Symmetric differentials and variations of Hodge structures

Yohan Brunebarbe

doi:10.1515/crelle-2015-0109

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Symmetric differentials and variations of Hodge structures

Yohan Brunebarbe

Published/Copyright: March 2, 2016

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 743

Abstract

Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.

A Integrable connections with logarithmic poles and Deligne’s canonical extension

A.1 Connections with logarithmic poles

A.1.1 Definitions

Definition A.1.

Let (X,D) be a log-pair and V be a holomorphic vector bundle on X. A connection on V with logarithmic poles along D is a 𝐂X-linear map of sheaves

∇:V→ΩX1⁢(log⁡D)⊗𝒪XV

which satisfies the Leibniz rule:

∇⁡(f⋅s)=f⋅∇⁡(s)+d⁢f⊗s,

where f is a local section of 𝒪X and s a local section of V.

Remark A.2.

In the particular case where D=∅ we recover the usual notion of connection on a holomorphic vector bundle. If (X,D) is a log-pair and V′ is a holomorphic vector bundle on X-D endowed with a connection ∇′ obtained by restricting to X-D a holomorphic vector bundle V on X endowed with a connection ∇ with logarithmic poles along D, then we say that ∇ has logarithmic poles along D with respect to the extension V. It is a property of the extension V.

If (V,∇) is a holomorphic vector bundle on X endowed with a connection with logarithmic poles along D, then for every p≥1 there exists a unique 𝐂X-linear map of sheaves

∇p:ΩXp⁢(log⁡D)⊗𝒪XV→ΩXp+1⁢(log⁡D)⊗𝒪XV

which satisfies the generalized Leibniz rule:

∇p⁡(ω⊗s)=d⁢ω⊗s+(-1)p⋅ω∧∇p⁡(s),

where ω is a local section of ΩXp⁢(log⁡D) and s a local section of V.

The curvature of the connection is the map F∇:∇1∘∇:V→ΩX2⁢(log⁡D)⊗𝒪XV. It is easily seen to be an 𝒪X-linear map.

Definition A.3.

The connection is called integrable if its curvature is zero.

A morphism between two holomorphic vector bundles (V1,∇1) and (V2,∇2) on X equipped with an integrable connection with logarithmic poles along D is a morphism of vector bundles ϕ:V1→V2 commuting with the connections. The holomorphic vector bundles on X equipped with an integrable connection with logarithmic poles along D form an abelian category VClog⁢(X,D).

To any map f:(Y,E)→(X,D) of log-varieties is associated an (additive) functor f*:VClog⁢(X,D)→VClog⁢(Y,E). We can also define the tensor product and internal hom of two elements in VClog⁢(X,D), extending the corresponding notions for holomorphic vector bundles.

Remark A.4.

If D=∅ and the connection ∇ is integrable, then the sheaf

V∇:=ker⁡(∇:V→ΩX1⊗𝒪XV)

is a complex local system. The classical Riemann–Hilbert correspondence says that for any complex manifold X, the functor (V,∇)↦V∇ between the category of holomorphic vector bundles on X endowed with an integrable connection and the category of complex local systems on X is an equivalence of categories. A quasi-inverse is defined by associating to any complex local system ℒ the holomorphic vector bundle V:=𝒪X⊗𝐂ℒ endowed with the connection ∇⁡(f⋅s)=d⁢f⊗s. These functors are compatible with the formation of tensor products, internal hom, dual and pull-back.

Finally, a complex local system ℒ on X (connected) is completely understood by its monodromy’s representation π1⁢(X,x)→GL⁢(ℒx) for any x∈X.

Example A.5.

Fix two integers 1≤p≤n. Let V be a 𝐂-vector space equipped with p endomorphisms N1,…,Np. Let (X,D) be the log-pair (Δn,⋃1≤k≤pDk) where

Dk=Δk-1×{0}×Δn-k⊂Δn.

Fix a system of coordinates x1,…,xn on Δn coming from the choice of a coordinate on Δ and consider the holomorphic vector bundle V:=V⊗𝐂𝒪X on X endowed with the connection with logarithmic poles along D defined by

∇=d-∑1≤k≤p12⁢i⁢π⋅Nk⋅d⁢xkxk

where Nk denotes by abuse of notation the induced endomorphisms of V.

The connection is integrable if and only if the Nk commute pairwise.

The group isomorphism π1⁢(Δ-{0})=𝐙, in which the counter-clockwise generator loop corresponds to 1∈𝐙, induces an isomorphism π1⁢(X-D)=𝐙p. When the connection is integrable, the monodromy of the associated complex local system on X-D along the element of π1⁢(X-D) corresponding to the k-th base-vector of 𝐙p through the preceding isomorphism is given by Tk=exp⁡(Nk).

By the classical Riemann–Hilbert correspondence, as a holomorphic vector bundle endowed with an integrable connection is completely determined up to isomorphism by its monodromy, all holomorphic vector bundles with an integrable connection on Δn-⋃1≤k≤pDk are obtained by restricting a connection with logarithmic poles along D of the precedent type.

A.1.2 Real and integral structures

Definition A.6.

A real (resp. integral) structure on an element (V,∇)∈VClog⁢(X,D) is a real (resp. integral) sub-local system ℒ of

V|U∇:=ker⁡(∇|U:V|U→ΩU1⊗𝒪UV|U).

such that

ℒ⊗𝐑𝐂=V|U∇ (resp.ℒ⊗𝐙𝐂=V|U∇).

A.1.3 Residue

Let (X,D) be a log-pair and (V,∇)∈VClog⁢(X,D). For any irreducible component Dk of D there is an associated Poincaré residue map

Rk:ΩX1⁢(log⁡D)→𝒪Dk.

It is an 𝒪X-linear map. The map (Rk⊗Id)∘∇ induces an 𝒪Dk-linear endomorphism: resDk⁢(∇)∈End⁡(V|Dk). The residue resDk⁢(∇) is an endomorphism of V|Dk as a vector bundle (i.e. it has constant rank). The endomorphisms resDk⁢(∇)x∈V⁢(x) for every x∈Dk have the same characteristic polynomials (cf. [7]). The residue is called nilpotent if the eigenvalues are all zero.

The full subcategory VClognil⁢(X,D) of VClog⁢(X,D) formed by vector bundles on X with an integrable connection with logarithmic poles along D and nilpotent residues is stable by all the functors considered above.

Example A.7 (Example A.5 continued).

The residue of ∇ on Dk is the endomorphism -12⁢i⁢π⋅Nk|Dk∈End⁡(V|Dk).

A.1.4 Deligne’s canonical extension

Theorem A.8 (Deligne, Manin, cf. [7]).

Let (X,D) be a log-pair and V a holomorphic vector bundle on X-D equipped with an integrable connection ∇ such that the corresponding complex local system V∇ has unipotent local monodromy around D. There exists a holomorphic vector bundle V~ on X extending V, unique up to unique isomorphism, called Deligne’s canonical extension, such that

the connection ∇ has logarithmic poles along D with respect to the extension V~,
the residues of ∇ with respect to the extension V~ are nilpotent.

Moreover, the association (V,∇)↦(V~,∇) defines a functor which is an equivalence between the category of holomorphic vector bundles on X-D equipped with an integrable connection such that the corresponding complex local system V∇ has unipotent local monodromy around D and the category VClognil⁢(X,D) of holomorphic vector bundles on X equipped with an integrable connection with logarithmic poles along D and nilpotent residues. This functor is exact and compatible with the formation of tensor products, internal hom, dual and pull-back along maps of log-varieties.

If (X,D) is a log-pair, we denote by VClognil⁢(X,D)𝐑 the category whose elements are triplets ℒ¯=(V,∇,ℒ) where (V,∇)∈VClognil⁢(X,D) and ℒ is a real structure on (V,∇). It is an 𝐑-linear abelian category with tensor products.

Corollary A.9.

The functor (V,∇,ℒ)↦ℒ is exact and compatible with the formation of tensor products, internal hom, dual and pull-back along maps of log-varieties, and defines an equivalence of categories between VClognil⁢(X,D)𝐑 and the category of real local systems on U with unipotent local monodromy around D.

A.1.5 Local description of integrable connections

Lemma A.10.

Fix two integers 1≤p≤n and consider the log-pair (X,D), where X=Δn and D=⋃1≤k≤pDk with Dk=Δk-1×{0}×Δn-k⊂Δn. Set U:=X-D. Let (V,∇) be a holomorphic vector bundle on X equipped with an integrable connection with logarithmic poles along D and nilpotent residues.

The action of π1⁢(U) on the complex local system V|U∇ extends to an action of π1⁢(U) on (V,∇) by automorphisms in the category VClognil⁢(X,D) (respectively in the category VClognil⁢(X,D)𝐑 if (V,∇) is endowed with a real structure). If Ti denotes the automorphism of (V,∇) image of ti, then
Ti|Di=exp⁡(-2⁢i⁢π⋅resDi⁢(∇)).
For any system of coordinates x1,…,xn on Δn coming from the choice of a coordinate on Δ, there exists a 𝐂-vector space V equipped with p commuting nilpotent endomorphisms N1,…,Np such that (V,∇) is isomorphic to the holomorphic vector bundle V⊗𝐂𝒪Δn endowed with the integrable connection with logarithmic poles along D defined by (cf. Example A.5)
∇=d-∑1≤k≤p12⁢i⁢π⋅Nk⋅d⁢xkxk.

Proof.

For (i) see [7, Proposition 3.11] and [11, Section 1]. Assertion (ii) is a direct application of Theorem A.8. ∎

B Resolution of singularities and extension of rational maps

B.1 Hironaka’s theorem

If f:X′→X is a birational morphism between complex algebraic varieties, there is a biggest open subset U of X such that f induces an isomorphism f-1⁢(U)→∼U. We call the closed subset Exc⁢(f):=X′-f-1⁢(U)⊆X′ the exceptional locus of f.

Theorem B.1 (Hironaka [19]).

Let X be a complex algebraic variety and Z be a proper closed subset of X. Then there exist a smooth algebraic variety X′ and a proper birational morphism f:X′→X such that the closed subset Exc⁢(f)∪f-1⁢(Z) is the support of a simple normal crossing divisor and Exc⁢(f)⊆f-1⁢(Xsing∪Z).

Corollary B.2.

Let g:Y→X be a proper holomorphic map between two smooth complex algebraic varieties X and Y. Let σ:X⇢Y be a rational section of g. Then there exist a smooth algebraic variety X′, a proper birational morphism Φ:X′→X and a morphism σ′:X′→Y such that σ∘Φ=σ′.

Moreover, given any proper closed subset Z of X, we can choose X′ and σ′ such that Φ-1⁢(Z) is a simple normal crossing divisor.

B.2 Proof of Lemma 3.2

To prove Lemma 3.2, we keep the notations of the statement. For any k, 1≤k≤rank⁡(E), we denote by π:Gr⁢(E,k)→X the relative Grassmannian of k-dimensional subspace of E, whose fiber over x∈X is the Grassmannian Gr⁢(Ex,k) of k-dimensional vector subspaces of Ex, and by Vk→Gr⁢(E,k) the tautological vector bundle of rank k over Gr⁢(E,k) (this naive definition will be sufficient for our purpose).

Denote by r the generic rank of ℱ. It follows from the hypothesis that there exists a rational section σ:X⇢Gr⁢(E,r) of π:Gr⁢(E,r)→X. By Lemma B.2, there exists a smooth complex algebraic variety X′ and a proper birational map f:X′→X such that the composition σ∘f:X′⇢Gr⁢(E,r) is actually a holomorphic map, therefore defining a holomorphic subvector bundle G of f*⁢E of rank r. We denote by 𝒢 the corresponding locally free sheaf. As the sheaf (f*⁢ℰ)/𝒢 is locally free and the composition f*⁢ℱ→f*⁢ℰ→(f*⁢ℰ)/𝒢 is generically zero, this composition is in fact everywhere zero, showing that the map f*⁢ℱ→f*⁢ℰ factors:

By definition, the map f*⁢ℱ→𝒢 is the identity on the preimage of the (open) subset in X where ℱ is a subvector bundle of ℰ. It follows from Lemma B.2 that X′ and f can be chosen such that f-1⁢(Z) is a simple normal crossing divisor.

C Segre classes and Segre forms

For any holomorphic vector bundle E on a compact complex manifold X, the total Segre class s⁢(E) is defined as the inverse of the total Chern class c⁢(E).

Let π:ℙ⁢(E)→X be the projective bundle of hyperplanes in E and 𝒪E⁢(1) be the quotient of π*⁢E by its tautological hyperplane subbundle. Set r=rank⁡(E).

Proposition C.1.

For any cohomology class α and any k≥0, we have

π*⁢((π*⁢α)∧(c1⁢𝒪E⁢(1)r-1+k))=sk⁢(E∨)∧α.

In particular,

sk⁢(E∨)=π*⁢(c1⁢𝒪E⁢(1)r-1+k).

Suppose now that E is endowed with a 𝒞∞ hermitian metric h. The Segre forms Sk⁢(E,h) are defined inductively from the Chern forms Ck⁢(E,h) by the relation

Sk⁢(E,h)+C1⁢(E,h)∧Sk-1⁢(E,h)+…+Ck⁢(E,h)=0.

Proposition C.2 (Guler [18]).

For any k≥0, we have

Sk⁢(E∨,h∨)=π*⁢(C1⁢(𝒪E⁢(1),h𝒪E⁢(1))r-1+k),

where h∨ is the dual metric of h and h𝒪E⁢(1) is the metric induced by h on 𝒪E⁢(1).

D Sakai’s dimension of a holomorphic vector bundle

D.1 Sakai’s dimension

Let X be a compact complex manifold and E be a holomorphic vector bundle on X. Sakai defined and studied in [30] a generalization for vector bundles of the Kodaira dimension of a line bundle (this is not really a generalization because Sakai’s dimension never equals minus infinity).

Let Σ⁢(X,E):=⊕m=0∞H0⁢(X,Sm⁢E). This is a commutative graded 𝐂-algebra.

Definition D.1.

Sakai’s dimension of E is by definition the number

σ⁢(X,E):=degtr𝐂⁢Σ⁢(X,E)-rank⁡(E).

It is an integer which belongs to {-rank⁡(E),…,0,…,dim⁡(X)}.

Remark D.2.

Let π:ℙ⁢(E)→X be the projective bundle of hyperplanes in E and 𝒪E⁢(1) be the tautological quotient line bundle. Then π*⁢𝒪E⁢(k)=Sk⁢E for every k≥0. In particular, Σ⁢(ℙ⁢(E),𝒪E⁢(1))=Σ⁢(X,E), and E is big if and only if Σ⁢(X,E)=dim⁡(X).

Remark D.3.

If there is a positive integer m0 such that dim⁡H0⁢(X,Sm0⁢E)>0, then the following estimate holds for large m:

α.mσ+rank⁡(E)-1≤h0⁢(X,Sm.m0⁢E)≤β.mσ+rank⁡(E)-1,

where α and β are positive numbers and σ=σ⁢(X,E). When E is big, we have the following stronger statement: there exist c>0 and j0≥0 such that

h0⁢(X,Sj⁢E)≥c.jdim⁡(X)+rank⁡(E)-1 for all ⁢j≥j0.

D.2 Logarithmic cotangent dimension

Definition D.4.

Let U be a smooth complex algebraic variety. The number

λ⁢(U):=σ⁢(X,ΩX1⁢(log⁡D))∈{-dim⁡(U),…,0,…,dim⁡(U)}

does not depend on the choice of a log-compactification (X,D) of U. It is called the logarithmic cotangent dimension of U.

Proposition D.5.

Let f:U→V be a holomorphic map between two smooth complex algebraic varieties.

If f is a finite étale cover or a proper birational morphism, then λ⁢(U)=λ⁢(V).
If f is dominant, then λ⁢(U)≥λ⁢(V)+(dim⁡(V)-dim⁡(U)).

Proof.

For compact U and V, the statements can be found in [30, Proposition 7 and Theorem 1]. The same proofs work for the general case, except when f is étale. In this case, the proof for non-compact U and V is a bit more subtle, so we briefly sketch it. Assume f:U→V is étale and let (X,D) and (Y,E) be good compactifications of U and V such that f extends to a morphism of log-pairs f:(X,D)→(Y,E). Let

ϕ:f*⁢ΩY1⁢(log⁡E)→ΩX1⁢(log⁡D)

be the associated map of 𝒪X-modules and Z be the closed subset of X where ϕ is not an isomorphism. This set is defined by the vanishing of det⁡(ϕ), in particular it is a divisor in X (here we use that the logarithmic cotangent bundles are locally free). By [20, Theorem 6.1.6], Z is contracted by f, i.e. codimY⁡(f⁢(Z))≥2.

We can now argue as in [35, Theorem 5.13]. As the logarithmic cotangent dimension increases in étale covers (this follows from the second case of the proposition, i.e. when f is dominant), we can assume from the beginning that f:U→V is a Galois étale cover with group G. If λ⁢(U)=-dim⁡(U), then the trivial inequality λ⁢(U)≥λ⁢(V) implies that λ⁢(U)=λ⁢(V). Suppose now that λ⁢(U)>-dim⁡(U). Let ω1,…,ωN be homogeneous elements of Σ⁢(X,ΩX1⁢(log⁡D)) such that Σ⁢(X,ΩX1⁢(log⁡D)) is an algebraic extension of the field generated over 𝐂 by the ωi. For every i, define the Sk⁢(ωi) by

∏g∈G(X-g*⁢ωi)=Xn+S1⁢(ωi)⋅Xn-1+⋯+Sn⁢(ωi),

where X is a variable and n is the order of G. The Sk⁢(ωi) are meromorphic symmetric forms on X with no poles on X-Z. As they are G-invariant by definition, they are pull-back of symmetric forms defined on Y-f⁢(Z) which extend to Y by Hartogs’ extension theorem (recall that ΩY1⁢(log⁡E) is locally free and codimY⁡(f⁢(Z))≥2). As the field 𝐂⁢(ω1,…,ωN) is a finite extension of the field generated over 𝐂 by the Sj⁢(ωi), we obtain that λ⁢(U)≤λ⁢(V). The other inequality is trivial. ∎

Acknowledgements

It is a great pleasure to thank Bruno Klingler for many enlightening discussions and his comments on a first version of this paper. I would like also to thank the referee for his help in improving the readability of the paper.

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Received: 2014-07-03

Revised: 2015-11-06

Published Online: 2016-03-02

Published in Print: 2018-10-01

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