Kodaira dimension and zeros of holomorphic one-forms, revisited
KKODAIRA DIMENSION AND ZEROS OF HOLOMORPHICONE-FORMS, REVISITED
MADS BACH VILLADSEN
Abstract.
We give a new proof that every holomorphic one-form on a smoothcomplex projective variety of general type must vanish at some point, firstproven by Popa and Schnell using generic vanishing theorems for Hodge modules.Our proof relies on Simpson’s results on the relation between rank one Higgsbundles and local systems of one-dimensional complex vectors spaces, and thestructure of the cohomology jump loci in their moduli spaces. Introduction
In [PS14], Popa and Schnell showed that any holomorphic one-form on a smoothprojective variety of general type must vanish at some point, a conjecture of Hacon-Kovács and Luo-Zhang [HK05; LZ05]. Wei [Wei20] later gave a slightly simplifiedargument (as well as a generalization to log-one-forms). Both proofs use thedecomposition theorem and various vanishing theorems for Hodge modules. Wegive a new approach using only classical Hodge theory, namely the rank one case ofSimpson’s correspondence between Higgs bundles and local systems, and his resultson the structure of cohomology jump loci of local systems. Our approach shouldthus be much more accessible than either of the two previous proofs.As in [PS14], we will prove the following more precise result.
Theorem 1.1 ([PS14, Theorem 2.1]) . Let X be a smooth complex projective varietyand f : X → A a morphism to an abelian variety. If H ( X, ω ⊗ dX ⊗ f ∗ L − ) (cid:54) = 0 forsome integer d ≥ and some ample line bundle L on A , then for every holomorphicone-form ω on A , the pullback f ∗ ω vanishes at some point of X . The following conjecture of Luo and Zhang [LZ05] follows as in [PS14]. Forvarieties of general type, this shows that every holomorphic one-form must vanishat some point.
Corollary 1.2 ([PS14, Conjecture 1.2]) . Let X be a smooth complex projectivevariety and W ⊆ H ( X, Ω X ) be a linear subspace such that every element of W \ { } is everywhere non-vanishing. Then dim W ≤ dim X − κ ( X ) . Our proof of Theorem 1.1 goes as follows. Let V = H ( A, Ω A ) and Z f = { ( x, ω ) ∈ X × V | f ∗ ω ( T x X ) = 0 } . The goal is to show that the restriction of the projection p : X × V → V to Z f issurjective.We borrow the idea in [PS14], going back to work of Viehweg-Zuo [VZ01], ofconstructing two separate sheaves on V . The first is an ambient sheaf coming from Mathematics Subject Classification. a r X i v : . [ m a t h . AG ] F e b MADS BACH VILLADSEN a cyclic cover of X , which we will show to be locally free. The second is a non-zerosubsheaf coming from X and supported on p ( Z f ) . As the subsheaf is necessarilytorsion free, it must have support equal to V , hence p ( Z f ) = V as desired.Let us outline the argument for why the ambient sheaf is locally free.After base change by an isogeny of A , we can assume that ( ω X ⊗ f ∗ L − ) ⊗ d hasa non-zero section s . Let Y be a resolution of singularities of the associated degree d cyclic cover of X branched along s , and consider the composition h : Y → A .The ambient sheaf is a higher direct image of a complex of sheaves on Y × V ,and the fibres of the complex over points in V are Dolbeault complexes of certainHiggs bundles on Y . Using Simpson’s results [Sim92; Sim93] relating Higgs bundlesto local systems and analyzing cohomology jump loci in the moduli space of localsystems, we show that the hypercohomology groups of these Dolbeault complexeshave constant dimension over V . This gives the result by Grauert’s theorem onlocally free direct images. Acknowledgements.
I would like to thank my advisor Christian Schnell for point-ing me to this topic, and for many enlightening remarks. I also thank Ben Wu forrelated discussions and useful comments on drafts of the paper.2.
The proof
Fix a smooth projective variety X over the complex numbers and a morphism f : X → A to an abelian variety throughout. Let V = H ( A, Ω A ) be the vector spaceof holomorphic one-forms on A , and let S = Sym V ∗ be the graded coordinate ring ofthe vector space V . For an integer i , let S • + i denote S as a graded module over itself,with grading shifted by i , and let C X, • be the complex of graded O X ⊗ S -modulesgiven by O X ⊗ S •− g → Ω X ⊗ S •− g +1 → · · · → Ω nX ⊗ S •− g + n , in degrees − n to , where n = dim X, g = dim A , and the differential is induced bythe map O X ⊗ V → Ω X given by φ ⊗ ω (cid:55)→ φf ∗ ω . In a basis ω , . . . , ω g of V withdual basis s , . . . , s g of S , the differential is given by θ ⊗ s (cid:55)→ g (cid:88) i =1 ( θ ∧ f ∗ ω i ) ⊗ s i s. We denote the associated complex of vector bundles on X × V by C X . Lemma 2.1 ([PS14, Lemma 14.1]) . The support of C X is equal to Z f = { ( x, ω ) ∈ X × V | f ∗ ω ( T x X ) = 0 } . For α ∈ Pic ( A ) , let C αX = C X ⊗ p ∗ f ∗ α and C αX, • = C X, • ⊗ f ∗ α , where p is theprojection X × V → X . The sheaves R i p ∗ C αX on V are then supported on p ( Z f ) for all i ; recall that we are trying to show that p ( Z f ) = V . We will show that thesesheaves are locally free for general α in Proposition 2.2 below. As we will see, thefibres of C αX over V are related to certain Higgs bundles on X .Recall that a Higgs bundle on X is a vector bundle E together with a morphismof coherent sheaves θ : E → Ω X ⊗ E , the Higgs field, satisfying θ ∧ θ = 0 . Given a ODAIRA DIMENSION AND ZEROS OF HOLOMORPHIC ONE-FORMS, REVISITED 3
Higgs bundle, we get a holomorphic Dolbeault complex E θ ∧ −−→ E ⊗ Ω X → · · · → E ⊗ Ω nX . Simpson’s non-abelian Hodge theorem [Sim92] associates to each Higgs bundle ( E, θ ) (satisfying some conditions on stability and Chern classes) a local system C ( E,θ ) of complex vector spaces, and shows that Dolbeault cohomology H k Dol ( X, E, θ ) = H k ( X, E θ ∧ −−→ E ⊗ Ω X → · · · → E ⊗ Ω nX ) , the hypercohomology of the Dolbeault complex, is isomorphic to the cohomology of C ( E,θ ) .We will only need the rank one case; see also the lecture notes [Sch13, Lectures17-18] for a concrete treatment of this case, and the associated Hodge theory.In the rank one case, a Higgs bundle is just a line bundle together with aholomorphic one-form. The stability condition in Simpson’s theorem is alwayssatisfied for line bundles, and the condition on Chern classes is simply that the firstChern class vanishes in H ( X, C ) ; let Pic τ ( X ) be the space of line bundles satisfyingthis condition. Let then M Dol ( X ) = Pic τ ( X ) × H ( X, Ω X ) , and let M B ( X ) denotethe moduli space of local systems of one-dimensional complex vector spaces on X .Then Simpson’s correspondence, mapping a rank one Higgs bundle to the associatedlocal system, takes the form of a real analytic isomorphism M Dol ( X ) ∼ = M B ( X ) .For each k and m , consider the cohomology jump loci Σ km ( X ) = {L ∈ M B ( X ) | dim H k ( X, L ) ≥ m } Σ km ( X ) Dol = { ( E, θ ) ∈ M Dol ( X ) | dim H k Dol ( X, E, θ ) ≥ m } of local systems and Dolbeault cohomology of Higgs bundles. These loci get mappedto each other under Simpson’s correspondence.Using this relationship, Simpson [Sim93] proves that every irreducible componentof these loci is a linear subvariety or, in his terminology, a translate of a triple torus(in fact a torsion translate, though we will not need that). A triple torus is a closed,connected, algebraic subgroup N of M B ( X ) such that the corresponding subgroupof M Dol ( X ) (which we will also refer to as a linear subvariety) is also algebraic (thisis equivalent to the usual definition, involving also the de Rham moduli space, by[Sim93, Lemma 2.1]). A linear subvariety is thus a subset of M B ( X ) of the form {L ⊗ N | N ∈ N } where N is a triple torus and L ∈ M B ( X ) a local system.Simpson [Sim93, Lemma 2.1] shows that a triple torus is of the form g ∗ M B ( T ) for a map g : X → T to an abelian variety, where g ∗ : M B ( T ) → M B ( X ) denotespullback of local systems. It follows that a linear subvariety in M Dol ( X ) is a translateof a subset of the form g ∗ Pic ( T ) × g ∗ H ( T, Ω T ) for g : X → T a morphism to anabelian variety. In particular, a linear subvariety is either the entire moduli space, ormaps to a proper subvariety of Pic ( X ) under the projection M Dol ( X ) → Pic ( X ) that forgets the Higgs field.The following proposition is the main new ingredient in the proof. Note thatthis proposition is valid for an arbitrary morphism f : X → A , not just those thatsatisfy the hypotheses of Theorem 1.1. Proposition 2.2.
For general α ∈ Pic ( A ) , the higher direct image sheaves R i p ∗ C αX are locally free on V for all i . MADS BACH VILLADSEN
Proof.
We will show that for general α , the dimensions of the hypercohomology H i ( X × { v } , C αX | X ×{ v } ) of C αX on fibres of p are constant in v . The result followsby a version of Grauert’s theorem on locally free direct images for complexes ofsheaves [EGAIII, Proposition 7.8.4]Note that for any fibre X × { v } of p for v ∈ V , the restriction of C αX to the fibreis the Dolbeault complex f ∗ α ∧ f ∗ v −−−→ f ∗ α ⊗ Ω X → · · · → f ∗ α ⊗ Ω nX . of the Higgs bundle ( f ∗ α, f ∗ v ) . The Dolbeault cohomology of these Higgs bundlesis governed by the cohomology jump loci Σ km ( X ) Dol , of which only finitely many arenonempty by algebraicity. Each irreducible component of the nonempty ones is alinear subvariety, so it suffices to show that for each linear subvariety S of M Dol ( X ) ,the set { α } × f ∗ V is either entirely contained in S or entirely disjoint from it, forgeneral α .Let then φ = f ∗ : M Dol ( A ) → M Dol ( X ) , and suppose S ⊂ M Dol ( X ) is a linearsubvariety. We observe that if N is a triple torus, then the connected componentof φ − ( N ) is again a triple torus; it follows that φ − ( S ) is either empty, or a finiteunion of linear subvarieties. If φ − ( S ) = M Dol ( A ) then { α } × f ∗ V ⊂ S for any α ∈ Pic ( A ) . If S is a proper subset of M Dol ( A ) , it suffices to take α to be outsidethe image of φ − ( S ) in Pic ( A ) under the projection M Dol ( A ) → Pic ( A ) . (cid:3) If we could show that one of the sheaves R i p ∗ C αX were nontrivial on V , underthe hypotheses of Theorem 1.1, we would now be done. Unfortunately we cannot,but instead we make use of a covering construction as in [PS14]. Lemma 2.3 ([PS14, Lemma 11.1]) . Suppose ω ⊗ dX ⊗ f ∗ L − has a nonzero sectionfor some d and some ample line bundle L on A . For an isogeny φ : A (cid:48) → A , define f (cid:48) : X (cid:48) → A (cid:48) by base change of f . For an appropriately chosen φ , there exists anample line bundle L (cid:48) on A (cid:48) such that ( ω X (cid:48) ⊗ f (cid:48)∗ L (cid:48)− ) ⊗ d has a nonzero section. Assume now the hypotheses of Theorem 1.1. Note that zero loci of one-forms arenot affected by étale covers, so if we can prove the theorem for f (cid:48) : X (cid:48) → A (cid:48) , thenthe desired conclusion also follows for f : X → A .In particular, replacing f by this f (cid:48) , we can now assume without loss of generalitythat B ⊗ d has a nonzero section s for B = ω X ⊗ f ∗ L − . Let Y be a resolution ofsingularities of the d -fold cyclic cover π : X d → X ramified along Z ( s ) , giving usthe following maps: Y X d XA φh π f By construction, X d = Spec (cid:76) d − i =0 B − i , so π ∗ π ∗ B = (cid:76) d − i = − B − i . This has asection in the i = 0 term, and the corresponding section of π ∗ B gives a morphism φ ∗ B − → O Y , an isomorphism away from Z ( s ) . Together with pullback of forms,this gives injective morphisms φ ∗ ( B − ⊗ Ω kX ) → Ω kY . As O X → φ ∗ O Y is injective,the corresponding morphisms B − ⊗ Ω kX → φ ∗ Ω kY ODAIRA DIMENSION AND ZEROS OF HOLOMORPHIC ONE-FORMS, REVISITED 5 on X are also injective.Note that we get a complex C Y, • of graded O Y ⊗ S -modules using the morphism h : Y → A , constructed in the same way that C X, • was constructed starting from f above Lemma 2.1.We give a slightly modified version of [PS14, Lemma 13.1]. Lemma 2.4.
The morphisms above induce a morphism of complexes of graded O X ⊗ S -modules B − ⊗ C X, • → R φ ∗ C Y, • Proof.
The morphisms φ ∗ ( B − ⊗ Ω kX ) → Ω kY commute with the differentials on Y ,giving φ ∗ ( B − ⊗ C X, • ) → C Y, • Using the projection formula and the morphism O X → R φ ∗ O Y , pushing forward to X gives the desired composition B − ⊗ C X, • → ( B − ⊗ C X, • ) ⊗ L R φ ∗ O Y → R φ ∗ C Y, • (cid:3) Proof of Theorem 1.1.
We must show that Z f surjects onto V under the projection p : X × V → V .Let α ∈ Pic ( A ) be a general element. Then Lemma 2.4 gives, after twisting by f ∗ α and pushing forward to V , a morphism R p ∗ ( p ∗ B − ⊗ C αX ) → R p ∗ C αY where p : X × V → X is the first projection, and p , by abuse of notation, is used forboth of the projections X × V → V and Y × V → V . Let F be the image of theinduced map R p ∗ ( p ∗ B − ⊗ C αX ) → R p ∗ C αY .As α is general, each R i p ∗ C αY is locally free by Proposition 2.2. In particular F is torsion free. Since C X is supported on Z f , F is supported on p ( Z f ) , so itsuffices to show that F is non-zero.Let k = g − n . Then C X,k = ω X and C Y,k = ω Y , and the morphism B − ⊗ C X,k → R φ ∗ C Y,k from Lemma 2.4 is just the morphism of sheaves f ∗ L = B − ⊗ ω X → φ ∗ ω Y constructed before the lemma. Indeed R φ ∗ ω Y = φ ∗ ω Y since φ is generically finite,by results of Kollár [Kol86].After twisting by α , the morphism B − ⊗ C αX,k → R φ ∗ C αY,k thus gets identifiedwith f ∗ ( L ⊗ α ) → φ ∗ ω Y ⊗ f ∗ α . For the graded S -module F • = H ( V, F ) , it followsthat F k ∼ = H ( X, f ∗ ( L ⊗ α )) since the pushforward to V preserves injectivity. But f ∗ ( L ⊗ α ) has non-zero sections; otherwise all sections of its pushforward f ∗ O X ⊗ L ⊗ α to A would vanish, which would imply that X is contained in a general translate ofa hyperplane section of A , a contradiction. Thus F is non-zero. (cid:3) References [EGAIII] Alexander Grothendieck. “Éléments de géométrie algébrique. III. Étudecohomologique des faisceaux cohérents. II”.
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Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
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