Alex Kuronya
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.
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We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.
By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X \times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) \to H^*(Y)$ is…
We show that a pseudoeffective R-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, giving an affirmative answer to a question of Peternell. We also give other positivity properties such subvarieties.
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.
The goal of this survey is to describe some recent results concerning the L 2 extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal…
Smooth subvarieties of Jacobians
本文研究了在光滑拟射影复变种上存在非光滑子簇线性组合的代数整数上同调类的新例子，其中一些例子的维数为6，是最小可能的。所考虑的类是非常一般曲线Jacobian上的最小上同调类。主要工具是复 bordism。
Realization of Cohomology Classes in Grassmannians
该文针对格拉斯曼流形中可由不可约子簇实现的上同调类展开研究，完成了低维低余维情形、特定格拉斯曼流形及相差正倍数情形下的上同调类分类，推进了相关代数几何问题的研究。
We show that the complex Radon transform realizes an isomorphism between the space of residual $\bar\partial$-cohomologies of a locally complete intersection subvariety in a linearly concave domain of ${\C}P^n$ and the space of holomorphic solutions of the associated homogeneous system of differential equations withconstant coefficients in the dual domain in $({\C}P^n)^*$.
In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.
We give a short, geometric proof of Graham's theorem on positivity in the equivariant cohomology of a flag variety, based on a transversality argument.
Given scheme-theoretic equations for a nonsingular subvariety, we prove that the higher cohomology groups for suitable twists of the corresponding ideal sheaf vanish. From this result, we obtain linear bounds on the multigraded Castelnuovo-Mumford regularity of a nonsingular subvariety, and new criteria for the embeddings by adjoint line bundles to be projectively normal. A special case of our work recovers the vanishing theorem of Bertram, Ein, and Lazarsfeld.
We give a complete description of the cohomology ring $A^*(\overline Z)$ of a compactification of a linear subvariety $Z$ of a torus in a smooth toric variety whose fan $Σ$ is supported on the tropicalization of $Z$. It turns out that cocycles on $\overline Z$ canonically correspond to Minkowski weights on $Σ$ and that the cup product is described by the intersection product on the tropical matroid variety $\operatorname{Trop}(Z)$.
We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs.
We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology.
We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.
We classify GV-subschemes of five-dimensional ppavs, showing that they exist only on Jacobians of curves and intermediate Jacobians of cubic threefolds, and confirming a conjecture of Pareschi--Popa in this case. The result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor.
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.
On Positivity for the Peterson Variety
本文探讨旗流形 $G/B$ 及其子变形上同调环中的几何正性，阐述了 Schubert 变形基的非负结构常数，并解释了该正性现象如何扩展至特定的幂零 Hessenberg 变形（Peterson 变形），同时提出了向其他 Hessenberg 变形扩展的开放问题。
We verify in an elementary way a result of Peterson for the maximal orthogonal and Lagrangian Grassmannians, and then find Vafa-Intriligator type formulas which compute their 3-point, genus zero Gromov-Witten invariants. Finally we study total positivity of the related Peterson varieties and show that Rietsch's conjecture about the total positivity holds for these cases.
We prove a positivity result in (T-)equivariant quantum cohomology of the homogeneous space G/P, generalizing Graham's positivity in equivariant cohomology.
An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties
本文通过延拓更小正则幂零 Hessenberg 子簇的 Poincaré 对偶，构造了正则幂零 Hessenberg 簇上同调环的加法基，并证明这些对偶线性无关。
We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …
In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let $\mathcal{P}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname{codim} B}χ(A\cap B, \mathcal{P}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersection…
Positivity determines the quantum cohomology of the odd symplectic Grassmannian of lines
针对 Picard 秩为 1 的奇辛线 Grassmann 簇 IG(2,2n+1)，本文提出正性条件，证明其量子上同调环在量子参数缩放下是唯一量子形变，并修正 Fulton 猜想。
We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equ…
We prove a conjecture of Dale Peterson on positivity in the multiplication in the T-equivariant cohomology of the flag variety. The theorem follows from a more general positivity result about the equivariant cohomology of varieties with actions of a solvable group with finitely many orbits. This more general result is an equivariant version of a theorem of Kumar and Nori.
The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the flag variety. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of $S_n$. An expli…
Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.
We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.
We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities f…
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。
We generalize a vanishing theorem for the cohomology of symmetric powers of the cotangent bundle of subvarieties of projective space due to Schneider. From this we deduce new vanishing results for Green-Griffiths jet differential bundles, generalizing results of Diverio and Pacienza-Rousseau.
In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincaré relations on a Jacobian variety.
Positivity in Weighted Flag Varieties
本文从李理论视角研究加权旗流形的环面等变上同调，证明其正性性质并推广了加权格拉斯曼流形及 A 型情形下的 Chevalley 公式、不动点限制与 GKM 描述等结果。
Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups
本文引入紧致平衡流形上埃尔米特全纯向量丛的P-临界联络概念，提出P-正性与P-稳定性数值条件刻画解的存在性，并证明其在等变情形与吹胀拉回下的保持性质。
Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton-Lazarsfeld for certain degrees.
Generalized Ueda Obstruction Classes and Applications to Non-semi-positivity of Line Bundles
引入沿复流形闭解析子簇的线束广义第一Ueda障碍类，允许奇异子簇和非平坦限制，通过Dolbeault分解和Chern曲率，类非零阻碍半正曲率光滑Hermite度量的存在，统一经典例子并给出新的nef非半正线束例子。
In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in X$. First we translate positivity properties of the Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows us to classify polarized minimal families of rational curves associated to Fano manifolds $X$ satisfying $ch_2(X)\geq0$ and $ch_3(X)\geq0$. As a first application, we provide sufficient conditions for these manifolds to be covered by…
We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson-Thomas invariants for a quiver with potential $(\tilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.
Tropical Feichtner-Yuzvinsky and positivity criterion for fans
本文建立单纯扇形 Chow 环与其典范紧化热带上同调中间次数部分的同构，证明热带版 Kleiman 丰富性判据，并给出拟阵 Chow 环的新表示。
In this paper, we study relations between positivity of the curvature and the asymptotic behavior of the higher cohomology group for tensor powers of a holomorphic line bundle. The Andreotti-Grauert vanishing theorem asserts that partial positivity of the curvature implies asymptotic vanishing of certain higher cohomology groups. We investigate the converse implication of this theorem under various situations. For example, we consider the case where a line bundle is semi-ample or big. Moreover, we show the converse implication holds on a projective surface without any assumptions on a line bun…
Splitting the cohomology of Hessenberg varieties and e-positivity of chromatic symmetric functions
本文利用遗忘映射的分解定理拆分正则半单 Hessenberg 簇上同调，组合刻画各部分 Frobenius 特征，并证明无差别图染色对称函数中长度为 2 的 $e_λ$ 系数非负。
Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps
本文研究带尖点的复双曲流形环面紧化上余切丛及其对数变体的正性，证明其丰沛性，并给出尖点深度大于4π时子簇一般型及最小体积趋于无穷的结论。
We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homol…
Positivity properties of Schur classes
本文将丰富和数值有效向量丛的Schur类正性结果推广到纯代数环境，证明具有Kähler包的射影丛环满足Hodge-Riemann关系，并由此推出拟阵Schur系数非负。
We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\C)$ relying in particular on the positivity of the structure constants, which are enumerative Gromov--Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterso…
Notes on the cohomology of partial Hessenberg varieties
本文研究部分旗簇中Hessenberg簇的上同调，通过将其与全旗簇中Hessenberg簇的上同调关联展开分析，该簇家族包含多种知名子簇，近年研究活跃且与超平面配置等领域有意外关联。
We observe that an interesting method to produce non-complete intersection subvarieties, the generalized complete intersections from L. Anderson and coworkers, can be understood and made explicit by using standard Cech cohomology machinery. We include a worked example of a generalized complete intersection Calabi-Yau threefold.
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.
We consider generalized complete intersection manifolds in the product space of projective spaces, and work out useful aspects pertaining to the cohomology of sheaves over them. First, we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces. We then prove an equivalence between configuration matrices of complete intersection Calabi-Yau manifolds. We also present a formula of the genus of curves in generalized complete intersection manifolds. Some of these curves arise as the fixed point locus of certain sym…
Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of dimension $n$. Demailly has introduced a very interesting invariant --- the Seshadri constant $ε(L,x)$ --- which in effect measures how positive $L$ is locally near a given point $x \in X$. For instance, Seshadri's criterion for ampleness may be phrased as stating that $L$ is ample if and only if there exists a positive number $e > 0$ such that $ε(L,x) > e$ for all $x \in X$, and if $L$ is VERY ample, then $ε(L,x) \ge 1$ for every $x$. We prove the somewhat surprising result that in each dimension $n$ there is a uniform…
We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. …
For a local complete intersection subvariety $X=V({\mathcal I})$ in ${\mathbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of ${\mathbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t=V({\mathcal I^t})$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of …
Positivity for toric Kac polynomials in higher depth
该研究证明更高深度下计数截断幂级数环上箭图秩1绝对不可分解表示的多项式系数非负，建立箭图表示计数与矩映射纤维射流计数的合冲恒等式，并验证了Wyss的两个渐近猜想。
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge met…
The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{ä}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation…
In this paper, we provide a Hodge-theoretic interpretation of Laurent phenomenon for general skew-symmetric quantum cluster algebras, using Donaldson-Thomas theory for a quiver with potential. It turns out that the positivity conjecture reduces to the certain statement on purity of monodromic mixed Hodge structures on the cohomology with the coefficients in the sheaf of vanishing cycles on the moduli of stable framed representations. As an application, we show that the positivity conjecture (and actually a stronger result on Lefschetz property) holds if either initial or mutated quantum seed i…
Positivity of automorphic vector bundles on unitary Shimura varieties
研究单位ary Shimura 变种上自动机向量丛的正性条件，基于签名数据和权重坐标，推广了Hilbert模形式和U(2)-Shimura变种的 ample cone 结果。
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