Qihong Xie
We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their counterexamples on ruled surfaces in positive characteristic.
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We give a counterexample and some conclusions for effective non-vanishing of Weil divisors on algebraic surfaces.
On vanishing of fundamental forms of algebraic varieties
本文利用线丛主部层研究代数簇基本形式，建立任意阶消没定理，并揭示其与高阶高斯映射和高阶切簇的联系。
We show that the homotopy invariant algebraic K-theory of Weibel vanishes below the negative of the Krull dimension of a noetherian scheme. This gives evidence for a conjecture of Weibel about vanishing of negative algebraic K-groups.
We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces 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.
Small changes in sections 4 and 5, results not affected
We give counterexamples to the Kawamata-Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface f:X-->C, then either C is a Tango curve or all of sections of f are ample.
For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor…
We investigate the vanishing of $H^1(X,\mathcal{O}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,Δ)$ over an algebraically closed field of positive characteristic $p$.
On Steenbrink vanishing for rational singularities in positive characteristic
本文证明正特征有理奇点的 Steenbrink 消失特殊情形，并由此推出强 F-正则三维簇及特征 p>41 的 klt 三维簇均满足 Steenbrink 消失。
We establish a generic vanishing theorem for surfaces in characteristic $p$ that lift to $W_2(k)$ and use it for surface classification of surfaces of general type with Euler characteristic 1 and large Albanese dimension.
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.
Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and o…
A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we give some concrete examples and properties of strongly liftable schemes. As an application, we prove that the Kawamata-Viehweg vanishing theorem in positive characteristic holds on any normal projective surface which is birational to a strongly liftable surface.
We generalize the construction of Raynaud of smooth projective surfaces of general type in positive characteristic that violate the Kodaira vanishing theorem. This corrects an earlier paper of the same purpose. These examples are smooth surfaces fibered over a smooth curve whose direct images of the relative dualizing sheaves are not nef, and they violate Kollár's vanishing theorem. Further pathologies on these examples include the existence of non-trivial vector fields and that of non-closed global differential 1-forms.
The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.
Generic Vanishing Fails for Surfaces in Positive Characteristic
本文证明在任意特征 p≥3 下存在不满足泛消失性的光滑曲面，并由此重新得到 Hacon 和 Kovács 在三维及以上构造泛消失性反例的结果。
We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata-Viehweg vanishing.
We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic $p>5$. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic $p>5$ are rational. We show that these theorems are sharp by providing counterexamples in characteristic $5$.
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.
On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic
本文研究正特征面上博戈莫洛夫不等式与miyaoka-sakai定理的等价性，并推导出相关消失定理及fujita猜想的reider类型结果。
We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When $k=\CC$ this implies a conjecture of Green and Lazarsfeld.
A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, first we prove that smooth toric varieties are strongly liftable. As a corollary, we obtain the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, we prove the Kawamata-Viehweg vanishing theorem for normal projective surfaces which are birational to a strongly liftable smooth projective surface. Finally, we deduce the cyclic cover trick over W_2(k), which can be used to construct a l…
We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces.
We study the $(k,s)$-positivity for holomorphic vector bundles on compact complex manifolds. $(0,s)$-positivity is exactly the Demailly $s$-positivity and a $(k,1)$-positive line bundle is just a $k$-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for $(k,s)$-positive vector bundles are proved and the vanishing theorems for $k$-ample vector bundles on projective algebraic manifolds are generalized to $k$-positive vector bundles on compact Kähler manifolds.
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。
Generic vanishing theory in positive characteristic
简化并改进了正特征一般消没理论的基本定理，证明了最大 Albanese 维数的正规簇满足 H^0(X, ω_X) ≠ 0，且若 Alb(X) 是普通的，则 S^0(X, ω_X) ≠ 0。
Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).
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…
Liftability and vanishing theorems for Fano threefolds in positive characteristic I
本文证明正特征光滑Fano三维流形可提升到Witt向量环，并满足Akizuki-Nakano消失、Hodge到de Rham谱序列E1退化及晶体上同调无挠性，针对$|-K_X||$非常丰沛且Picard群由$\omega_X$生成的情形。
A smooth scheme X over a field k of positive characteristic is said to be strongly liftable over W_2(k), if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we first deduce the Kummer covering trick over W_2(k), which can be used to construct a large class of smooth projective varieties liftable over W_2(k), and to give a direct proof of the Kawamata-Viehweg vanishing theorem on strongly liftable schemes. Secondly, we generalize almost all of the results in [Xie10, Xie11] to the case where everything is considered over W(k), the ring of Witt vectors of k.
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.
Mumford vanishing for threefolds in positive characteristic
研究三重拟正则簇在正特征下的Mumford消失定理，证明当K_X不大的情况下L为大除子或-L_K_X为拟正则且L的数值维数为二时，H^1(X,-L)=0。
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 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 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.
On the stability of vanishing cycles of étale sheaves in positive characteristic
研究正特征下平展层消失圈的稳定性，表明光滑曲面上其对测试函数的依赖一般仅取决于有限阶喷流，猜想高维也成立，还研究了消失圈稳定性最强的层的类别及相关性质。
Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of Denef-Jacobs. As an application we obtain a strong vanishing result for candidate poles of p-adic and motivic Igusa zeta functions.
We characterize K. Kato's log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.
Koszul modules with vanishing resonance in algebraic geometry
本文利用有限长度 Koszul 模分次分量的一致消失结果，推动 Green 猜想相关方法在代数簇加厚、曲线模空间、K3 曲线枚举及退化轨迹中的应用，并证明秩 2 正向量丛稳定性由共振决定。
We investigate compact complex manifolds of dimension three and second Betti number $b_2(X) = 0$. We are interested in the algebraic dimension $a(X)$, which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic $χ_{\mathrm{ top}}(X) $ equals the third Chern class $c_3(X)$ by a theorem of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex manifold with $b_2(X) = 0$ and $a(X) > 0$, then $c_3(X) = χ_{\rm top}(X) = 0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X)…
Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new suffic…
We consider Calabi-Yau threefolds $X$ over an algebraically closed field $k$ of characteristic $p>0$ that are not liftable to characteristic $0$ or liftable ones with $p=2$. It is unknown whether Kodaira vanishing holds for these varieties. In this paper, we give a lower bound of $h^1(X, L^{-1})=\dim_k H^1(X, L^{-1})$ if $L$ is an ample divisor with $H^1(X, L^{-1})\ne{0}$. Moreover, we show that a Kodaira type vanishing holds if $X$ is a Schröer variety or a Schoen variety, which extends the similar result given in my previous paper for the Hirokado variety. We show that such kind of vanishing…
Cohomology of the differential fundamental group of algebraic curves
本文研究特征零域上光滑射影曲线的微分基本群，通过Tannakian对偶定义，探讨带联络向量丛的de Rham上同调与对应表示的群上同调的关系，并得到消没与非消没结果。
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
Brauer--Manin obstruction from failure of Kodaira vanishing in positive characteristic
研究正特征全局域上 Kodaira 消失定理失效的光滑射影曲面的 Brauer-Manin 集，特别应用于 Raynaud 曲面。
Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length …
On the Kawamata-Viehweg vanishing theorem for log Calabi-Yau surfaces in large characteristic
本文证明了当$B$具有标准系数时，Kawamata-Viehweg消灭定理对于大特征代数闭域上的log Calabi-Yau曲面$(X, B)$成立。
In this article, we study the possibility of producing a Calabi-Yau threefold in positive characteristic which is a counter-example to Kodaira vanishing. The only known method to construct the counter-example is so called inductive method such as the Raynaud-Mukai construction or Russel construction. We consider Mukai's method and its modification. Finally, as an application of Shepherd-Barron vanishing theorem of Fano threefolds, we compute $H^1(X, H^{-1})$ for any ample line bundle $H$ on a Calabi-Yau threefold $X$ on which Kodaira vanishing fails.
Generic vanishing on homogeneous spaces in arbitrary characteristic
本文在任意特征下研究同构空间上的通用消失现象，扩展了Schürmann-Simpson-Wang在特征零情况下的定理，并在有限域上给出了迹函数恒等式和Lang-Weil估计。
We give an algebraic proof valid in arbitrary characteristic for the known equivalence between (strongly) slope semistable vector bundles with vanishing discriminant and vanishing determinant and numerically flat bundles. We also address a related question of Misra.
Liftability and vanishing theorems for Fano threefolds in positive characteristic II
本文研究正特征域上光滑 Fano 三维簇的提升性与上同调性质，证明当 Picard 数或 Fano 指数大于 1 时其具有拟 F-分裂性，从而在除特定极丰沛情形外建立 Akizuki–Nakano 消失、Hodge–de Rham 谱序列 E1-退化及晶体上同调无挠性。
In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or $ln-exp$ definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated.
A well-known theorem of W. Fischer and H. Grauert states that analytic fiber spaces with all fibers isomorphic to a fixed compact connected complex manifold are locally trivial. Motivated by this result, we show that if $k$ is an algebraically closed field of infinite transcendence degree over its prime field, then every smooth projective family over a reduced $k$-scheme of finite type with isomorphic fibers having reduced automorphism group schemes is locally trivial in the étale topology. We do so by reducing the problem to the case when the base is a smooth integral curve, and then, using t…
We generalize the Generic Vanishing theorem by Hacon and Patakfalvi in the spirit of Pareschi and Popa. We give several examples illustrating the pathologies appearing in the positive characteristic setting.
We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfaces
Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic
本文证明了正特征曲面上大类的Kawamata-Viehweg消失定理，并由此建立Reider型定理与态射延拓定理，进而用有理函数刻画任意基域上平面曲线的非奇异有理点。
We investigate the notion of the $p$-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic $p$ and we study some of their properties. We present a structure theorem for the $p$-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such $p$-divisors are reduced.
A note on Kodaira vanishing on surfaces
本文利用几何稳定性条件，为光滑复曲面上的Kodaira消失定理提供新证明，并重新证明了Xie关于正特征反例的刻画结果。
In this paper we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.
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