Indranil Biswas, Tomas L. Gomez
Let $M_X(r,ξ)$ be the moduli space of stable vector bundles, on a smooth complex projective curve $X$, of rank $r$ and fixed determinant $ξ$ such that $°(ξ)$ is coprime to $r$. If $E$ is a vector bundle $M_X(r,ξ)$ whose restriction to every Hecke curve in $M_X(r,ξ)$ is trivial, we prove that $E$ is trivial.
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We define Hecke transformation for orthogonal bundles over a compact Riemann surface. Using the cycles on a moduli space of orthogonal bundles given by Hecke transformations, we prove that the projectivized Picard bundle on the moduli space is stable.
We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.
Let $X$ be a smooth projective curve of genus $g \ge 3$ and let $M_0$ be the moduli space of semistable bundles over $X$ of rank 2 with trivial determinant. Three different desingularizations of $M_0$ have been constructed by Seshadri \cite{Se1}, Narasimhan-Ramanan \cite{NR}, and Kirwan \cite{k5}. In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in \ci…
In this paper, we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of "universal" Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the "wonderful," or De Concini--Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametriz…
Hecke modifications of vector bundles
本文综述向量丛 Hecke 修改的起源、文献脉络与显式描述进展，并给出射影直线及椭圆曲线相关情形的新分类结果。
Hecke transformation for orthogonal bundles over curves
本文在光滑射影曲线上的正交丛模空间中，利用拉格朗日子模的初等变换定义Hecke变换，证明Tyurin对偶定理的类比成立，并详细研究了秩为2、3、4、6的特殊情形。
Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $\mathrm{deg}\,L>0$. Let $\mathbf{M}$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let $\mathbf{M}_{X}$ be the moduli space of stable rank 2 $L$-twisted Higgs bundles with determinant $\mathcal{O}(-x)$ for some $x\in X$ on $X$. We construct a cycle in the product of a stack of rational maps from nonsingular curves to $\mathbf{M}_{X}$ and $\mathrm{Pic}^{2\mathrm{deg}\,L}(X)$ by using Hecke modifications of a stable $L$-twisted Higgs bundle in …
Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles
本文研究了特征为2的代数闭域上光滑射影曲线X的模2向量丛模空间中的Frobenius分层，并证明其中存在一个由Hecke曲线覆盖的分层。
Hall algebras and Hecke modifications of vector bundles
本文研究光滑射影曲线上向量丛的 Hecke 修正，利用 Hall 代数给出完整分类并证明无分歧环面自守形式空间平凡。
Let X be a proper scheme and Z a prestack over X equipped with a flat connection. We give a local-to-global description of D-modules on the prestack S(Z) of flat sections of Z. Examples of S(Z) include the moduli stacks of principal G-bundles and de Rham local systems on X. We show that the category of D-modules is equivalent to the category of ind-coherent sheaves which are equivariant with respect to infinitesimal Hecke groupoids parametrized by finite subsets of X. We describe a number of applications to geometric representation theory and conformal field theory, including a derived enhance…
We study the parametrization of the moduli space Bun_2(C)_L of rank 2 bundles over a curve C with fixed determinant, provided by Hecke modifications at fixed points of the trivial bundle. This parametrization is closely related to the Tyurin parametrization of vector bundles over curves. We use it to parametrize the Hitchin and KZB systems, as well as lifts of the Beilinson-Drinfeld D-modules. We express a generating series for the lifts of the Beilinson-Drinfeld operators in terms of a "quantum L-operator" \ell(z). We explain the relation to earlier joint work with G. Felder, based on paramet…
We prove that the moduli space of stable maps of degree 2 to the moduli space of rank 2 stable bundles of fixed determinant O(-x) over a smooth projective curve of genus g>2 has two irre- ducible components which intersect transversely. One of them is Kir- wan's partial desingularization of the moduli space of rank 2 semistable bundles with determinant isomorphic to O(y-x) for some y in X. The other component is the partial desingularization of PHom(sl(2)^*;W)//PGL(2) for a vector bundle W of rank g over the Jacobian of X. We also show that the Hilbert scheme H, the Chow scheme C of conics in …
We define functors on the derived category of the moduli space M of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space M, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.
Spectral Data of Special Orthogonal Higgs Bundles and Hecke Modification
该研究利用Hecke修改的新方法，完整自洽地计算了$\text{SO}_{2n+1}$等类型希钦纤维化一般纤维中希格斯丛的谱数据，澄清了文献中对应关系的疑点，还计算了连通分支数并论证了纤维中的朗兰兹对偶性。
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism …
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.
Projectivity of good moduli spaces of vector bundles on stacky curves
本文利用良模空间存在性判据证明栈曲线上半稳定向量丛模栈具有射影良模空间，并给出线丛无基点性的有效界，从而为抛物丛模空间提供了新的有效构造。
We prove that the normalized Poincaré bundle on the moduli space of stable rank $r$ vector bundles with a fixed determinant on a smooth projective curve $X$ induces a family of nef vector bundles on the moduli space. Two applications follow. We show that when the genus of $X$ is large, the derived category of $X$ is embedded into the derived category of the moduli space for arbitrary rank and coprime degree, which extends the results of Narasimhan, Fonarev-Kuznetsov, and Belmans-Mukhopadhyay. As the second application, we construct a family of ACM bundles on the moduli space. A key ingredient …
Recent work on the log minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program. Our main result is the construction of a universal moduli space of slope semi-stable sheaves which compactifies the moduli space of vector bundles over the moduli space of pseudo-stable curves.
We construct a compactification of the universal moduli space of semistable principal $G$-bundles over $\overline{\textrm{M}}_{g}$, the fibers of which over singular curves are the moduli spaces of $δ$-semistable singular principal G-bundles.
We discuss some relations of moduli of sheaves on rational surfaces by using universal extensions. These are a generalization of Maruyama's method to construct Uhlenbeck compactification of moduli of vector bundles.
We examine the moduli of framed holomorphic bundles over the blowup of a complex surface, by studying a filtration induced by the behavior of the bundles on a neighborhood of the exceptional divisor.
In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the…
The variety of minimal rational tangents associated to Hecke curves was used by J.-M.Hwang [8] to prove the simplicity of the tangent bundle on the moduli of vector bundles over a curve. In this paper, we use the tangent maps of the symplectic and orthogonal Hecke curves to prove an analogous result for symplectic and orthogonal bundles. In particular, we show the nondegeneracy of the associated variety of minimal rational tangents, which implies the simplicity of the tangent bundle on the moduli spaces of symplectic and orthogonal bundles over a curve. We also show that for large enough genus…
The line bundles on the moduli stack of principal bundles on families of curves
研究连通约化群G在任意光滑曲线族上主丛模叠的Picard群，确定其线丛结构。
We prove that the moduli space of mathematical instanton bundles on ${\Bbb P}^3$ with $c_2=5$ is smooth.
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.
We consider the moduli space of stable vector bundles on curves embedded in P_2 with Hilbert polynomial 3m+1 and construct a compactification of this space by vector bundles. The result is a blow up of the Simpson moduli space M_{3m+1}(P_2).
On the smoothness of moduli spaces for quiver bundles
本文探讨光滑复射影曲线上有限箭袋向量丛模空间的光滑性问题。
We generalize the construction of M. Lieblich for the compactification of the moduli stack of $\PGL_r$-bundles on algebraic spaces to the moduli stack of Tanaka-Thomas $\PGL_r$-Higgs bundles on algebraic schemes. The method we use is the moduli stack of Higgs version of Azumaya algebras. In the case of smooth surfaces, we obtain a virtual fundamental class on the moduli stack of $\PGL_r$-Higgs bundles. An application to the Vafa-Witten invariants is discussed.
We present a systematic study of involutions on the moduli space of $G$-Higgs bundles over an elliptic curve $X$, where $G$ is complex reductive affine algebraic group. The fixed point loci in the moduli space of $G$-Higgs bundles on $X$, and in the moduli space of representations of the fundamental group of $X$ into $G$, are described. This leads to an explicit description of the moduli spaces of pseudo-real $G$-Higgs bundles over $X$.
We compute Chow groups of moduli spaces of rank 2 vector bundles on curves with determinant of odd degree in terms of generators and relations.
In this paper we construct the moduli of semistable principal bundles with structure group a semisimple group, over nonsingular curves in positive characteristic, with good bounds on the prime.
Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct algebraic stacks and moduli schemes for such bundles.
We compute the Brauer group of the moduli stack of stable PGL(r)-bundles on a curve $X$ over an algebraically closed field of characteristic zero. We also show that the Brauer group of such a moduli stack coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of stable PGL(r)-bundles on $X$.
Isomorphisms between moduli stacks of vector bundles with fixed determinant
该研究通过曲线同构、向量丛对偶和线束拉回的组合，分类了具有固定行列式向量丛的模堆之间的同构，并比较了模堆与模空间的自同构群。
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.
In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.
The line bundles on moduli stacks of principal bundles on a curve
该研究针对代数闭域上仿射约化代数群的情形，确定了光滑射影曲线上主G-丛模栈的Picard群，推进了代数几何中模栈线丛的相关研究。
Universal Chern classes on the moduli of bundles
本文针对曲线稳定丛模空间非精细（秩与次数不互素）的情况，构造泛上同调类，证明曲线与丛模栈乘积上的泛丛的部分陈类可提升至曲线与稳定丛模空间的乘积上。
The Segre-Verlinde correspondence for the moduli space of stable bundles on a curve
该研究建立光滑射影曲线上半稳定向量丛模空间的经典Verlinde数与模空间上自然泛复形Segre数的对应关系，为相关代数几何研究提供新视角。
A morphism of the moduli functor of admissible semistable pairs to the Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface, is constructed. It is shown that these functors are isomorphic, and main components of moduli scheme for semistable admissible pairs are isomorphic to main components of the Gieseker -- Maruyama moduli scheme
We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F regular type. As a consequence, all higher cohomology of theta line bundle vanish. During the proof, we develop a method to estimate codimension, and consider the infinite grassmannians for parabolic $G$ bundles.
Let $X$ be a compact Riemann surface of genus $g\geq 3$ and let $\mathcal{M}_{\mathrm{par}}$ be the moduli space of semistable parabolic bundles over $X$. Let $\mathcal{M}_{\mathrm{parH}}$ denote the moduli space of semistable parabolic Higgs bundles over $X$. In this article, we study the Picard group of $\mathcal{M}_{\mathrm{par}}$ and $\mathcal{M}_{\mathrm{parH}}$.
We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.
Let M(j) denote the moduli space of bundles on the blown-up plane which restrict to the exceptional divisor as O(j)+O(-j). We show that there is a topological embedding of M(j) into M(j+1).
Twistor Hecke eigensheaves in genus 2
本文遵循已有策略，在亏格2的光滑射影曲线上构造了沿wobbly轨迹带奇点的平坦联络，验证了相关$D$-模为Hecke特征层，并基于非阿贝尔Hodge对应与Hitchin纤维揭示了局部系统与谱簇的联系。
We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under …
Mathematical instanton bundles of rank 4 and $c_2=2$ on ${\mathbb P}^4$ have a smoothquasiprojective moduli space, which is shown via a direct GIT construction. A complete classification of jumping lines of these vector bundles is obtained. The duals of these bundles are described. Dimension computations and irreducibility proofs for some "interesting" subsets of the moduli space are presented. Restrictions of vector bundles from ${\mathbb P}^4$ to ${\mathbb P}^3$ are shown to produce mathematical instanton bundles on ${\mathbb P}^3$.
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.
We prove a canonical identifications of the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras.
We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss …
We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.
Hecke cycles on moduli of vector bundles and orbital degeneracy loci
本研究从一般三向量出发构造轨道退化轨迹型法诺流形，证明其定义亏格二曲线的秩3半稳定向量丛模空间中的Hecke线族，建立其与奇模空间的同构关系，并得到亏格19的K3曲面。
We define a new class of holomorphic (n-1)-bundles on the n-dimensional complex projective space, that contains the Tango bundles and their stable Horrock's pull-backs; we show that these bundles are invariant under small deformations and that their isomorphism classes correspond to smooth points of moduli spaces.
Let $X$ be a smooth irreducible projective curve with an involution $σ$. A vector bundle $E$ over $X$ is called anti-invariant if there exists an isomorphism $σ^*E\rightarrow E^*$. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over $X$.
In this paper we proof the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over $\overline{M}_{g}$ to the construction of the universal moduli space of swamps.
By means of a Fourier-Mukai transform we embed moduli spaces of stable bundles on an algebraic curve C as isotropic subvarieties of moduli spaces of mu-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new examples of special Lagrangian submanifolds.
We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.
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