This article is concerned with the convexity properties of universal covers of projective varieties. We study the relation between the convexity properties of the universal cover of X and the properties of the pullback map sending vector bundles on X to vector bundles on its universal cover. Our approach motivates a weakened version of the Shafarevich conjecture. We prove this conjecture for projective varieties X whose pullback map identifies a nontrivial extension of a negative vector bundle $V$ by the trivial line bundle with the trivial extension. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map of vector bundles is almost an imbedding. Our methods also give a new proof of the vanishing of the first cohomology for negative vector bundles $V$ over a compact complex manifold $X$ whose rank is smaller than the dimension of X.
We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseaux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and we prove a weak form of our conjecture for the nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our main tools is Jonsson's limit theorem for tropical varieties.
The goal of this note is to affirm a local version of the conjecture of Nisse--Sottile [NS16] on higher convexity of complements of tropical varieties, while providing a family of counter-examples for the global Nisse--Sottle conjecture in any codimension and dimension higher than 1. Moreover, it will be shown that, surprisingly, this family also provides a family of counter-examples for the generalized Hodge conjecture for positive currents in these dimensions, and gives rise to further approximability obstruction.
Atiyah proved that the moment map image of the closure of an orbit of a complex torus action is convex. Brion generalized this result to actions of a complex reductive group. We extend their results to actions of a maximal solvable subgroup.
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.
In this paper, we clarify the mistakes made in the former article entitled "Vanishing Theorems on Toric Varieties in Positive Characteristic". On the other hand, we use the positive characteristic method to reprove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties over a field of arbitrary characteristic, where concerned Weil divisors are torus invariant.
We prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of Kawamata's theorems: a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the Albanese map when the Frobenius stable Kodaira dimension is zero.
In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \mathbb{Q}-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.
We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the Bott-Steenbrink-Danilov vanishing theorem.
We continue our study on smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying $χ(X, ω_X)=0$. We formulate a conjectural characterization of such varieties and prove this conjecture when the Albanese variety has only three simple factors.
For perverse sheaves K on abelian varieties X defined over a finitely generated field F we prove that the Euler-Poincare characteristic (defined for the extension of K to the algebraic closure of F) is non-negative.
In [GHKK18], Gross-Hacking-Keel-Kontsevich discuss compactifications of cluster varieties from "positive subsets" in the real tropicalization of the mirror. To be more precise, let $\mathfrak{D}$ be the scattering diagram of a cluster variety $V$ (of either type -- $\mathcal{A}$ or $\mathcal{X}$), and let $S$ be a closed subset of $\left(V^\vee\right)^{\text{trop}}(\mathbb{R})$ -- the ambient space of $\mathfrak{D}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its $\mathbb{N}$-dilations define an $\mathbb{N}$-graded subalgebra of $Γ(V, \mathca…
The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.
In this article we prove some strong vanishing theorems on K3 surfaces. As an aplication of them, we obtain higher syzygy results for K3 surfaces and Fano varieties.
We correct the proof and slightly strengthen a Kodaira-type vanishing theorem for singular varieties originally due to Jaffe and the first author. Specifically, we show that if $L$ is a nef and big line bundle on a projective variety of characteristic zero, the $i^{\text{th}}$ cohomology of $L^{-1}$ vanishes for $i$ in a range determined by the depth and dimension of the singular locus.
We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.
Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context. We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to $GL_n$ and $G_2$. This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur's inequality for $GL_n$ and $G_2$ respectively. It is known that the latter imply the non-emptiness of some affine Deligne-Lu…
We present an approach to Green-Lazarsfeld's generic vanishing combining gaussian maps and the Fourier-Mukai transform associated to the Poincarè line bundle. As an application we prove the Generic Vanishing Theorem for all normal Cohen-Macaulay subvarieties of abelian varieties over an algebraically closed field. This paper is a reworking of part of arXiv:math/0310026v2 [math.AG], which contained a mistake.
We establish a, and conjecture further, relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of certain sheaf cohomology. The main ingredient is the Generic Vanishing criterion of math.AG/0608127, based on the Fourier-Mukai transform.
In this Note we prove the vanishing of (twisted) Koszul cohomology groups $K_{p,1}$ of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property $(N_p)$.
The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $Θ$ a polarization of type $(1,...,1,2)$, then a general pair $(X,Θ)$ is log canonical
This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is applied to the construction of holomorphic functions on the universal covering of the complex projective variety .
The purpose of this paper is to show that the third unramified cohomology with divisible coefficients of a smooth projective geometrically rational threefold over a finite field must vanish under $\Z_{\ell}$-exactness Hard Lefschetz condition.
We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.
We extend slightly the results of Evens-Mirković, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.
We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.
We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in di…
This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Kodaira dimension. Finally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay.
We use Cox's description for sheaves on toric varieties and results about the local cohomology with respect to monomial ideals to give a characteristic free approach to vanishing results on arbitrary toric varieties. As an application, we give a proof of a strong form of Fujita's conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.
We consider a complete nonsingular variety $X$ over $\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes Ω_X^j(\log D)\big) = 0$ for any nef line bundle $L$ on $X$ and all $i < j - c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the vanishing of $H^i(X, L \otimes Ω_X^j)$ for $L$ ample and $i > j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety.
This paper is about sheaf cohomology for varieties (schemes) in characteristic $p>0$. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre's vanishing theor…
We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.
Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal smooth projective variety $X$ of Kodaira dimension $κ(X)=\dim X-1$ admits a holomorphic 1-form without zero if and only if there is a smooth morphism from $X$ to an elliptic curve. Furthermore, for a general smooth projective variety (not necessarily minimal) $X$ of Kodaira codimension one, we give a structure theorem for $X$ given that $X$ admits a holomorphi…
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor's book, we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues' theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. In application, we obtain the description of the local components of cohomological automorphic representations of certains unitary groups, a global Jacquet-Lang…
Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with optimal slice properties in any given $G_0$-orbit. For an open $G_0$- orbit $D$, given $p$ in the boundary of $D$, a variety $Y\setminus D$ containing $p$ with maximal dimension with respect to the compact cycles in $D$ is constructed. The method of incidence varieties then yields information on the complex geometry of the associated cycle space. In particular, ho…
The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.
We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree.
Consider a root system $R$ and the corresponding toric variety $V_R$ whose fan is the Weyl fan and whose lattice of characters is given by the root lattice for $R$. We prove the vanishing of the higher cohomology groups for certain line bundles on $V_R$ by proving a purely combinatorial result for root systems. These results are related to a converse to Mazur's Inequality for (simply-connected) split reductive groups.
A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.
Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd…
Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.
We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.
Let E be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section $s \in Γ(\cal E)$ whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X - r: = n -r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.
We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is b…
