V. Balaji, János Kollár
The Mehta-Ramanathan theorem ensures that the restriction of a stable vector bundle to a sufficiently high degree complete intersection curve is again stable. We improve the bounds for the "sufficiently high degree" and propose a possibly optimal conjecture.
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We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
Restriction of stable bundles on a jacobian of genus 2 to an embedded curve
本文的目的是描述从2维Jacobian $X$上具有小$c_2$的稳定秩2丛的模空间到嵌入$X$中的亏格2曲线$C$上的稳定秩2丛的模空间的限制映射.
A stratified bundle is a vector bundle which is a D-module. We show that regular singularity of stratified bundles on smooth varieties in positive characteristic is preserved by pullback and that regular singularity can be checked on curves, if the ground field is large enough.
Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.
Using Bridgeland stability conditions we give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.
The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$, let $X$ be a smooth projective variety over $k$, let $H$ be a very ample line bundle on $X$ and let $E$ be a semistable (resp. stable) principal $G$-bundle on $X$ w.r.t. $H$. The main result of this paper is that the restriction of $E$ to a general smooth curve which is a complete intersection of ample hypersurfaces of sufficiently high degree's is again semi…
It is known that the cotangent bundle $Ω_Y$ of an irreducible Hermitian symmetric space $Y$ of compact type is stable. Except for a few obvious exceptions, we show that if $X \subset Y$ is a complete intersection such that $Pic(Y) \to Pic(X)$ is surjective, then the restriction $Ω_{Y|X}$ is stable. We then address some cases where the Picard group increases by restriction.
Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles E which do not enjoy this property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J. 96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew. Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Mat…
In this article we deduce criteria for the splitting and the triviality of vector bundles, by restricting them to partially ample divisors. This allows to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with a number of examples. For products of minuscule homogeneous varieties, our results allow to test the splitting of vector bundles by restricting them to products of Schubert 2-planes. The triviality criteria obtained inhere are particularly suited to Frobenius split varieties, whose splitting is defined by a section in the anti-canonical li…
Restriction of stable rank two vector bundles in arbitrary characteristic
设$X$是定义在具有任意特征的代数闭域上的光滑簇，$Ø_X(H)$是$X$上的非常充分的线丛.我们证明了对于秩为2的半稳定$X$ -丛$E$，存在一个仅依赖于$Δ(E).H^{\dim(X)-2}$和$H^{\dim(X)}$的整数$m$，使得在$|mH|$中将$E$限制为一个通约数也是半稳定的.作为推论，我们得到了任意特征曲面的有界性结果，以及Bogomolov定理和Kodaira消灭定理的弱形式。
The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n. A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every…
Deformation to the normal bundle and blow-ups via derived Weil restrictions
本文通过导出Weil限制方法，研究了导出代数几何中法向丛的变形和吹胀，推广了之前在准光滑情况下的研究成果。
Restriction of the Poincaré bundle to a Calabi-Yau hypersurface
设$\cMx$是连通Riemann曲面$X$上秩为$n\geq 3$且行列式为$ξ$的稳定向量丛的模空间，$n$与$d(ξ)$互质.设$D$是$\cMx$的Calabi-Yau超曲面。用$U_D$表示通用束对$X\times D$的限制。结果表明，对于任何$x\in X$，$(U_D)_x$到$x\times D$的限制是稳定的。此外，对于一般曲线，$D$上半稳定层的模空间的连通分支，包含$(U_D)_x$，同构于$X$。它还表明，$U_D$是稳定的任何极化，和连接组件的模空间的半稳定层超过$X\times D$，包含$U_D$，是同构的雅可比矩阵。此外，这是极化簇的同构，因此这样的模空间决定了Reimann曲面。
Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
本文研究梳状曲线上半稳定向量丛限制到各分量时的半稳定性判据，并推广核丛半稳定性结果至一般模空间分量情形。
For $n\geq 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let $\mathcal{O}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in\vert \mathcal{O}_M(d_i)\vert$ with $d_i\geq 2$. We prove a vanishing theorem for twisted holomorphic forms on $Y$. As an application, we show that the tangent bundle $T_Y$ of $Y$ is stable. Moreover, if $X$ is a smooth hypersurface of degree $d$ in $Y$ such that the restriction $Pic(Y)\rightarrow Pic(X)$ is surjective, we establish some e…
Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz fnctions
本文讨论余切丛Ω_{ \PP^N }对椭圆曲线C \subset \PP^N的可能限制.本文将这一正特征应用于分次截环R = \bigoplus_{n \in \NN} Γ（C，n）中齐次R_+准素理想\subset R的Hilbert-Kunz函数的计算.
We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.
Seshadri constants, gonality of space curves and restriction of stable bundles
我们定义了空间曲线的Seshadri常数，并考虑了估计它的方法，然后我们证明了它决定了曲线的角性。我们使用一个论点的基础上博戈莫洛夫的不稳定性定理的三倍。同样的方法也被用于研究P^3上的稳定向量丛在曲线和曲面的限制下的行为。
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
Semistability and restrictions of tangent bundle to curves
我们考虑所有的复射影流形X满足以下三个条件中的至少一个：1。存在一对$(C ,φ)$，其中$C$是紧致连通Riemann曲面，$φ: C\to X$是全纯映射，使得拉回$φ^*TX$不是半稳定的. 2.簇$X$允许一个阿贝尔簇覆盖的代数。3.尺寸$\dim X \leq 1$。我们猜想所有的复射影流形都是上述类型，并证明了下列类是上述类型。i）具有有限基本群的所有$X$。ii）所有$X$，使得从投影线到$X$存在非常数态射。iii）所有的$X$使得典范线丛$K_X$为正或负或$c_1(K_X) \in H^2(X, {\mathbb Q})$消失。iv）所有投影曲面。
Picard bundles and Twisted Picard bundles on the Jacobian of a curve
研究了曲线雅可比上皮卡束的限制及其稳定性，揭示了稳定ACM束的构造与theta divisor的关系。
Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I d…
Modular vector bundles on hyperkähler manifolds of Debarre-Voisin type
本文针对非常一般的Debarre-Voisin四维簇，证明了Gr(6,10)商丛限制的Schur函子是半稳定模向量丛，确定了原子丛形式，并计算得到非平凡一维Ext群的例子。
Extensions of vector bundles and rationality of certain moduli spaces of stable bundles
本文证明了某些稳定的秩3向量丛可以写成线丛和稳定的秩2丛的扩张。作为应用，我们证明了射影平面P^2上稳定秩3丛的某些模空间的合理性。
Weil restriction, normal bundles and motivic Thom spaces
本文通过Weil限制研究法丛与动机Thom空间，证明Weil限制保持向量丛并扩展法丛结果，进而建立与动机Thom空间的相容性，并在有理系数动机上同调中证明Weil限制映射与范数映射一致且保持Thom类。
Positivity for toric vector bundles
证明了完备复曲面簇上的等变向量丛是nef的当且仅当它对每个不变曲线的限制分别是nef的或充足的。进一步证明了Nef复曲面向量丛在每一点都有非零的整体瓣，并推导出基础向量丛是平凡的当且仅当它对每一条不变曲线的限制是平凡的.我们应用我们的方法和结果来研究，特别是，向量丛M_L，出现作为核的评价地图上的部分L，当L是一个丰富的线丛。我们给出的例子扭曲的束是充足的，但不是全球性的。
We study stable rationality properties of conic bundles over rational surfaces.
Stable vector bundles on algebraic surfaces
证明了代数曲面上任意秩的稳定向量丛的存在性结果，并确定了有理直纹曲面上稳定丛的某些模空间的双有理结构。
Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so.
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
Semistable reduction for G-bundles on curves
证明了曲线上几乎任意特征的主丛的一个半稳定约化定理。对于例外群，我们需要对特征进行一些小的显式限制。
We construct a vector bundle $E$ on a smooth complex projective surface $X$ with the property that the restriction of $E$ to any smooth closed curve in $X$ admits an algebraic connection while $E$ does not admit any algebraic connection.
Brill-Noether Theory for stable vector bundles
本文综述了代数曲线上向量丛的Brill-Noether理论的主要结果。
We clarify the undecided case $c_2 = 3$ of a theorem of Ein, Hartshorne and Vogelaar [Math. Ann. 259 (1982), 541--569] about the restriction of a stable rank 3 vector bundle with $c_1 = 0$ on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by $-1$); it has $c_3 = 6$. There are also some exceptions with $c_3 = 2$ (plus, of course, their duals). We also prove, for completeness, the basic properties of the corresponding moduli spac…
A recent paper of Totaro develops a theory of $q$-ample bundles in characteristic 0. Specifically, a line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\geq m_0$ implies $H^i(X,\mathcal{F}\otimes \mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension 1 subscheme $D$ of $X$ suc…
We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable bundle, then a large enough direct sum $V^{\oplus n}$ has ample deformations unless there is an obvious numerical reason why it cannot. Previous work in this area has mostly focused on rank two bundles and relied primarily on classical constructions such as the Serre construction. In contrast, we use recent advances in moduli of vector bundles to obtain strong…
Restriction theorems for homogeneous bundles
本文证明了对于不可约表示$τ:GL(n)\to GL(W)$，相应的齐次${\bf P}_k^n$ -向量丛$W_τ$在限制于${\bf P}_k^n$中的任意光滑二次或任意光滑三次时是强半稳定的，其中$k$分别是特征$\neq 2,3$的代数闭域.特别是$W_τ$是半稳定的时候限制到一般超曲面的程度$\geq 2$和强半稳定的时候限制到$k$ -通用超曲面的程度$\geq 2$。
We prove that if $X$ is the total space of an elliptic principal bundle $π:X\ra B$ which is non-Kähler, then the restriction of any torsion-free sheaf on $X$ to the general fiber of $π$ is semi-stable.
We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. We discover examples of moduli spaces of rank 2 stable bundles on surfaces of Picard rank one with arbitrarily many connected components.
Poincare polynomials of moduli spaces of stable bundles over curves
给定有限域上的一条曲线，计算其上秩和次数不一定互质的稳定丛的个数，并应用此结果计算曲线上稳定丛的模空间的虚Poincare多项式.给出了虚Hodge多项式和模的一个类似公式。
Stable bundles on 3-fold hypersurfaces
利用单子，我们在3重超曲面上构造了一大类秩为2和3的稳定丛，并研究了所有可能的稳定向量丛的Chern类的集合。
We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.
For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.
For an abelian surface $A$, we explicitly construct two new families of stable vector bundles on the generalized Kummer variety $K_n(A)$ for $n\geqslant 2$. The first is the family of tautological bundles associated to stable bundles on $A$, and the second is the family of the "wrong-way" fibers of a universal family of stable bundles on the dual abelian surface $\widehat{A}$ parametrized by $K_n(A)$. Each family exhibits a smooth connected component in the moduli space of stable bundles on $K_n(A)$, which is holomorphic symplectic but not simply connected, contrary to the case of K3 surfaces.
Rank-3 stable bundles on rational ruled surfaces
本文比较了有理直纹曲面上秩为3的向量丛关于不同的充分因子稳定的模空间。我们还讨论了这些模空间的不可约性、单合理性和合理性。
We observe that the proof of the Bogomolov stable restriction theorem can be adapted to give an ampleness criterion for globally generated rank 2 vector bundles on certain surfaces. This applies to the Lazarsfeld-Mukai bundles, to congruences of lines in P^3, and possibly to the construction of surfaces with ample cotangent bundle (help welcome!).
Frobenius inverse image of the semi-stable boundary in the moduli space of vector bundles
研究了一类稳定的秩为2的具有平凡行列式的向量丛，其Frobenius拉回在亏格g>1的一般曲线上是不稳定的。在亏格2中，我们应用最近的结果有关的θ因子与束B的局部精确微分形式，我们推导出一个计划理论描述的Frobenius逆像的半稳定边界的模空间。
Some stable vector bundles with reducible theta divisors
设C是亏格g的曲线，L是C上的2g次线丛.设M是从具有纤维的平凡丛H^0（C，L）到L的赋值映射的核.我们证明了当L足够一般时，秩g丛M及其外幂是稳定的，并且允许一个可约的θ因子.
A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For X of genus at least 5, we construct stable vector bundles over X of rank $r$ for all $r \geq 5$, with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the appendix, Raynaud's original example of a stable rank 2 vector bundle with reducibl…
Let $π: X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $π$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,θ)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= π^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,ψ)$ of degree $d$ on $C$ whose pullback under $π$ is isomorphic to $(V,θ)$. As a consequence, we get an…
Asymptotically Z-stable bundles over projective surfaces
本文研究了多项式曲面上渐近Z稳定的向量丛的存在性，提出了一种构造秩3严格渐近Z稳定向量丛的方法，并给出了新的例子及Hoppe准则的推广。
We extend a result by Fulton-Harris-Lazarsfeld in Brill-Noehter theory of line bundles and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to the Brill-Noehter locus of stable bundles inside the moduli space of higher rank stable vector bundles on a smooth projective algebraic curve. We give some consequences of this extended result.
Exterior power of stable vector bundle destabilized by Frobenius pull-back
本文研究了在正特征域上，稳定向量丛经弗罗贝尼乌斯拉回后其外幂不再半稳定的性质。
Stratifying the moduli space of stable vector bundles by decomposition type
本文研究正规射影簇上稳定向量丛模空间按分解类型的分层结构，并在亏格至少为2的光滑射影曲线上给出各层的大多精确维数估计，进而应用于素到p可平凡化稳定丛闭包的维数估计。
We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.
In this paper we construct new examples of stable bundles of rank 2 of small degree with 4 sections on a smooth irreducible curve of maximal Clifford index. The corresponding Brill-Noether loci have negative expected dimension of arbitrarily large absolute value.
For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable (it does not admit any nonzero nilpotent Higgs field).
We discuss stability conditions for all rank-2 ample vector bundles on Hirzebruch surfaces with the second Chern class less than 7.
We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, th…
Let X be a smooth projective curve of genus $g\geq 2$ defined over an algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a stable bundle on $X_{1}$ with $I(E)= (p-1)(2g-2)$ if and only if E is the direct image of some stable bundle W on $X$.
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