Samuel Lundqvist
We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.
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Cohomology of projective schemes: From annihilators to vanishing
我们提供的Castelnuovo-Mumford正则性方面的“定义方程”的界限，通过使用元素，湮灭一些上同调模块，宫崎，内格尔，Schenzel和Vogel的作品的启发。这些零化子中的元素由联络理论或紧闭理论提供。我们的结果适用于任何特征。
In this paper, we establish several results related to vanishing theorems for Mather-Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffith-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather-Jacobian multiplier ideals.
Vanishing theorems and universal coverings of projective varieties
本文给出了复射影簇上负向量丛的第一上同调为零的一个新的证明，如果丛的秩小于基的维数。类似的论证被应用于复射影簇的泛覆盖上的全纯函数的构造。
Cohomological vanishing on blown-up projective spaces
本文利用复几何初等技巧，证明了在不超过 $n+1$ 个点处爆破射影空间 $\mathbb{P}^n$ 上线丛的精确上同调消没结果。
We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue of the footprint bound, and a version of it that is suitable for estimating the number of points of a projective algebraic variety over a finite field. An application to Serre's inequality for the number of rational points of projective hypersurfaces over finite fields is included
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.
On projective threefolds with two-dimensional space of vanishing cycles
本文完整分类了复数域上消没循环空间维数为2的光滑射影三维簇，以及光滑射影曲面上第二陈类为3的秩2非常充足向量束。
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof…
Vanishing of weight one syzygies of projective varieties
本文探究射影簇上权1 Koszul上同调的消失条件，结合(M_q)与(N_p)性质推导相关推论，深化了对极小自由消解的认知。
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 describe generating functions for arbitrary-genus Gromov-Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing results for these invariants. They suggest deep conjectures relating Gromov-Witten invariants of symplectic manifolds to the energy of pseudo-holomorphic maps and the expected dimension of their moduli space.
Let X be a smooth projective threefold, and let A be an ample line bundle such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective, the adjoint bundle $K_X+A$ has global sections. We also give a very short proof of the Beltrametti-Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint bundle $K_X+2A$ has global sections.
We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.
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…
Vanishing theorems for projective morphisms between complex analytic spaces
本文讨论复解析空间射影态射的消失定理，从最小模型理论观点建立了Kollár纲领的终极推广，对最小模型理论起关键作用。
Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties
本文在多重分次Castelnuovo–Mumford正则性框架下，证明射影簇的通用上同调消失定理，推广前期工作，刻画射影空间乘积线丛的极小正则性，重证Green定理并计算超椭圆曲线典范像的Betti表。
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a…
The classical Kodaira Vanishing Theorem states that Hi(X, ωX \otimes L) = 0 for i > 0, where X is a smooth projective variety over C and L is an ample line bundle on X. We prove an analogous vanishing result under the assumption that X is a semi-smooth projective variety over C. This is done by passing to the normalization of X, which is a smooth projective variety when X is semi-smooth. The classical vanishing result is proved using a cyclic cover. Therefore we include a proof that the cyclic cover of a demi-normal variety is again demi-normal.
Vector Fields on Smooth Threefolds Vanishing on Complete Intersections
紧致Kaehler流形上具有非空零轨迹的向量场的存在性及其零轨迹的性质对流形的几何有重要影响。例如，J. Wahl证明了在投射正规簇X的充分因子上为零的向量场的存在性意味着X是这个因子上的锥。如果X是光滑的，则X同构于n维射影空间。本文首次尝试将Wahl定理推广到更高余维的情形：给定一个复光滑三重射影空间X和X上的一个向量场在一条不可约和约化曲线上消失，该曲线是两个充分因子的概型论交，X同构于三维射影空间或三维二次曲面.
Numerical equivalence defined on Chow groups of Noetherian local rings
本文在Noether局部环的Chow群或Grothendieck群上定义了一个数值等价的概念，它类似于光滑投射簇上的数值等价.在较弱的条件下，证明了Chow群模数值等价是有限维${\Bbb Q}$ -向量空间，与光滑射影簇的情形一样.局部环上的数值等价与光滑投射簇上的数值等价有着密切的联系。例如，如果Grothendieck的标准图为真，则可以证明（局部环的）Chow群的模数值等价为零。利用数值等价理论，定义了数值Roberts环的概念.证明了具有正特征的科恩-麦考利局部环是数值罗伯茨环的充要条件是有限投射维数的极大准素理想的Hilbert-Kunz重数总是等于它的长度。数值上Roberts环满足交重数的零性质.我们将证明另一种特殊情况下的消失交重数使用本地化陈特征消失。
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.
Convexity of coverings of projective varieties and vanishing theorems
本文研究了投射簇的泛覆盖的凸性。研究了X的泛覆盖的凸性与将X上的向量丛发送到其泛覆盖上的向量丛的拉回映射的性质之间的关系。我们的方法激发了一个弱化版本的Shafarevich猜想。我们证明了这个猜想的投射簇X的拉回映射确定一个非平凡的扩张的负向量丛$V$的平凡线丛的平凡扩张。证明了如下关键性结果：若投射簇的泛覆盖不含非常数全纯函数，则向量丛的拉回映射几乎是嵌入。我们的方法还给出了秩小于维数X的紧致复流形$X$上负向量丛$V$的第一上同调为零的新证明。
Linear syzygies of projective space
研究线性合系q=1情形下Veronese簇合系的Ein-Lazarsfeld猜想，证明在最高和次高阶线性合系上与猜想一致的消失性命题。
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.
Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology
本文证明混合特征概形的母题上同调满足Weibel消没的母题精细化、射影丛公式等预期性质，并将其与Milnor K理论比较，从而把Elmanto–Morrow的结果推广到非域上概形。
An Application of a Log Version of the Kodaira Vanishing Theorem to Embedded Projective Varieties
给定CP^n中的一个嵌入光滑投射簇Y，我们证明了沿着Y具有高重数，但在Y附近具有相对低的次数和对数标准的超曲面的存在意味着对于Y的理想层的某些幂的扭曲，高上同调消失。当Y是一个泛行列式簇（即二次Veronese、Segre嵌入或P^1的Grassmannian）或一条被一个高次完全线性级数嵌入的曲线时，这一方法可以用来得到相当非平凡的消失。在行列式的情况下，关键是使用完全对象理论。虽然完全二次曲面和线性映射的理论产生了所需的结果，但完全斜形式的相应理论似乎以前没有制定出来。在附录中获得了预期的结果。
We prove the Kawamata-Viehweg vanishing and another Kodaira-type vanishing for projective toric surfaces over arbitrary fields.
Define a line bundle L on a projective variety to be q-ample, for a natural number q, if tensoring with high powers of L kills coherent sheaf cohomology above dimension q. Thus 0-ampleness is the usual notion of ampleness. Intuitively, a line bundle is q-ample if it is positive "in all but at most q directions". We show that q-ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that q-ampleness is a Zariski open condition, which is not clear from the definition. We also show that a …
Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a…
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.
In this paper, we study the cohomology of vector bundles on projective space defined as kernels or cokernels of general maps $V_1 \to V_2$, where the $V_i$ are direct sums of line bundles or certain exceptional bundles. We prove an asymptotic cohomology vanishing theorem. We characterize the stability of general Steiner bundles on projective space. We also give a criterion for a Steiner bundle to be ample.
We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.
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.
Finite automorphism groups of hypersurfaces via Bott vanishing
本文通过Bott消失定理给出上同调判据，证明驯特征下光滑Deligne-Mumford栈中超曲面的自同构群概形离散，并应用于环面轨道簇，获得拟光滑加权射影超曲面及射影空间乘积中超曲面的有限性结果，同时指出方法失效的例外窗口。
We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathbf{M}$ according to the Hilbert function $H$ and classify all possible Hilbert functions $H$ of such bundles. For each $H$, we describe a stratification of $\mathbf{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.
We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles of projective semi-log canonical pairs over curves, which implies the projectivity of the moduli spaces of stable varieties. It is worth mentioning that we do not use the theory of variation of (mixed) Hodge structure.
We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hy…
Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,Ω^i_X\otimes L)=0$ for every $j>0$, $i\geq 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational morphism $π\colon X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^jπ_*(Ω^i_X\otimes A)$ is zero for every $j>0$ and $i\geq 0$.
In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening…
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.
Multiplier Ideals, Vanishing Theorem and Applications
本文的目的是综述乘子理想的代数性质，并举例说明它们在经典射影几何中的一些应用。
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…
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.
Kawamata-Viehweg vanishing for toric varieties
本文针对射影环面簇上的 klt 配对 $(X,B)$，证明相应的 Kawamata–Viehweg 消灭定理。
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.
Effective non-vanishing and Mukai type conjecture
本文引入Fano簇的不变量，提出刻画射影空间乘积的Mukai型猜想，并证明Ambro-Kawamata有效非消失性猜想蕴含该猜想。
Elementary counterexamples to Kodaira vanishing in prime characteristic
本文给出了Kodaira素特征消失在射影n-空间（n>2）中点和面的关联簇上作为自然丛的Frobenius拉回的射影化而构造的初等反例.
Bott Vanishing via Hodge Theory
本文利用与川又-维韦格消失定理相容的条件，为射影环面簇上的博特消失定理提供了新证明，并将其推广至混合霍奇模块情形，同时构造了双有理态射下相对博特消失定理的反例。
Vanishing of cohomology in infinitely ramified towers
本文通过使用近旁周期的反常性（全局形式），给出了Esnault关于射影空间的“模拟”Frobenius覆盖塔中可构造层的上同调消失结果的新证明。
Kodaira-type Vanishings via Non-abelian Hodge Theory
本文借助非阿贝尔霍奇理论，利用混合扭量$\boldsymbol{\frak{D}}$-模推广斋藤消没定理，结合$\boldsymbol{\frak{Q}}$-除子推广为川又-维赫韦格型消没定理，还证明了有效整体生成定理与射影态射的相对消没定理。
Certain degenerations of surfaces in fake weighted projective $3$-space
本文给出了伪加权射影3-空间中某些一般非退化曲面具有Picard数大于1的判别准则，这些代数曲面为一般型，通过考虑沿边的退化并跟踪几何亏格，利用消失上同调类构造了与典范除子不成比例的有理Picard类。
In this paper we establish a Nadel-type vanishing theorem on a projective manifold $X$ concerning the asymptotic multiplier ideal sheaf.
Let X be a smooth complex projective variety of dimension 4 and let L be an ample line bundle on X. In this paper, we study a natural number m such that h^{0}(m(K_{X}+L))>0 for any polarized 4-folds (X,L) with κ(K_{X}+L)\geq 0.
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。
We apply Weiss calculus to compute the number of topological complex vector bundles of rank $n-2$ with vanishing Chern classes over $\mathbb{C}P^n$ for $n>3$, as given by the list $1, 1, 12, 2, 1, 3, 2, 2, 3, 1, 4, 6, 1, 1, 6, 2, 1, 3, 4, 2, 3, 1, 2, 6$, where the $i$-th entry in this list is the number of such bundles whenever $n$ is congruent to $i$ modulo $24$, starting with $i = 0$. Similarly, the number of rank $n-1$ bundles with vanishing Chern classes over $\mathbb{C}P^n$ for $n>2$ is $2$ when $n$ is odd and $1$ when $n$ is even.
Characterization of Hermitian symmetric spaces by fundamental forms
证明了在既不包含射影空间也不包含超二次曲面作为分支的射影空间中等变嵌入的Hermite对称空间的基本形式是射影空间的局部子流形.利用一些不变的理论性质的基本形式和世石的工作线性微分方程的有限型，我们减少证明消失的某些斯宾塞上同调群。用Kostant关于李代数上同调的调和理论检验了消失。
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。
When X is a singular complex projective variety, we prove a Kodaira type vanishing theorem generalizing results of Navarro Aznar and others. This is proved by extending the Deligne-Illusie decomposition to the Du Bois complex. We also give a new definition of Frobenius amplitude (introduced in math.AG/0202129) using ultraproducts.
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…
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