A Nonvanishing Theorem for Q-divisors

On the abundance theorem for numerically trivial canonical divisors in positive characteristic

In this paper, we prove the abundance theorem for numerically trivial canonical divisors on strongly $F$-regular varieties, assuming that the geometric generic fibers of the Albanese morphisms are strongly $F$-regular.

Base point free theorems--saturation, b-divisors, and canonical bundle formula--

We reformulate base point free theorems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the base point free theorem in Fukuda's paper: On numerically effective log canonical divisors.

Higher direct images of log canonical divisors and positivity theorems

In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollár's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita-Kawamata's semi-positivity theorem. By this semi-positivity theorem, we generalize Kawamata's positivity theorem and apply it to the study of a log canonical bundle formula. The final section is an appendix, which is a res…

Adelic divisors on arithmetic varieties

In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski decompositions for arithmetic divisors on arithmetic surfaces, to the case of the adelic arithmetic divisors.

Singularities of divisors on flag varieties via Hwang's product theorem

We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety $X=G/P$ and an effective $\mathbb{Q}$-divisor $D$ stable with respect to a Borel subgroup the pair $(X,D)$ is Kawamata log terminal if and only if $\lfloor D\rfloor=0$.

Vanishing theorems for linearly obstructed divisors

We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus. We extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, we describe linear systems of divisors in $\mathbb{P}^n$ blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus.

Singularities of the Prym theta divisor

For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym varieties. Applications of this theorem to cubic threefolds, and Prym varieties of dimension five, are also considered.

Effective nonvanishing, effective global generation

We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.

Infinitesimal Torelli Theorem for regular surfaces with very ample canonical divisor

The article proves the Infinitesimal Torelli theorem for surfaces subject to the following conditions: 1) the canonical bundle of a surface is ample and generated by its global sections, 2)the geometric genus $p_g \geq 4$, 3) the irregularity $q=0$ . The main novelty is a realization of the Kodaira-Spencer classes lying in the kernel of the cohomology cup-product controlling the derivative of the period map of weight 2 in the category of the coherent sheaves of a surface.

Decomposition, purity and fibrations by normal crossing divisors

We give a simple geometric proof of the decomposition theorem in terms of Thom-Whitney stratifications by reduction to fibrations by normal crossings divisors over the strata and explain the relation with the local purity theorem an unpublished result of Deligne and Gabber.

On projective varieties with nef anticanonical divisors

We prove a structure theorem for projective varieties with nef anticanonical divisors.

Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor

We prove a structure theorem for the differential operator in the 0-term of the ${\cal V}$-filtration with respect to a free divisor. Using this theorem, we give a formula for the logarithmic de Rham complex in terms of ${\cal V}_0$-modules. We also give a sufficient condition for perversity of the logarithmic de Rham complex.

Tropical Independence I: Shapes of Divisors and a Proof of the Gieseker-Petri Theorem

We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.

Logarithmic Comparison Theorems

In this paper we study the comparison between the logarithmic and the meromorphic de Rham complexes along a divisor in a complex manifold. We focus on the case of free divisors, starting with the case of locally quasihomogeneous divisors, and we explain how D-module theory can be used for this comparison.

Log abundance of the moduli b-divisors of lc-trivial fibrations

We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. The result follows from a theorem on the restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms, which allows us to treat $\mathbb{R}$-divisors and proper morphisms possibly with disconnected fibres. We also prove a theorem on extending a finite cover over a closed subvariety to that over a variety in arbitrary characteristic.

Hodge ideals for Q-divisors: birational approach

We develop the theory of Hodge ideals for Q-divisors by means of log resolutions, extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteria and a global vanishing theorem, as well as other analogues of standard results from the theory of multiplier ideals, and we derive a new local vanishing theorem. The connection with the V-filtration is analyzed in a sequel.

Ample subvarieties and q-ample divisors

We introduce a notion of ampleness for subschemes of higher codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.

Nef and abundant divisors, semiampleness and canonical bundle formula

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.

Relations among divisors on the moduli space of curves with marked points

We prove relations among the classes of certain divisors on the moduli spaces of curves with marked points, generalizing the Brill-Noether Ray Theorem of Eisenbud and Harris.

The motive of a smooth Theta divisor

We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $Θ$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $Θ$.

Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), R…

A quick proof of nonvanishing for asymptotic syzygies

We give a quick new approach to the main cases of the nonvanishing theorems of first and third authors concerning the asymptotic behavior of the syzygies of a projective variety as the positivity of the embedding line bundle grows. Specifically, we present a surprisingly elementary and concrete proof of the asymptotic nonvanishing of Veronese syzygies, and we obtain effective results for arithmetically Cohen-Macaulay varieties. The idea is that one can reduce the statements to some simple computations with monomials.

Ample Weil divisors

We define and study positivity (nefness, amplitude, bigness and pseudo-effectiveness) for Weil divisors on normal projective varieties. We prove various characterizations, vanishing and non-vanishing theorems for cohomology, global generation statements, and a result related to log Fano.

Remarks on the cone of divisors

We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of the numerical linear systems in order to simplify the argument on the Zariski decompositions.

Logarithmic vanishing theorems for effective $q$-ample divisors

Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,Ω^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$

Duality of Gauss-Manin systems associated to linear free divisors

We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these D-modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.

Logarithmic Comparison Theorem and Euler homogeneity for free divisors

This paper has been withdrawn by the author since the proof of Lemma 8 is not correct.

相干概形上的除子与齐性空间

Divisors on coherent schemes and homogeneous spaces

本文研究相干基上齐性空间上除子的正性与可延展性，发展了局部相干概形上的估值除子理论，建立了有效估值除子与秩为一的反射sheaf的精确对应关系，推导出非Noetherian的Ramanujam-Samuel定理。

关于弱丰沛除子的Kodaira--Saito消没定理

On the Kodaira--Saito vanishing theorem of weakly ample divisor

本文在光滑射影簇上，针对层上同调意义下弱丰沛的既约有效除子，引入并证明了相应的Kadaira--Saito消没定理。

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.

Vanishing and injectivity for R-Hodge modules and R-divisors

We prove the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in \cite{Wu15}. In particular, the injectivity generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for R-Hodge modules in the normal crossing case.

Smooth contractible threefolds with hyperbolic $\mathbb{G}_{m}$-actions via ps-divisors

The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group $\mathbb{G}_{m}\simeq\mathbb{C}^{\text{*}}$, using the language of polyhedral divisors developed by Altmann and Hausen as generalization of $\mathbb{Q}$-divisors.

对数底数变换定理与log极子的光滑下降

Logarithmic base change theorem and smooth descent of positivity of log canonical divisor

本文基于对数底数变换定理，证明了log极子正性通过光滑投影映射下降，并推广了Viehweg-Zuo的结果，同时验证了Popa关于光滑代数纤维空间对数Kodaira维数超加性的猜想。

Finiteness theorems for complements of large divisors

We prove finiteness results on integral points on complements of large divisors in projective varieties over finitely generated fields of characteristic zero. To do so, we prove a function field analogue of arithmetic finiteness results of Corvaja-Zannier and Levin using Wang's function field Subspace Theorem. We then use a method of Evertse-Győry for concluding finiteness of integral points over finitely generated fields from known finiteness results over number fields.

On Smooth Divisors of a Projective Hypersurface

We prove an effective bound for the degree of a smooth divisor of a hypersurface of P^n, n>4 (projective space over an algebraically closed field of characteristic zero). Our result follows from a strong (since the degree of the divisor is not involved) generalization of the "Speciality theorem" of Gruson-Peskine, which we prove to hold for any smooth, subcanonical, codimension two, projective verieties of dimension at least three.

Frobenius split anticanonical divisors

In this note I extend two theorems of Sommese regarding abelian varieties to arbitrary characteristic; that an abelian variety cannot be an ample divisor in a smooth projective variety and that a cone over an abelian variety of dimension at least two is not smoothable.

Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…

The Grothendieck-Lefschetz theorem for normal projective varieties

We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an isomorphism for a general member $Y\in |L|$ provided that $\dim{X}\geq 4$. This is a generalization of the Grothendieck-Lefschetz Theorem, for divisor class groups of singular varieties.

Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties

In this paper, we study obstructions to the Dirichlet property by two approaches: density of non-positive points and functionals on adelic R-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic R-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.

关于纤维化到曲线上的除子的收缩定理

A contraction theorem for divisors fibering over a curve

本文针对纤维化到曲线上的 Q-Cartier 除子，给出了双亚缅收缩存在的充分条件，并作为推论恢复了紧 Kähler 三维簇的收缩结果。

A Lefschetz (1,1) Theorem for normal projective complex varieties

We prove an analogue of the Lefschetz (1,1) Theorem characterizing cohomology classes of Cartier divisors (or equivalently first Chern classes of line bundles) in the second integral cohomology. Let $X$ be a normal complex projective variety. We show that the classes of Cartier divisors in $H^2(X,Z)$ are precisely the classes $x$ such that (i) the image of $x$ in $H^2(X,C)$ (cohomology with complex coefficients) lies in $F^1 H^2(X,C)$ (first level of the Hodge filtration for Deligne's mixed Hodge structure), and (ii) $x$ is Zariski-locally trivial, i.e., there is a covering of $X$ by Zariski o…

超凯勒簇上有效锥的 Lefschetz 型定理

Lefschetz-type theorems for the effective cone on Hyperkähler varieties

本文证明超凯勒簇有效锥的 Lefschetz 型定理，即光滑 ample 除子诱导有效锥同构，并推广至例外除子，从而计算射影化余切丛等有效锥。

Vanishing for Hodge ideals of $\mathbb{Q}$-divisors

Mustaţă a and Popa introduce the notion of Hodge ideals for an effective $\mathbb{Q}$-divisor $D$ and prove a vanishing theorem for Hodge ideals, which generalizes Nadel vanishing for multiplier ideals. However, their proof needs an extra assumption on the existence of $\ell$-roots of the line bundle $\mathscr{O}_X(\ell D)$, which is not necessary for Nadel vanishing. In this paper, we prove that vanishing for Hodge ideals still holds even without this assumption.

Witt除子层的零化定理与一个反例

Vanishing and a counterexample for Witt divisorial sheaves

本研究完善正特征完美域上光滑射影簇的Witt除子层对偶理论，得到更简洁同构，证明了曲面nef且big除子对应Witt除子层的Ramanujam型零化定理，并给出二维下Kawamata-Viehweg型零化的反例。

通过拟稳定除子的图的 Torelli 定理

A Torelli theorem for graphs via quasistable divisors

本文证明图的拟稳定除子偏序集可完全决定图的双连通分支（至多收缩分离边），从而给出 Torelli 定理的组合版本，并将其推广到热带曲线。

A Chebotarev-type density theorem for divisors on algebraic varieties

Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition behaviour of geometrically integral Cartier divisors. As an application, we classify Galois covers among all finite branched covers of a given normal geometrically integral variety $X$ over $k$ by the decomposition behaviour of points of a fixed codimension $r$ with $0 < r < \dim X$.

Constructing new ample divisors out of old ones

We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for constructing new ample divisors. We then propose a conjecture relating continued fractions approximations and Seshadri-like constants of line bundles over rational surfaces. By applying our algorithm recursively we verify our conjecture in many cases and obtain new asymptotic estimates on these constants. Finally, we explain the intuition behind the gluing theo…

Divisorial Cohomology Vanishing on Toric Varieties

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.

超越 b-除子 II -- 单调性定理

Transcendental b-divisors II -- Monotonicity theorem

本文发展了 nef b-除子的一般交理论，扩展了可移动交理论，定义了 b-除子的限制体积，并证明了交积的单调性定量版本，从而得出新的当前体和上同调类体积不等式。

A Riemann-Roch theorem for edge-weighted graphs

We prove a Riemann-Roch theorem for real divisors on edge-weighted graphs over the reals, extending the result of Baker and Norine for integral divisors on graphs with multiple edges.

Minimal model theorem for toric divisors

Minimal model conjecture for a proper variety $X$ is that if $κ(X)\geq 0$, then $X$ has a minimal model with the abundance and if $κ=-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture for a $\D$-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of ambient toric variety. Furthermore, for such a divisor $X$ with $κ(X)\geq 0$ we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of…

Linear free divisors and the global logarithmic comparison theorem

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is r…

Teissier's problem on inequalities of nef divisors over an arbitrary field

Boucksom, Favre and Jonsson establish in [4] an analog of Diskant's inequality in convex geometry for nef and big line bundles on a complete algebraic variety over an algebraically closed field of characteristic zero (Theorem F [4]), from which they deduce a solution (Theorem D [4]) to a problem of Teissier (Problem B [29]) on proportionality of nef divisors. In this paper we extend these results to a complete variety over an arbitrary field k.

射影klt对具有nef反典范除子的结构定理

Structure theorem for projective klt pairs with nef anti-canonical divisor

本文证明射影klt对在有限拟平展覆盖下具有局部平凡有理连通纤维化，基为数值平凡典范除子的klt簇，推广了光滑情形并应用于分解klt Calabi-Yau型簇。