Ernesto C. Mistretta
We show the stability of certain syzygies of line bundles on curves, which we call transforms, and are kernels of the evaluation map on subspaces of the space of global sections. For the transforms constructed, we prove the existence of reducible theta divisors, in the cases where the slope is integer.
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Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher-codimensional generalization of Ueda's result, we give a sufficient condition for the existence of a non-singular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.
Ueda's theory is a theory on a flatness criterion around a smooth hypersurface of a certain type of topologically trivial holomorphic line bundles. We propose a codimension two analogue of Ueda's theory. As an application, we give a sufficient condition for the anti-canonical bundle of the blow-up of the three dimensional projective space at $8$ points to be non semi-ample however admit a smooth Hermitian metric with semi-positive curvature.
We classify smooth complex projective varieties $X \subset \proj^N$ of dimension $2s+1$ containing a linear subspace $Λ$ of dimension $s$ whose normal bundle $N_{Λ/X}$ is numerically effective.
Let N be the moduli space of stable rank 2 quasiparabolic vector bundles of fixed degree on the projective line with 2g+1 marked points, where g>1, and stability is with respect to the weights {0,1/2} at each marked point. In this note we show that N is isomorphic to the variety of (g-2)-dimensional linear subspaces of P^{2g}, contained in the intersection of two quadrics. The proof relies on the work of Bhosle on the relation among quasiparabolic vector bundles on P^1 and invariant vector bundles on hyperelliptic curves, and the description by Bhosle and Ramanan of the moduli space of stable …
Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces
本文研究K3曲面上单纯层模空间，通过合冲丛与扩张丛构造，证明其诱导态射为局部闭嵌入，并建立原模空间拉格朗日/迷向代数子空间到另一分支同类子空间的对应关系。
Affine Subspace Concentration Conditions
本文定义了格点多面体的新型仿射子空间集中条件，并证明其在光滑和反射多面体中成立，核心方法涉及canonical扩展的斜率稳定性及Fano toric变种上的延伸类。
Linear subspaces of the intersection of two quadrics via Kuznetsov component
本文通过 Kuznetsov 分量将两个二次曲面交集中的线性子空间与超椭圆曲线上的向量丛对应，给出 Desale-Ramanan 分类的新证明，并构造了到稳定丛模空间的闭嵌入。
Let $C$ be a chain-like curve having $n$ smooth components and $n-1$ nodes, where $n \geq 2$. Let $E$ be a vector bundle on $C$ and $V \subseteq H^0(E)$ be a linear subspace generating $E$. We investigate the (semi)stability of the kernel bundle $M_{E,V}$ associated to $(E,V)$.
Let M be the moduli space of SO(r)-bundles on a curve, and L the determinant bundle on M. We define an isomorphism of H^0(M,L) onto the dual of the space of r-th order theta functions on the Jacobian of C. This isomorphism identifies the map M -->|L|* defined by the linear system |L| with the map M -->|r Theta| which associates to a quadratic bundle (E,q) the Theta divisor of the vector bundle E . The two components M+ and M- of M are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for Sp(2r)-bundles.
This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier-Mukai kernel ${\cal E}$ on a product $B\times X$ by a unitary flat line bundle on the fiber $X$. When this fiber is a complex surface and ${\cal E}$ is a holomorphic 2-bundle, the result can be interpreted …
Stability of syzygy bundles over certain curves of compact type
本研究针对紧型曲线$C$上由线性子空间$V \subseteq \mathrm{H}^0(C, E)$生成的向量丛$E$，探讨其关联合冲丛$M_{E,V}$在特定紧型曲线上的稳定性问题。
Moduli of Higgs Coherent Systems with stable underlying vector bundles over a curve
本文研究亏格至少为2的复射影曲线上的希格斯相干系统，引入其模问题，并构造了底层向量丛稳定时的精细模空间。
Stability of kernel bundles on projective bundles over curves
该研究针对亏格g≥3的光滑曲线C上的射影丛X，在C、X及生成子空间V为一般且𝒪_X(1)足够丰裕的条件下，证明了系统(𝒪_X(1),V)的核丛是𝒪_X(1)-稳定的。
The Desale-Ramanan Theorem is an isomorphism between the moduli space of rank two vector bundles over complex hyperelliptic curve and the variety of linear subspaces in an intersection of two quadrics. We prove a real version of this theorem for the moduli space of real vector bundles over a real hyperelliptic curve. We then apply this result to study the topology of the moduli space, proving that it is relatively spin and identifying the diffeomorphism type for genus two curves. Our results lay the groundwork for future study of the quantum homology of these moduli spaces.
Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace
本文应用 Chow-Mumford 线丛理论，证明了沿线性子空间爆破的射影空间在例外除子上的形变到法锥测试构型具有 $\K$-半稳定性。
Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the coherent systems depend on a parameter $τ$. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. For small values of $τ$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles con…
Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bund…
In this note, we obtain a number of results related to the hard Lefschetz theorem for pseudoeffective line bundles, due to Demailly, Peternell and Schneider. Our first result states that the holomorphic sections produced by the theorem are in fact parallel, when viewed as currents with respect to the singular Chern connection associated with the metric. Our proof is based on a control of the covariant derivative in the approximation process used in the construction of the section. Then we show that we have an isomorphsim between such parallel sections and higher degree cohomology. As an applic…
Normal Bundle Splitting Strata of Rational Curves in Toric Varieties
本文研究环面簇中有理曲线法丛分裂型的分层，通过Cox构造和相对上同调态射给出非平衡性数值判据，并应用于射影空间爆破，与已有结果等价。
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles…
Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $λ_i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in C$ and linear subspaces $U_i\subseteq E|P_i$ such that $\dim (U_i)= λ_i$ for all $i$ and the evaluation map $H^0(E)\to \oplus _{i=1}^{s} U_i$ has maximal rank (A. Atanasios). We prove that $C$ satisfies strong interpolation if either $C$ is a linearly normal elliptic curve or $C$ is a general embedding of degree $d\ge (5n-8)g+2n^2-5n+4$ of a smooth curve $X$ of gen…
Given a vector bundle $E$ on a complex reduced curve $C$ and a subspace $V$ of $H^0(E)$ which generates $E$, one can consider the kernel of the evaluation map $ev_V:V\otimes \mathcal{O}_C\to E$, i.e. the {\it kernel bundle } $M_{E,V}$ associated to the pair $(E,V)$. Motivated by a well known conjecture of Butler about the semistability of $M_{E,V}$ and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known …
We study, in the setting of algebraic varieties, finite-dimensional spaces of functions V that are invariant under a ring D^V of differential operators, and give conditions under which D^V acts irreducibly. We show how this problem, originally formulated in physics (Kamran-Milson-Olver), is related to the study of principal parts bundles and Weierstrass points (Laksov-Thorup), including a detailed study of Taylor expansions. Under some conditions it is possible to obtain V and D^V as global sections of a line bundle and its ring of differential operators. We show that several of the published …
Special K-stability and positivity of CM line bundles
该研究证明了参数化极大变异特殊K-稳定簇的真族上CM线丛的丰沛性，并应用该结论推导出曲线上一致绝热K-稳定klt-平凡纤维化粗模空间的任意真子空间具有射影性。
We determine all the Fourier-Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by the positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier-Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Using these results we draw…
I define an algebraic metric and closed 3-form on a subspace $\mathcal{M}$ of the moduli of $G$-bundles on a complex projective curve $C$, and show that the resulting two-dimensional $σ$-model with target $\mathcal{M}$ has a zero-curvature formulation.
Sections of Hodge bundles I: Global theory and applications to period maps
本文通过变形论和矩阵表示两种方法，研究Hodge包的全局截面，证明提升后的周期映射像包含在复欧几里得子空间中，并构造Calabi-Yau类型流形的Teichmüller空间的全局复仿射结构。
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), …
We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are $K$-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.
We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic subspaces in the fibers as P^1-bundles over the discriminant double cover. When the P^1-bundle is suitably stable, we deduce effective estimates for the heights of sections over finite fields satisfying various approximation conditions. We also discuss the behavior of the spaces of sections as the base of the fibration acquires singularities.
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.
We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibres degenerate along t…
We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra $\mathcal{H}_{G,C}$. In the case $G=\rm{GL}_2$, we construct a commutative subalgebra in $\mathcal{H}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G=\rm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. I…
This paper considers curves in Grassmannians which are themselves immersed in projective space by the Plucker map. It is shown that for a generic vector bundle of high enough degree, the image curve lies in a proper linear subvariety of this projective space and satisfies good conditions on syzygies as a curve in this subspace. For very small degree and generic vector bundle, the curve is non- degenerate.
When is the matroid Schubert variety $\mathbb{Q}$-Gorenstein?
该研究针对拟阵舒伯特簇$Y_V$，运用运算乔伊上同调证明其上线丛的限制性质，给出其为戈伦斯坦与$\funnybb{Q}$-戈伦斯坦的组合学刻画，并构造了三类不同性质的实例。
Fano 4-fold quiver moduli from subspace quivers
研究在特定维数向量假设下子空间箭图表示的模空间，通过 GIT 商分类出四个法诺四维流形，用箭图模技术描述其几何，其中两个已知，两个新，为大皮卡秩法诺四维流形分类提供有趣例子。
A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
该论文将代数向量丛的各类欧拉类等同，建立其整性结果与局部指标公式，应用拓扑欧拉数计算射影空间完全交上d-平面算术计数对应的欧拉类。
A `coherent system' $(\Cal E,V)$, consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Based on the usual notion in Geometric Invariant Theory, a notion of slope stability has been defined for such objects (by Le Potier, and also by Rhagavendra and Vishwanath). In this paper we show that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (i.e. the Hermitian-Einstein equation with an additional zeroth order term), and the other is an or…
Vanishing Cohomology of Dominant Line Bundles for Real Groups
本文将Broer关于线束高阶上同调消失的定理推广至实约化代数群，通过结合典范丛计算与Grauert-Riemenschneider消失定理证明了特定权锥下的上同调消失，并以此刻画奇点及给出Kostant-Rallis结果的新证明。
Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to …
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