M. Aprodu, J. Nagel
We completely revised the paper after the referee's comments. In the new version, we replaced two erroneous examples, studied a link with earlier work of Koh and Stilmann, and strengthened the main theorem.
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We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.
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 $(N_p)$.
We give a sufficient condition to study the vanishing of certain Koszul cohomology groups for general pairs $(X,L)\in W^r_{g,d}$ by induction. As an application, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices to check all cases with the Brill-Noether number $ρ=0$.
Equivariant Koszul Cohomology of Canonical Curves
本文研究了典范曲线的等变Koszul上同调，结合等变欧拉特征和黎曼-罗奇定理，推导出显式公式。
In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it holds for a generic canonical or Prym-canonical binary curve of genus $g+1$. We also verify the Green and Prym-Green conjectures for generic canonical and Prym-canonical binary curves of low genus ($6\leq g\leq 15$, $g\neq 8$ for Prym-canonical and $3\leq g\leq 12$ for canonical).
We study the deformation-obstruction theory of Koszul cohomology groups of $g^r_d$'s on singular nodal curves. We compute the obstruction classes for Koszul cohomology classes on singular curves to deform to a smooth one. In the case the obstructions are nontrivial, we obtain some partial results for generic vanishing of Koszul cohomology groups.
Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_λ$ and $L_λ$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.
In this work we prove the so called dimension property for the cohomological field theory associated with a homogeneous polynomial W with an isolated singularity, in the algebraic framework of arXiv:1105.2903. This amounts to showing that some cohomology classes on the Deligne-Mumford moduli spaces of stable curves, constructed using Fourier-Mukai type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in arXiv:math/0011032 for certain characteristic classes of Koszul matrix factorizations of 0. To reduce to this…
Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $Δ_X \subset X \times X$.
It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.
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 describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
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。
A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul cohomology on the Hilbert scheme of points on the surface. In this note we construct representations of the Koszul complex on Grassmann varieties; Voisin's setup arises as the inverse image of one of the maps. Using a different map, we give a substantially shorter proof of Voisin's result for K3 surfaces of even sectional genus.
Hochschild cohomology of the second kind: Koszul duality and Morita invariance
本文研究了第二种Hochschild上同调的定义、性质及其在Koszul对偶和Morita等价下的不变性，展示了其在不同范畴中的应用。
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the…
Stable sheaf cohomology and Koszul--Ringel duality
本文建立稳定层上同调与Koszul--Ringel对偶的联系，证实周期猜想，发现钩子划分Schur函子$\Ext$群的不变性，并给出非零稳定上同调度数的锐界。
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.
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 study the relationship between positivity of line bundles restricted to complete intersection subvarieties and the vanishing of higher cohomology groups. Based on this connection we prove generalizations of the vanishing theorems of Kawamata-Viehweg and Fujita that are valid for not necessarily big line bundles.
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 show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.
We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial $f$ and the pole order filtration $P$ on the cohomology of the open set $U=\PP^n \setminus D$, with $D$ the hypersurface defined by $f=0$. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration $P$ in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial $f$. The case of a nodal hypersurface $D$ is treated in terms of the…
We use representation theory and Bott's theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a biproduct we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations.
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.
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.
Vanishing Theorem for Hodge Ideals on smooth hypersurfaces
本文通过Koszul型消解证明光滑超曲面的弱Bott消失定理，并由此建立有效Cartier除子对应Hodge理想的消失定理，推广了Mustata和Popa的已有工作。
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。
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describ…
Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus
本文提出了一种张量展开方法，用于研究secant种群的行列式方程，并证明了2×2矩阵乘法张量的边界秩为7。
Consider a complex Stein manifold X and a subanalytic relatively compact Stein open subset U of X.. We prove the vanishing on U of the holomorphic temperate cohomology.
We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it …
We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in k[x,y,z]. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local complete intersection. The proof uses a characterization of complete intersections due to Herzog. When I is saturated, we relate our theorem to results of Weyman and of Simis and Vasconcelos. We conclude with an example of how our theorem fails for four generated local complete intersections in k[x,y,z] and we discuss generalizations to higher dimensions.
This paper has been withdrawn by the author due to a crucial sign error in Theorem 3.4.
Under the generic situation, the cohomology with the coefficients in the local system on complements of hypersurfaces vanishes except in the highest dimension. Our problem is of when the local system cohomology does not vanish. In the case of arrangements of hyperplanes, many examples were founded. In this paper, we shall generalize their examples to hypersurfaces. We obtain that hypersurfaces given by some linear system have non-vanishing local system cohomologies.
We provide bounds on the Castelnuovo-Mumford regularity in terms of ``defining equations'' by using elements that annihilates some cohomology modules, inspired by works of Miyazaki, Nagel, Schenzel and Vogel. The elements in these annihilators are provided either by liaison or by tight closure theories. Our results hold in any characteristic.
Surjectivity of some local cohomology map and the second vanishing theorem
本文研究了完全分枝正则局部环中第二消失定理的部分解答，基于局部上同调理论中的 surjective 元素理论进行证明。
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 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.
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 $G_{31}$.
We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen-Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients, and obtain new generalizations.
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