Therefore the converse is false.

3. A Nonabelian Analogue of b_{1} = 2q.

The role of b_{1} will be played by the dimension of an irreducible component M of M_{B}(X, n), and q by the dimension of the image of the corresponding component of M_{Dol} under the Hitchin map. It will be shown that dim M = 2dim h(M) as expected. This will yield refinements of proposition 2.4.

I want to prove a stronger statement which is an analogue of the equality dim G = 2 dim G^{10} for a sub-Hodge structure G ⊆ H^{1}(X). Observe that H^{1}(X, ℂ) carries the structure of a left module over the ring of quaternions ℍ = ℝ ⊕ ℝi ⊕ ℝj ⊕ ℝk, where multiplication by i is given by the usual complex structure and multiplication by j is given by v ↦ i^{p−q}v̄ on H^{pq}. A sub-Hodge structure of H^{1}(X, ℂ) would be a quaternionic submodule. H^{1}(X, ℝ) is invariant under j, and the restriction of j to H^{1}(X, ℝ) is compatible, via the exponential map, with the complex structure on M_{U}(X, 1) = H^{1}(X, U(1)) given by its identification with P ic^{τ} (X). This suggests the following nonabelian analogue to the notion of a sub-Hodge structure. Call a subset N_{Dol} ⊂ M_{Dol}(X, n) (respectively N_{B} ⊂ M_{B}(X, n)) quaternionic provided that it is a closed analytic reduced subvariety and such that the corresponding closed subset of M_{B}(X, n) (respectively M_{Dol}(X, n)) is also a reduced analytic subvariety. Then the main result is:

Theorem 3.1. If N ⊆ M_{Dol}(X, n) is an irreducible quaternionic set, then dim N = 2dim h(N).

The proof will be postponed until the fifth section. First, I want to discuss some basic examples and applications.

Example 3.2. Let f : X → Y be a morphism of smooth projective varieties. The pullback of a quaternionic subset of M_{B}(Y, n) to M_{B}(X, n) is quaternionic. In particular, f^{∗}M_{B}(Y, n) is quaternionic. The corresponding operation for Higgs bundles is given as follows. If (E, θ) ∈ M_{Dol}(Y, n), then its pullback is the pair consisting of f^{∗}E with the Higgs field given as the composite

f^{∗}E → f^{∗}Ω_{Y}^{1} ⊗ f^{∗}E → Ω_{X}^{1} ⊗ f^{∗}E.

Example 3.3. The cohomology support loci are given by

Σ_{m,B}^{k}(X, n) = {V ∈ M_{B}(X, n) | dim H^{k}(X, V) ≥ m}

These will be shown to be quaternionic in section 6.

Applying the theorem to the first example, where f : X → Y is map of smooth projective varieties, yields a refinement of proposition 2.4:

Corollary 3.4.

∑_{i=1}^{n} dim im[H^{0}(S^{i}Ω_{Y}^{1} ) → H^{0}(S^{i}Ω_{X})] ≥ \frac{1}{2}dim f^{∗}M_{B}(Y, n)

Corollary 3.5. If π_{1}(X) has an infinite number of nonisomorphic semisimple representations into GL_{n}(ℂ) factoring through π_{1}(Y), then f^{∗}H^{0}(S^{i}Ω_{Y}^{1} ) ≠ 0 for some 1 ≤ i ≤ n. In particular, the map f^{∗}H^{0}(S^{n!}Ω_{Y}^{1} ) ≠ 0 is nonzero.

4. Quaternionic geometry and Lagrangian maps

This section is largely expository. I want to give a quick introduction to quaternionic (or hyperkähler ) geometry, and to some of Verbitsky’s work in particular. A nice discussion of some related ideas and examples can be found in [H3]. See also [H1], [F], [S4], [V1], [V2] and references contained therein.

A quaternionic (or hypercomplex) manifold is a C^{∞}-manifold X with two complex structures I and J which induce the same real analytic structure on X and satisfies IJ = −J I. Setting K = IJ gives an action of the quaternions ℍ on the tangent bundle. Any quaternionic vector space is naturally a quaternionic manifold. A morphism of quaternionic manifolds is C^{∞} map which is holomorphic with respect to I, J and K (it suffices to check holomorphicity with respect to any two).

Lemma 4.1. Let V_{1} and V_{2} be two finite dimensional quaternionic vector spaces, and let U_{i} ⊆ V_{i} be open neighbourhoods of the origin with their induced quaternionic structures. Any morphism f : U_{1} → U_{2} satisfying f(0) = 0 is the restriction of an ℍ-linear map.

Proof. (Deligne, see [S3]). Choose a point x ∈ U_{1}. After identifying the tangent space at x and f(x) with V_{1} and V_{2}, the differential gives an ℝ-linear map df_{x} : V_{1} → V_{2} By assumption

df_{x}I = Idf_{x}, df_{x}J = J df_{x}

so in fact df_{x} is ℍ-linear. Thus df can be viewed as a C^{∞} map from U_{1} to Hom_{ℍ}(V_{1}, V_{2}). Differentiating again yields the Hessian, which is a symmetric ℝbilinear form H_{x} : V_{1} × V_{1} → V_{2}. H_{x} is ℍ-linear in one variable and therefore in both. Now for the punchline:

KH_{x}(α, β) = IH_{x}(J α, β) = H_{x}(J α, Iβ) = J H_{x}(α, Iβ) = −KH_{x}(α, β)

Therefore H_{x} = 0 and the lemma follows immediately.

A quaternionic submanifold of a quaternionic manifold is a C^{∞} submanifold such that the inclusion is a morphism. More generally a quaternionic subvariety Y of a quaternionic manifold X is a reduced real analytic subvariety whose complexified ideal is locally defined by both I and J holomorphic functions.

Corollary 4.2. Any quaternionic submanifold of a quaternionic vector space V is a translate of a linear subspace. A quaternionic subvariety is a union of subamanifolds.

A hyperkähler manifold is a C^{∞} manifold with a Riemannian metric g and two anticommuting complex structures I and J, such that g is Kähler with respect to both of these structures.

Proposition 4.3. ([V1, 6.5]) If X is hyperkähler then the underlying real analytic structures associated to I and J coincide. Therefore X is a quaternionic manifold.

Theorem 4.4. (Verbitsky [V2]). Let (X, x) be a germ of a hyperkähler manifold. The the germ of quaternionic subvariety (Y, x) is the union of a finite number of germs of quaternionic submanifolds.

Here is an outline of the proof: Let R be the local ring of real analytic functions of (X, x). Given a complex structure L on X, let O_{L} be the local ring of Lholomorphic functions. There is a natural inclusion O_{L} ⊂ R ⊗ ℂ which splits: the complement is the ideal generated by L-antiholomorphic functions vanishing at x. Let ϕ be the composition of local homomorphisms:

O_{I} ↪ R ⊗ ℂ → O_{J} ↪ R ⊗ ℂ → O_{I}

A direct calculation shows that the induced endomorphism on the cotangent space m/m^{2} of O_{I} is a homothety associated to a scalar λ ∈ ℂ^{∗} which is not a root of unity. After a change of variables, one can arrange I-holomorphic coordinates so that ϕ(x_{i}) = λx_{i}. Let I be the ideal of Y in O_{I}. Then I(R ⊗ ℂ) is also generated by J -holomorphic functions. Therefore ϕ(I) ⊂ I and this implies that I is homogeneous, and thus (Y, x) is isomorphic to the germ of its tangent cone. The tangent cone is a quaternionic subvariety of a quaternionic vector space, and therefore by 4.2 is a union of manifolds.

Corollary 4.5. The normalization of a quaternionic subvariety of a hyperkähler manifold with respect to I coincides with the normalization with respect to J and it is smooth. Furthermore it inherits a hyperkähler structure from the ambient manifold.

Given a hyperkähler manifold X, set ω_{L}(α, β) = g(Lα, β) for L = I, J , K. These are just the Kähler forms associated to the complex structures. In particular, they define (real) symplectic structures on X. The form ω_{J} + \sqrt{−1}ω_{K} defines an Iholomorphic symplectic structure on X.

Let (X, ω) be a real or holomorphic symplectic manifold. A closed submanifold Y ⊂ X is Lagrangian if the tangent spaces of Y are maximal isotropic subspaces of the tangent spaces of X with respect to the symplectic pairing induced by ω. A map h : X → B (C^{∞} or holomorphic according to the category) to manifold B will be called Lagrangian if its differential has maximal rank, and all its fibers are Lagrangian submanifolds. Note that a Lagrangian map is the same thing as a completely integrable system. A complete discussion of these notions would take us too far afield, see for example [GS] for further details.

If ω is only defined on a dense open subset of U′ ⊆ X, a map h : X → B will be called generically Lagrangian if there is a dense nonsingular open set U ⊂ B such that h^{−1}(U) ⊆ U′ and h^{−1}(U) → U is Lagrangian, the complement of the largest such U will be called the discriminant.

Lemma 4.6. Let X be a Kähler manifold with a real C^{∞} symplectic structure ω (not necessarily equal to the Kähler form). If h : X → B is a proper Lagrangian holomorphic map, then the connected components of the fibers are complex tori.

Proof. After Stein factorization, the fibers can be assumed to be connected. Standard results in symplectic geometry [GS, page 353] show that any fiber F is diffeomorphic to a torus. It follows easily that the Albanese map

F → H^{1}(F, ℝ)^{∗}/H_{1}(F, ℤ) ≅ H^{0}(F, Ω_{F}^{1} )^{∗}/H_{1}(F, ℤ)

is a diffeomorphism and therefore a biholomorphism. Note that the isomorphism H^{1}(F, ℝ) ≅ H^{0}(F, Ω_{F}^{1} ) is only place where Kähler condition on X is used, so the lemma holds under considerably weaker hypotheses.

Let (X, g, I, J) be a hyperkähler manifold. We will usually take I as the preferred complex structure with holomorphic symplectic structure given by ω_{J} + \sqrt{−1}ω_{K}. In particular, a Lagrangian map h : X → B will be assumed to be Iholomorphic and Lagrangian with respect to the indicated symplectic structure. Note that such a map is also Lagrangian with respect to the real symplectic structure ω_{J}, consequently the above lemma applies.

Theorem 4.7. Let (X, g_{X}, I, J) and (Y, g_{Y} , I, J) be a hyperkähler manifolds Suppose that h : X → B is a Lagrangian map. If f : Y → X is a finite quaternionic morphism such that g_{Y} = f^{∗}g_{X} (away from critical points). Then h ∘ f : Y → B′ is a a generically Lagrangian map, where the image B′ = h(f(Y)) is endowed with the reduced analytic structure.

Proof. For any x ∈ X, V_{x} = ker dh_{x} is a maximal isotropic subspace of the tangent space T_{x} with respect to ω_{J}. Thus J V_{x} is orthogonal to V_{x} with respect to g_{X}. Since

dim J V_{x} = dim V_{x} = dimT_{x}/2,

T_{x} = V_{x} ⊕ J V_{x}. Let y ∈ Y be a general point, then identify T_{y} with a quaternionic subspace of T_{f(y)} and ker d(h ∘ f)_{y} with V_{f(y)} ∩ T_{y}, and so T_{y} = ker d(h ∘ f)_{y} ⊕ J ker d(h ∘ f)_{y}. Therefore ker d(h ∘ f)_{y} is a maximal isotropic subspace. So the general fibers of h ∘ f are Lagrangian.

Corollary 4.8. If in the above notation h ∘ f is proper, then the connected components of its general fibers are complex tori.

5. Integrability of Hitchin’s map

In this section, I will prove a strengthening of theorem 3.1, namely that the restriction of Hitchin’s map to a quaternionic subset is integrable.

Proposition 5.1. Let C ⊂ X be a curve obtained as a complete intersection of divisors associated to an Nth power of L, with N >> 0. Then the restriction maps M_{B}(X, n) → M_{B}(C, n) and M_{Dol}(X, n) → M_{Dol}(C, n) are injective and compatible with the homeomorphisms of the associated analytic spaces.

Proof. This is well known to the experts, so I will merely indicate the main ideas. The first part follows from the Lefschetz hyperplane theorem. For the second part, first note that the restriction of a semistable Higgs bundle is semistable thanks to Simpson’s generalization of the Mehta-Ramanathan theorem [S1], so the map is well defined. As for injectivity, choosing N >> 0 guarantees that

H^{1}(E_{1}^{∗} ⊗ E_{2} ⊗ I_{C}) = 0

for any pair of Higgs bundles on X. Thus any isomorphism of their restrictions can be lifted to map of the Higgs bundles which can be seen to be an isomorphism (using, for example, the fact that polystable bundles are direct sums of simple bundles). The last part is just a restatement of the naturality of the correspondence.

Remark 5.2. The images of M_{Dol}^{s}(X, n) and M_{B}^{irr}(X, n) under restriction lie in the corresponding subsets for C.

Theorem 5.3. (Hitchin[H1]) Let C be a smooth projective curve. Then the stable locus M_{Dol}^{s,an}(C, n) is smooth and carries a natural hyperkähler structure (g, I, J). Where I is the usual complex structure and J is the structure induced from the identification M_{Dol}^{s,an} ≅ M_{B}^{irr,an}.

Remark 5.4. Hitchin stated this only when n = 2, although his proof presumably works for higher rank bundles. In any case, Fujiki [F] has established a much more general result.

Let

S_{n}(X) = ⨁_{i=1}^{n} H^{0}(X, S^{i}Ω_{X}^{1})

As noted Hitchin’s map h = h_{X} is proper, in fact more is true:

Theorem 5.5. ( Hitchin[H2]) When dimX = 1, h_{X} is surjective and generically Lagrangian with respect to the hyperkähler structure on M_{Dol}^{s,an}.

Suppose that a curve C ⊆ X has been chosen as in 5.1, then a commutative diagram can be obtained:

M_{Dol}(X, n) ↪ M_{Dol}(C, n)
h_{X} ↓  h_{C} ↓
S_{n}(X) →^{i} S_{n}(C)

While i need not be injective, we do have:

Lemma 5.6. The map h_{X}(M_{Dol}(X, n)) → h_{C}(M_{Dol}(C, n)) is finite.

Proof. This follows from the properness of all the other maps in the diagram.

Consequently, fibers of h_{X} are components of fibers of i ∘ h_{X}. Putting the above results together yields the main theorem:

Theorem 5.7. Let X be a smooth projective variety. If Ñ is a connected component of the normalization of a quaternionic subset N ⊂ M_{Dol}(X, n) (with its reduced structure), and h̃ : Ñ → h(Ñ) the natural map. If the image of Ñ in M_{Dol}(X, n) meets the stable locus M_{Dol}^{s}, then h̃ is generically Lagrangian. In particular the connected components of its general fibers are abelian varieties of dimension half that of dim Ñ

Proof. Let C ⊆ X be a complete intersection curve of high degree. Then N is a quaternionic subvariety of M_{Dol}(C, n) by the previous results. Therefore the theorem follows from 4.7.

Theorem 3.1 is a consequence of:

Corollary 5.8. Let N be an irreducible quaternionic subvariety of M_{Dol}(X, n). Then an irreducible component F of any fiber of h|_{N} satisfies dimF = dim N/2.

Proof. First assume that N meets M_{Dol}^{s}, then the equality holds for the general fibers. By upper semicontinuity of dimensions, it is enough to prove dimF ≤ dim Ñ/2. Let N_{B} denote N with the complex structure inherited from M_{B}(X, n). N_{B} is Stein, therefore H_{i}(N, ℤ) = 0 for i > dim N [N]. Since the map h|_{F} is projective 2.3, the fundamental class of F defines a nonzero element of H_{2dim(F)}(Ñ, ℤ), and this forces the inequality.

In general, there is a partition n = ∑_{i} n_{i} and a commutative diagram

N_{1} × … N_{k}  → N
↓  ↓
h(N_{1}) × … h(N_{k}) → h(N)

such that the horizontal maps are finite, and where N_{i} ⊂ M_{Dol}(X, n_{i}). are irreducible quaternionic varieties meeting the M_{Dol}^{s}(X, n_{i}). So the general case follows.

Remark 5.9. The theorem can also be used to recover a result of Biswas [B, 8.2] about Poisson commutivity of higher dimensional Hitchin maps.

6. Cohomology support loci

I want to discuss another natural example of quaternionic varieties, namely the cohomology support loci. The first task is it introduce a notion of cohomology for Higgs bundle. Given a Higgs bundle (E, θ), define H^{i}(E, θ) to be the ith hypercohomology of the complex:

E ⟶^{θ∧} Ω_{X}^{1} ⊗ E ⟶^{θ∧} Ω_{X}^{2} ⊗ E …

An alternative interpretation is as follows. The cotangent bundle π : T_{X}^{∗} → X has a tautological section Θ ∈ H^{0}(π^{∗}Ω_{X}^{1}) which induces a Higgs field on the push forward of any coherent O_{T_{X}^{∗}} -module E (of course this won’t be a bundle unless E is finite and flat over O_{X}). Conversely, any Higgs bundle arises this way [S1]. Then the ith cohomology of the Higgs bundle E is isomorphic to Ext^{i}(O_{X}, E), where O_{X} is identified with the structure sheaf of the zero section of T_{X}^{∗}; this can be seen by computing Ext via a Koszul resolution of O_{X} [A2].

We define the cohomology support loci as:

Σ_{m,Dol}^{k}(X, n) = {[(E, θ)] ∈ M_{Dol}(X, n) | dim H^{k}(E, θ) ≥ m}

We can make an analogous definition for M_{B}. Given a representation ρ : π(X) → Gl_{n}(ℂ), there exists (up to isomorphism) a unique rank n locally constant sheaf, or local system, with ρ as its monodromy representation. So we can, and will, view M_{B}(X, n) as a moduli space of semisimple local systems on X. Let

Σ_{m,B}^{k}(X, n) = {V ∈ M_{B}(X, n) | dim H^{k}(X, V) ≥ m}

A family of Higgs bundles on X parameterized by T is a vector bundle E on X × T, with a section Θ of p_{X}^{∗}Ω_{X}^{1} ⊗ End(E) satisfying Θ ∧ Θ = 0. M_{Dol} is only a coarse moduli space, so there may not be a universal family of Higgs bundles. However Simpson’s construction gives a bit more. Namely M_{Dol}(X, n) is a quotient, in the sense of geometric invariant theory, of a locally closed subscheme Q_{n} of an appropriate Quot or Hilbert scheme. X × Q_{n} will in fact carry a family of Higgs bundles (E, Θ) such that its restriction (E, Θ)|_{q} to a slice X × {q} is semistable and corresponds to the image of q in M_{Dol}(X, n). Over M_{Dol}^{s}(X, n) it possible to find cross sections to Q_{n} → M_{Dol}(X, n) locally in the etale topology. The following is an almost immediate consequence:

Proposition 6.1. Let S be a set of semistable Higgs bundles on X. Suppose that for every T and semistable family of Higgs bundles (E, Θ) parameterized by T, the set S_{T} = {t | (E, Θ)|_{t} ∈ S} is Zariski closed. Then the set S_{M} of all m ∈ M_{Dol} which possess a semistable representative in S is Zariski closed.

Proof. S_{M} is constructible because it is the image of S_{Q} under the canonical map. Given a limit point s of S_{M}, there is an irreducible curve C such that s ∈ C and C − {s} ⊂ S_{M}. As Q → M_{Dol} is surjective, there is an irreducible curve C′ ⊂ Q and finite map C′ → C. One obtains a family of Higgs bundles on C′ by restriction. S_{C′} = C′ since it is closed and contains the preimage of C − {s}. Therefore s ∈ S_{M} and so S_{M} is closed.

Proposition 6.2. Σ_{m,Dol}^{k} is Zariski closed in M_{Dol}. Σ_{m,B}^{k} is is Zariski closed in M_{B}. Σ_{m,Dol}^{k} and Σ_{m,B}^{k} coincide under the correspondence between M_{Dol}^{an} and M_{B}^{an}.

Proof. Note that if (E_{i}, θ_{i}), i = 1, 2 are equivalent semistable bundles with (E_{1}, θ_{1}) polystable, then

dimH^{k}(E_{1}, θ_{1}) ≥ dimH^{k}(E_{2}, θ_{2})

by subadditivity of cohomology. Therefore the first statement follows from proposition 6.1 and the semicontinuity theorem for cohomology [EGA III, 7.7.5, 7.7.12] The second statement is proved in [A1]. The last part follows from [S1, 2.2].

Corollary 6.3. Σ_{m,B}^{k} is quaternionic.

7. Characteristic Cones and Products of Higgs Bundles

Let (E_{1}, θ_{1}), … (E_{k}, θ_{k}), be a collection of Higgs bundles. I will define the Ω_{X}^{k}- Higgs bundle (E_{1}, θ_{1}) ∧ (E_{2}, θ_{2}) ∧ … (E_{k}, θ_{k}) to be the pair consisting of the tensor product of the E_{i}’s and the composite

E_{1} ⊗ … ⊗ E_{k} ⟶^{θ_{1}⊗...⊗θ_{k}} Ω_{X}^{1} ⊗ E_{1} ⊗ … ⊗ Ω_{X}^{1} ⊗ E_{k} →

(Ω_{X}^{1})^{⊗k} ⊗ E_{1} ⊗ … ⊗ E_{k} → Ω_{X}^{k} ⊗ E_{1} ⊗ … ⊗ E_{k}.

Fix a point x ∈ X. Then a Higgs bundle (E, θ) determines a linear map θ_{x} : T_{x} → End(E_{x}), where E_{x} denotes the fiber. Consider the characteristic cone

C_{x}^{nilp}(E, θ) = {v ∈ T_{x} | θ_{x}(v) is nilpotent }.

Given a set S of Higgs bundles, let C_{x}^{nilp}(S) = ∩_{s∈S} C_{x}^{nilp}(s). As x varies, C_{x}^{nilp}(S) forms an algebraic family, thus dim X−dim C_{x}^{nilp}(S) attains a maximum for general x; let c(S) = c_{X}(S) be this number, and let c(n) = max_{S}c(S) as S ranges over the irreducible components of M_{Dol}(X, n). I will refer to this as the nth characteristic dimension (it is helpful to think of this as the dimension of a fictitious variety).

Lemma 7.1. The following holds:

(a) c(1) = dim a(X).

(b) c(n) ≥ c(m) if m ≤ n

(c) c(n) = 0 if and only if M_{B}(X, n) is finite.

Proof. (a) follows from the comments at the end of section 1 which shows that when S = M_{Dol}(X, 1), C_{x}^{nilp}(S) = ker(da_{x}). (b) follows by taking direct sums. One direction of (c) follows from the arguments of the previous section, if M_{B} is infinite, then there must exist a Higgs bundle (E, θ) with h(E, θ) ≠ 0, so its characteristic cone must be nontrivial at some point. For the converse, note that an isolated point of M_{Dol} must be fixed under the ℂ^{∗}-action: (E, θ) ↦ (E, tθ), consequently θ must be nilpotent.

Lemma 7.2. Let M be an irreducible component of M_{Dol}(X, n). Then k ≤ c(M) if and only if there exists (E_{1}, θ_{1}), … (E_{k}, θ_{k}) ∈ M such that h((E_{1}, θ_{1}) ∧ … ∧ (E_{k}, θ_{k})) ≠ 0.

Proof. Suppose k ≤ c(M). Let x ∈ X be a general point. Then k is less than or equal to the codimension of C_{x}^{nilp}(X, n). By hypothesis, a Higgs bundle (E_{1}, θ_{1}) ∈ M_{Dol}(X, n) can be chosen so that θ_{1,x} has a nonzero eigenvalue λ_{1} ∈ T_{x}^{∗}. Pick 0 ≠ v_{1} ∈ ker(λ_{1}), then choose (E_{2}, θ_{2}) with an eigenvalue λ_{2} at x such that λ_{2}(v_{1}) ≠ 0. In this way, one obtains a collection of Higgs bundles (E_{1}, θ_{1}), … (E_{k}, θ_{k}), with linearly independent eigenvalues λ_{i} at x. Therefore h((E_{1}, θ_{1}) ∧ … (E_{k}, θ_{k})) ≠ 0 because it has a nonzero eigenvalue λ_{1} ∧ … ∧ λ_{k}.

Conversely, if h((E_{1}, θ_{1}) ∧ … (E_{k}, θ_{k})) ≠ 0, then for general x,

C_{x}^{nilp}(E_{1}, θ_{1}) ⫌ C_{x}^{nilp}(E_{1}, θ_{1})∩C_{x}^{nilp}(E_{2}, θ_{2}) ⫌ … C_{x}^{nilp}(E_{1}, θ_{1})∩…∩C_{x}^{nilp}(E_{k}, θ_{k}).

This implies k ≤ c(M).

There is a global characterization of the characteristic dimension (at least when M_{B} is sufficiently large).

Theorem 7.3. Let S be an irreducible component of M_{Dol}(X, n). Then (a) ⇒ (b) ⇔ (c), and these are equivalent if dim S ≥ 2nk.

(a) c(S) ≥ k.

(b) Let x ∈ X is a general point, and Z → Z′ ↪ X any desingularization of a subvariety containing x with dim Z′ > dim X − k. Then S has a positive dimensional image in M_{Dol}(Z, n).

(c) Let x ∈ X is a general point, and Z → Z′ ↪ X any generically finite map of a smooth variety on to a subvariety containing x with dim Z′ > dim X − k. Then S has a positive dimensional image in M_{Dol}(Z, n).

The proof will be given elsewhere.

8. Shafarevich Maps

In this section, I want to relate the nth characteristic dimension to the dimension of an appropriate Shafarevich variety.

Let Γ be a finitely generated group. Given a set of representations S, let K_{S}(Γ) = ∩_{ρ∈S} ker(ρ). As conjugate representations have the same kernel, there is no harm in taking S to be set of conjugacy classes of representations. Let K_{n}(Γ) = K_{S}(Γ) when S is the set of all semisimple representations into GL_{n}(ℂ) and let K(Γ) be the intersection of all subgroups of finite index. Then

Lemma 8.1. (a) K_{1}(Γ) = [Γ, Γ]

(b) There are inclusions K_{n+1}(Γ) ⊆ K_{n}(Γ)

(c) ∩_{n} K_{n}(Γ) = K(Γ).

Proof. (a) is straight forward. (b) follows from the fact that any representation can extended to one of higher rank by adding a trivial summand. Finally, the inclusion ∩_{n} K_{n}(Γ) ⊇ K(Γ), follows from the fact that any finite group has faithful finite dimensional representation. The converse is the more interesting direction. Given a representation ρ : Γ → GL_{n}(ℂ), it can be factored as Γ → GL_{n}(R) ⊂ GL_{n}(ℂ) where R is a finitely generated ℤ-algebra. Let G be the product of the finite groups GL_{n}(R/m) as m varies over all maximal ideals. The canonical map GL_{n}(R) → G is easily seen to be injective, and so the lemma follows.

Given a normal subgroup H ⊂ π_{1}(X), Kollár [K1, K2] has introduced the notion of a Shafarevich map. It is a rational map sh^{H} : X −−→ Sh^{H}(X) with connected fibers such that that if p : Z → X is a map of smooth projective varieties whose image is not contained in a specific countable union of proper subvarieties of X, then sh^{H} ∘ p is constant if and only if π_{1}(Z) has finite image in π_{1}(X)/H. (It may be helpful on first reading to take this to mean that for most subvarieties Z′ = p(Z) ⊂ X, Z′ gets contracted to a point in Sh^{H}(X) if and only if ...) I will often work with a specific biregular model of sh^{H}, which may require replacing X by a blow up. Let sh^{S} = sh^{K_{S}(π_{1}(X))} and sh^{n} = sh^{K_{n}(π_{1}(X))} and ŝh = sh^{K(π_{1}(X))}. Lemma 8.1 implies that sh^{n} factors through sh^{n+1} and ŝh.

Given a subset S ⊆ M_{B}(X, n), let S_{Dol} be the corresponding subset of M_{Dol}(X, n).

Theorem 8.2. Given a subset S ⊆ M_{B}(X, n), and a general point x ∈ X, the tangent space at x to the fiber of sh^{S} containing x lies in C_{x}^{nilp}(S_{Dol}).

The proof follows easily from:

Lemma 8.3. If p : Z → X is a map from a smooth projective variety such that π_{1}(Z) has finite image in π_{1}(X)/K_{S}, then for every Higgs bundle (E, θ) ∈ S_{Dol}, p^{∗}θ = 0.

Proof. The hypothesis implies that the representation corresponding to p^{∗}(E, θ) is unitary.

Corollary 8.4. dim Sh^{n}(X) ≥ c(n)

Corollary 8.5. If c(n) = dim X, then X, Sh^{n}(X) and Ŝh(X) are birational, in particular X has generically large algebraic fundamental group (see [K2] for the definition).

9. Powers of the canonical bundle

I now come to the main point, which is to apply these methods to produce sections of powers of the canonical bundle ω_{X}. The first result is an immediate corollary of lemma 7.2.

Proposition 9.1. Let n > 0. Then for each 0 < k ≤ c(n), there exists 1 ≤ i ≤ n^{k} such that h^{0}(S^{i}Ω_{X}^{k}) ≠ 0. In particular, if c(n) = dim X then the ith plurigenus P_{i}(X) ≠ 0 for some positive i ≤ n^{dimX}

Note that an Abelian variety satisfies the above hypothesis. To get P_{i}(X) ≥ 2 requires more conditions.

Proposition 9.2. Suppose that M is an irreducible component of M_{Dol}(X, n) such

that c(M) = dim X and dimM ≥ 2 ( \begin{matrix}n + dim X \\ n \end{matrix} ) − 2, then P_{N}(X) ≥ 2 for some

N < n^{dim X}.

Proof. Lemma 7.2 shows that there is a nonzero section of some ω^{⊗N} given as a component of the characteristic polynomial of a wedge product of Higgs bundles lying in M. This section will be nonzero at a general point y ∈ X. It is enough to find another nonzero section vanishing at y. A dimension count (using the hypothesis plus 3.4) shows that there is a nonnilpotent Higgs bundle (E_{1}, θ_{1}) ∈ M such that h(E_{1}, θ_{1}) vanishes at y. Now choose a second general point x ∈ X, and repeat the argument used in lemma 7.2 to find a sequence of Higgs bundles (E_{i}, θ_{i}) ∈ M with linearly independent eigenvalues at x. Let (E, Θ) = (E_{1}, θ_{1})∧…, then the Nth component of h(E, Θ) is the desired section.

For the next result, I will be considering a pair consisting of X and a generically finite map of a smooth variety i : Z → X. If (E, θ) ∈ M_{Dol}(X, n) then by construction the restriction (E, θ)|_{Z} = i^{∗}(E, θ) arises from an Ω_{X}^{1}|_{Z}-Higgs bundle, and the same goes for Ω^{k}-valued Higgs bundles. If c_{X}(n) = dim X, then c_{Z}(n) = dim Z provided that Z contains a general point of X. By slight modification of the above arguments, one gets:

Proposition 9.3. Suppose that c_{X}(n) = dim X. If Z ⊂ X contains a general point of X, then

dim [H^{0}(S^{i}Ω_{X}^{k}) → H^{0}(S^{i}Ω_{Z}^{k})]

is nonzero for some positive i ≤ n^{dim Z} and any k ≤ dim Z. If in addition there is an irreducible component M ⊆ M_{Dol}(X, n) such that c_{X}(M) = dim X and the

dimension of the image of M in M_{B}(Z, n) is at least 2 ( \begin{matrix}n + dim Z \\ n \end{matrix} ) − 2 then for

some N < n^{dim X},

dim[H^{0}(S^{N}Ω_{X}^{dim Z}) → H^{0}(ω_{Z}^{⊗N})] ≥ 2.

Products of Abelian varieties with curves of large genus satisfy the hypothesis of 9.2. To obtain varieties of general type, it will be necessary to make a more stringent hypothesis. I will say that the nth characteristic cones are generically separated provided that there exists a nonempty open set U ⊂ X and an irreducible component M of M_{Dol}(X, n) such that for any pair of distinct points x, y ∈ U and vectors v ∈ T_{x}, w ∈ T_{y}, there exists a Higgs bundle (E, θ) ∈ M with v ∈ C_{x}^{nilp}(E, θ) and w ∉ C_{y}^{nilp}(E, θ).

Proposition 9.4. If the nth characteristic cones of X are generically separated, then for some N < n^{dimX}, the global sections of ω^{⊗N} separate points over a nonempty open subset of X. In particular, X is of general type.

Proof. Given two distinct general points x and y, it is enough to construct a section of some ω^{⊗N} which vanishes at x but not at y. Choose vectors v ∈ T_{x} and w_{1} ∈ T_{y}. By hypothesis, there is a Higgs bundle (E_{1}, θ_{1}) ∈ M such that v ∈ C_{x}^{nilp}(E_{1}, θ_{1}) and w_{1} ∉ C_{y}^{nilp}(E_{1}, θ_{1}). Choose w_{2} ∈ C_{y}^{nilp}(E_{1}, θ_{1}) if it is nonempty, otherwise choose w_{2} arbitrarily. Let (E_{2}, θ_{2}) ∈ M be a Higgs bundle such that v ∈ C_{x}^{nilp}(E_{2}, θ_{2}) and w_{2} ∉ C_{y}^{nilp}(E_{2}, θ_{2}), and so on. Then (E, Θ) = (E_{1}, θ_{1}) ∧ … (E_{dimX}, θ_{dimX}) is a Higgs bundle which is nilpotent at x and nonnilpotent at y. Therefore, for some N which can be chosen uniformly with respect to x and y, the Nth component of h(E, Θ) vanishes at x but not at y.

In order, for this result to be useful, some reasonable criteria will be needed to guarantee that the characteristic cones are generically separated. As an example of what I am hoping for, let me point out that a variety with c(1) = dim X (i.e. with a generically finite Albanese map) is of general type if and only if it has generically separated first characteristic cones. Note that when the nth characteristic cones are generically separated, X will have generically large fundamental group in the sense of Kollár [K1, K2]. Under these conditions he obtains much better estimates on the lower plurigenera such as P_{2}(X) ≥ 1, and P_{4}(X) ≥ 2. However, the present methods seems to give better control on the location of the zeros of the sections and better asymptotic results.

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14 DONU ARAPURA

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(1996) Department of Mathematics, Purdue University, West Lafayette IN 47907