Non-vanishing theorem for generalized log canonical pairs with a polarization

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics

We prove a relative Kawamata Viehweg vanishing type theorem for birational morphisms. We use this to prove a Grauert Riemenschneider theorem over log canonical threefolds without zero dimensional log canonical centers, in residue characteristic p greater than five. In large enough residue characteristics, we prove a Grauert Riemenschneider theorem over threefold log canonical singularities with standard coefficients. These vanishing theorems can also be used to study the depth of log canonical singularities at non log canonical centers, as well as the singularities of the log canonical centers…

关于反典范丛的两个非消失性结果

Two Non-Vanishing results concerning the Anti-Canonical Bundle

本文研究了klt三维fold对在反log典范丛为nef时的非消失性问题，证明了其Kodaira维数非负，并通过建立任意维度均成立的等变非消失性结果来实现这一目标。

Vanishing theorems of Kodaira type for Witt canonical sheaves

Given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf. We also give a generalisation of this vanishing theorem to one of Kawamata-Viehweg type.

Kawamata-Viehweg vanishing and quint-canonical map of a Complex threefold

This paper proves that the 5-canonical map of a smooth minimal 3-fold is birational when the geometric genus is bigger than 2. A combination of the results in this paper and that of Ein-Lazarsfeld-Lee, the possible exceptional cases are very limited. Unfortunately, no counter examples have been found up to now.

Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties

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.

Some non-vanishing results on log canonical pairs of dimension 4

Let $(X,Δ)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $Δ$ an effective $\mathbb{Q}$-divisor, and $K_X+Δ$ nef. We give a non-vanishing criterion for $K_X+Δ$ in dimension $n$ with $X$ uniruled, assuming various conjectures of LMMP in dimensions (up to) $n-1$ or $n$, and a semi-ampleness criterion in the irregular case. In particular, we obtain that if $X$ is a uniruled $4$-fold, then $κ(K_X+Δ)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+Δ$ is semi-ample.

Remarks on the non-vanishing conjecture

We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for rationally connected varieties under assuming Shokurov's ACC conjectures.

Bogomolov-Sommese vanishing on log canonical pairs

Let (X, D) be a projective log canonical pair. We show that for any natural number p, the sheaf (Omega_X^p(log D))^** of reflexive logarithmic p-forms does not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds p. This generalizes a classical theorem of Bogomolov and Sommese. In fact, we prove a more general version of this result which also deals with the orbifoldes géométriques introduced by Campana. The main ingredients to the proof are the extension theorem of Greb-Kebekus-Kovács-Peternell, a new version of the Negativity lemma, the Minimal Model Program, and a resid…

Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic

We show that the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface in large characteristic unless the Iitaka dimension of the round-down of the log canonical divisor is not equal to two. As an application, we prove that a log resolution of a pair of a normal projective surface and a reduced divisor in large characteristic lifts to the ring of Witt vectors when the Iitaka dimension of the log canonical divisor is less than or equal to zero. Moreover, we give explicit and optimal bounds on the characteristic unless their Iitaka dimensions are equal to zero.

Non-vanishing of the Central Derivative of Canonical Hecke L-functions

In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congrue…

A vanishing theorem for the canonical blow-ups of Grassmann manifolds

Let $\mathcal T_{s,p,n}$ be the canonical blow-up of the Grassmann manifold $G(p,n)$ constructed by blowing up the Plücker coordinate subspaces associated with the parameter $s$. We prove that the higher cohomology groups of the tangent bundle of $\mathcal T_{s,p,n}$ vanish. As an application, $\mathcal T_{s,p,n}$ is locally rigid in the sense of Kodaira-Spencer.

A local version of the Kawamata-Viehweg vanishing theorem

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

Kawamata-Viehweg vanishing as Kodaira vanishing for stacks

We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as Kodaira vanishing for stacks.

On Kawamata--Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic

In this paper, we prove a Kawamata--Viehweg type vanishing theorem for smooth Fano threefolds, canonical del Pezzo surfaces and del Pezzo fibrations in positive characteristic.

A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation of the canonical ring for the case of general type. After my lectures many people asked me for a copy of the slides which I used for my lectures. Since my slides were quite sketchy because of the time limitation for the lectures, I promised to post later on a preprint server my detailed notes from which my slides were extracted. Here are my detailed notes giving the techniques and the proof.

Non-Cohen-Macaulay canonical singularities

The purpose of this note is to show that in characteristic two there exists a smooth Fano variety for which the square of the anti-canonical bundle violates Kodaira vanishing. This is used in turn to construct non-Cohen-Macaulay canonical singularities in characteristic two.

On the non-vanishing conjecture and existence of log minimal models

We prove that the non-vanishing conjecture and the log minimal model conjecture for projective log canonical pairs can be reduced to the non-vanishing conjecture for smooth projective varieties such that the boundary divisor is zero.

A vanishing theorem for log canonical pairs

Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld in the smooth case and by Chardin--Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.

Vanishing and injectivity theorems for Hodge modules

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number

Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.

Very ampleness of the canonical bundle of surfaces of type (1, 2, 2) on abelian threefolds

The present work deals with the canonical map of smooth, compact complex surfaces of general type in a polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces …

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.

除子的通用消失猜想

Generic vanishing conjecture for divisors

受Pareschi和Popa的通用消失猜想启发，本文证明了余维数为1情形的对应类比。

Fundamental theorems for semi log canonical pairs

We prove that every quasi-projective semi log canonical pair has a quasi-log structure with several good properties. It implies that various vanishing theorems, torsion-free theorem, and the cone and contraction theorem hold for semi log canonical pairs.

Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties

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.

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.

A vanishing theorem on fake projective planes with enough automorphisms

For every fake projective plane $X$ with automorphism group of order 21, we prove that $H^i(X, 2L)=0$ for all $i$ and for every ample line bundle $L$ with $L^2=1$. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle $K$. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.

Introduction to the log minimal model program for log canonical pairs

We describe the foundation of the log minimal model program for log canonical pairs according to Ambro's idea. We generalize Kollár's vanishing and torsion-free theorems for embedded simple normal crossing pairs. Then we prove the cone and contraction theorems for quasi-log varieties, especially, for log canonical pairs.

曲面上一个广义的非消失定理

A generalized non-vanishing theorem on surfaces

本文证明了 $\mathbb Q$-因子曲面的反正则丛数值有效与伪有效等价，并通过建立伪有效除子极化曲面的数值非消失定理回答了 Fontanari 的问题。

On generic vanishing for pluricanonical bundles

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.

有限域上阿贝尔簇的极化：通过典范提升

Polarizations of abelian varieties over finite fields via canonical liftings

本文通过典范提升和范畴等价，显式计算了有限域上同源类中阿贝尔簇的极化同构类，并实现了相应算法。

正特征下曲面的Steenbrink型消没

Steenbrink-type vanishing for surfaces in positive characteristic

本文研究正特征域上曲面的Steenbrink型消没问题，证明对数典范且p>5或F-纯的曲面对满足该消没，且满足该消没的有理曲面奇点是F-内射的。

On a class of symplectic $4$-orbifolds with vanishing canonical class

A study of certain symplectic $4$-orbifolds with vanishing canonical class is initiated. We show that for any such symplectic $4$-orbifold $X$, there is a canonically constructed symplectic $4$-orbifold $Y$, together with a cyclic orbifold covering $Y\rightarrow X$, such that $Y$ has at most isolated Du Val singularities and a trivial orbifold canonical line bundle. The minimal resolution of $Y$, to be denoted by $\tilde{Y}$, is a symplectic Calabi-Yau $4$-manifold endowed with a natural symplectic finite cyclic action, extending the deck transformations of the orbifold covering $Y\rightarrow …

Openness results for uniform K-stability

Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close to the original polarization.

On direct images of pluricanonical bundles

We show that techniques inspired by Kollár and Viehweg's study of weak positivity, combined with vanishing theorems for log-canonical pairs, lead to new consequences regarding generation and vanishing properties for direct images of pluricanonical bundles. We formulate the strongest such results as Fujita conjecture-type statements, which are then shown to govern a range of fundamental properties of direct images of pluricanonical and pluriadjoint line bundles, like effective vanishing theorems, weak positivity, or generic vanishing.

第一陈类消失的Vaisman流形

Vaisman manifolds with vanishing first Chern class

本文研究了第一陈类消失的紧致Vaisman流形，证明具有非正Bott-Chern类的流形具有典范度量、拟正则性及形变稳定性，并指出Calabi-Yau Vaisman流形满足Beauville-Bogomolov分解且具有挠典范丛。

On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

We consider a family of slightly extended version of the Raynaud's surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H^1(X, Z^{-1})\ne 0. The surfaces are at least normal but smooth under a special condition. We compute the cohomologies H^i(X, Z^n), for intergers i and n, and study their (non-)vanishing. Finally, we give a fairly large family of non Mumford-Szpiro type polarizations Z_{a,b} with Kodaira non-vanishing.

Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T\subset X$ of certain codimension. In some cases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge struc…

Relative critical exponents, non-vanishing and metrics with minimal singularities

In this article we prove a non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments involve many ideas from Y.-T. Siu's analytic proof of the finite generation of the canonical ring. An important technical tool is the notion of relative critical exponent of two closed positive currents with respect to a measure.

Green's canonical syzygy conjecture for generic curves of odd genus

We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus $g$ with fixed gonality $d$ was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus.

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic …

Polarizations of Prym Varieties of pairs of coverings

To any pair of coverings $f_i: X \ra X_i, i = 1,2$ of smooth projective curves one can associate an abelian subvariety of the Jacobian $J_X$, the Prym variety $P(f_1,f_2)$ of the pair $(f_1,f_2)$. In some cases we can compute the type of the restriction of the canonical principal polarization of $JX$. We obtain 2 families of Prym-Tyurin varieties of exponent 6.

Ordinary varieties with trivial canonical bundle are not uniruled

We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.

A Witt Nadel vanishing theorem for threefolds

In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_q$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.

典范丛公式与Schnell关于某类代数纤维空间的猜想

Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell

本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Kollár-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degre…

A simply connected numerical Godeaux surface with ample canonical class

We prove that a recent construction of a numerical Godeaux surface due to P. Craighero and R. Gattazzo is simply connected, and show how to realize their construction as a double plane. By proving that the surface contains no (-2)-curves, we obtain that this is the first example of a simply connected surface with vanishing geometric genus and ample canonical class.

The characteristic cycles and semi-canonical bases on type $A$ quiver variety

In this article we study a conjecture of Geiss-Leclerc-Schr{ö}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.

关于扭曲的 Kawamata 半正定性和广义 canonical 环的有限生成性

On twisted Kawamata's semi-positivity and finite generation of generalized canonical rings

本文提出了一种扭曲的 Kawamata 半正定性定理，并应用于广义对 canonical 环的有限生成性问题。

第二陈类消失的极小紧 Kähler 流形的丰度定理

Abundance theorem for minimal compact Kähler manifolds with vanishing second Chern class

本文证明极小紧 Kähler 流形上第二陈类消失等价于余切丛 nef 且典范丛数值维数为 0 或 1，并由此推出典范丛半丰富，同时揭示 Iitaka 纤维化纤维变化与余切丛半正性的联系。

A Vanishing Theorem and Asymptotic Regularity of Powers of Ideal Sheaves

Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as $ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.

A vanishing theorem of Koll'ar-Ohsawa type

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi. As an application, we prove a vanishing theorem of Koll'ar-Ohsawa type by combining the L^2-method for the dbar-equation.

The canonical sheaf of Du Bois singularities

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $π_*ω_{X'}(G) \simeq ω_X$ for a log resolution $π: X' \to X$, where $G$ is the reduced exceptional divisor of $π$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for …

An Application of a Log Version of the Kodaira Vanishing Theorem to Embedded Projective Varieties

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain twists of powers of the ideal sheaf of Y. This is applied to get rather non-trivial vanishing in case Y is a universal determinantal variety (i.e. quadratic Veronese, Segre embedding or Grassmannian of P^1's) or a curve embedded by a complete linear series of high degree. The key in the determinantal cases is to use the theory of complete objects. While the theo…