Lawrence Ein, Bo Ilic, Robert Lazarsfeld
We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow (almost) exponentially in the dimension of X.
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Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.
We construct examples to show that having nef cotangent bundle is not preserved under finite ramified covers. Our examples also show that a projective manifold with Stein universal cover may not have nef cotangent bundle, disproving a conjecture of Liu-Maxim-Wang.
We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample cotangent bundle. We discuss the conjecture that the analogous statement should hold in the projective space. Finally, we present a construction due to Bogomolov of varieties with ample cotangent bundle as linear sections of a product of varieties with big cotangent bundle.
In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an étale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.
Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.
Positivity of the cotangent sheaf of singular Calabi-Yau varieties
该研究推广了前人成果，证明正规射影klt Calabi-Yau簇及不可约全纯辛簇的切层与自反余切层非伪有效，并给出了余维2带奇性的低维Calabi-Yau簇实例。
Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $Ω_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$ be the Zariski closure of the set of all $K$-rational points of $X$, endowed with its reduced induced structure. We prove that there is a projective variety $X_0$ over $k$ and a finite and surjective $K^{\rm sep}$-morphism $X_{0,K^{\rm sep}}\to R_{K^{\rm sep}}$, which is birational when ${\rm char}(K)=0$.
The Shafarevich conjecture for varieties with globally generated cotangent
该研究证明了在温和数值条件下，具有全局生成余切丛的簇满足Shafarevich猜想。
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
In this short note, we will gives several remarks on rational points of varieties whose cotangent bundles are generated by global sections. For example, we will show that if the sheaf of differentials of a projective variety X over a number field k is ample and generated by global sections, then the set of k-rational points of X is finite.
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
Following the program of algebraic Frobenius splitting begun by Kumar and Littelmann, we use representation-theoretic techniques to construct a Frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic. We also show that this splitting is the same as one of the splittings constructed by Kumar, Lauritzen, and Thomsen.
In this paper we study the cohomology of tensor products of symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. We give several applications. First we prove a non-vanishing result, then we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Our main application is the construction of varieties with ample cotangent bundle, prov…
In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties…
In this paper we study the problem of density in (1,3] for the Chern ratio of surfaces with ample cotangent bundle. In particular we prove density in (1,2) by constructing a family of complete intersection surfaces in a product of varieties with big cotangent bundle. We also analyse the case of complete intersections in a product of curves of genus at least 2.
Remarks on the positivity of the cotangent bundle of a K3 surface
本文利用 Bayer-Macrì 的结果计算了光滑射影 K3 曲面射影化余切丛的伪有效锥与 nef 锥，并构造光滑曲线显式族证明余切丛限制非半稳定，从而给出 Campana-Peternell 问题的反例。
We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties.
In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and high codimension from the known results on hypersurfaces. Then we prove a existence theorem for jet differentials that generalizes a theorem of S. Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove th…
Remarks on the positivity of the cotangent bundle of an Enriques surface
本文研究了Enriques曲面余切包在特定曲线限制下的半稳定性，并给出了破坏其稳定的曲线族构造。
A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GL_n
本文证明了 GL_n 旗丛余切丛的 Frobenius 分裂具有最高次数，给出了分量函数公式并证明其在例外轨迹上消失，进而得出与支配权相关线丛的高阶上同调群为零。
Let F be a function field of one variable over an algebraically closed field of characteristic zero, X a geometrically irreducible smooth projective variety over F, and L a line bundle on X. In this note, we will prove that if the contangent bundle of X is ample and X is non-isotrivial, then there are a proper closed algebraic set Y of X and a constant A > 0 such that h_L(P) <= A d(P) + O(1) for all P \in X(\bar{F}) \ Y(\bar{F}), where h_L(P) is a geometric height of P with respect to L and d(P) is the geometric logarithmic discriminant of P. As corollary of the above height inequality, we can…
The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum of at least n general sufficiently ample hypersurfaces.
A permutation is called covexillary if it avoids the pattern $3412$. We construct an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. As applications of this embedding, we show that the characteristic cycles of covexillary Schubert varieties are irreducible, and provide a new proof of Lascoux's model computing Kazhdan-Lusztig polynomials of vexillary permutations. Combining the above embedding with earlier work of the author on the conormal varieties of Grassmannian Schubert varieties, we develop an algebraic criterion identifying the conormal varie…
We prove a formula which compares intersection numbers of conormal varieties of two projective varieties and their dual varieties. When one of them is linear, we can recover the usual Plucker formula for the degree of the dual variety. The basic strategy of the proof is to study a category of Lagrangian subvarieties in the cotangent bundle of a projective space under a birational transformation.
For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.
We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.
Lefschetz-type theorems for the effective cone on Hyperkähler varieties
本文证明超凯勒簇有效锥的 Lefschetz 型定理，即光滑 ample 除子诱导有效锥同构，并推广至例外除子，从而计算射影化余切丛等有效锥。
A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.
Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A
针对A型特殊线性群的偏Flag簇余切丛上的向量丛，通过与Bott--Samelson簇上同调的比较得到高阶上同调消失等结论，进而给出相关函数环的生成元集并证明其仿射化具有终端奇点。
We study for rationally connected varieties $X$ the group of degree 2 integral homology classes on $X$ modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies that this group is trivial for any rationally connected variety.
Hodge microsheaves on cotangent bundles and plumbings
本文引入Hodge微层范畴，研究缠绕层的Hodge理论版本，并应用于代数簇环路空间上同调的Hain Hodge结构及Ginzburg代数的Koszul对偶。
It is shown that for a perverse sheaf $K$ on an abelian variety $X$ the integers $i$ for which the cohomology $H^i(X,K)$ does not vanish define an interval in the number line (under certain conditions on the field of definition of $K$)
Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $μ:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $μ(T^*_XX_w)$. Inspired by the…
A Remark on Stability Conditions on Smooth Projective Varieties
本文证明了平滑射影簇的有界导出范畴总能接纳Bridgeland意义下的稳定性条件。
In this paper we study the connection between rigid sheaves and separable-exceptional objects on Fano varieties over arbitrary fields. We give criteria for a rigid vector bundle on a Fano variety to be the direct sum of separable-exceptional bundles and for a separable-exceptional vector bundle to be part of a full separable-exceptional collection.
In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space. This answers in the negative (at least in the category of terminal varieties) a question posed by Krylov.
This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-Kähler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.
We study the Lefschetz standard conjecture on a smooth complex projective variety X. In degree 2, we reduce it to a local statement concerning deformations of vector bundles on X. When X is hyperkähler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension 2.
In this paper we define an action of the Weyl group on the quiver varieties $M_{m,\grl}(d,v)$ with generic $(m,\grl)$. To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectness for the smooth quiver variety $M(d,v)$ and normality for $M_0(d,v)$ in the case of a quiver of finite type and $d-v$ a regular weight.
In this paper, we firstly point out, by a counter example, that Proposition 6.4 of Section 6 in Bump's book (Algebraic Geometry) is error, and then give a correct statement with proof. We finally point out a gap in the proof of Theorem 3, in Chapter I Section 8, of Mumford's red book, and indicate a way to complete it.
Dynamics of Projectivized Toric Vector Bundles
该研究聚焦环面簇上射影丛的满自同态动力学，通过结构定理、动力次数控制分析及显式转移函数法，完成分类、验证猜想并证明相关丛无非自同构交换满自同态，推进代数几何动力学研究。
We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).
In this note, using methods introduced by Hacon, McKernan and Xu, we study the accumulation points of volumes of varieties of log general type. First, we show that, if the set of boundary coefficients $Λ$ is DCC, closed under limits and contains 1, then also the corresponding set of volumes is DCC and closed under limits. Then, we consider the case of $ε$-log canonical varieties, for $0 < ε< 1$. In this situation, we prove that, if $Λ$ is finite, then the corresponding set of volumes is discrete.
We conjecture the equality of the numerical and Kodaira dimensions $ν_1^*(X)$ and $κ_1^*(X)$ for the cotangent bundle of compact Kähler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $ν_1^*(X)=κ_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite étale cover of $X$. The proof rests on the Beauville-Bog…
Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same card…
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…
We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.
We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-Kähler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.
Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties
本文研究Iwahori-Coulomb分支在局部化equivariant量子上同调中的多项式性质，通过稳定包络和Demazure-Lusztig元素计算作用，并应用到旗流形切丛的量子上同调中。
We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of p…
We present a simple description of moduli spaces of torsion-free D-modules (``D-bundles'') on general smooth complex curves X, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties. Namely, we show that the moduli of D-bundles form twisted cotangent bundles to stacks of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of ``perverse vector bundles'' on T^*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes (T^*X)^…
Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of $q$-difference equations holomorphic in $\boldsymbol{z}$ with poles in $\boldsymbol{a}$ to a basis of solutions holomorphic in $\boldsymbol{a}$ with poles in $\boldsymbol{z}$. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this …
We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomor…
Let us say that a curve $C\subset\mathbb P^3$ is osculating self-dual if it is projectively equivalent to the curve in the dual space $(\mathbb P^3)^*$ whose points are osculating planes to~$C$. Similarly, we say that a $k$-dimensional subvariety $X\subset\mathbb P^{2k+1}$ is osculating self-dual if its second osculating space at the general point is a hyperplane and $X$ is projectively equivalent to the variety in $(\mathbb P^{2k+1})^*$ whose points are second osculating spaces to $X$. In this note we show that for each $k\ge 1$ there exist many osculating self-dual $k$-dimensional subvarieti…
We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.
Catanese and Franciosi defined a semispecial tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor and by a 2-torsion line bundle. A slope zero tensor is instead a section of the nm-th symmetric power of the cotangent bundle twisted by m times the anticanonical divisor. With these definitions we have: Theorem 1. The universal cover of X is the polydisk iff 1) holds. 1) X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding hypersurface in the projectivized tangent space is reduce…
We study the cones of q-ample divisors on smooth complex varieties. In favourable cases, we identify a part where the closure of this cone and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties.
We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the cotangent bundle over an invariant Lagrangian subvariety coincide. This implies that the complexity and rank of the Lagrangian subvariety are equal to the half of the corank and to the defect of the Hamiltonian variety, respectively. This result generalizes a theorem of Panyushev on the complexity and rank of a conormal bundle. A simple elementary proof of this theo…
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