Ajneet Dhillon, Nicole Lemire
We find upper bounds for the essential dimension of various moduli stacks of $\sln$-bundles over a curve. When $n$ is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.
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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.
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 define the moduli problem of Hitchin pairs over Deligne-Mumford Stack and prove this moduli problem is represented by a separated and locally finitely presented algebraic space, which is considered as the moduli space of Hitchin pairs over Deligne-Mumford stack.
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 give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points.
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…
We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.
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 will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case.
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.
We determine the Artin-Mazur étale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the étale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an étale homotopy point of view.
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).
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.
Ample vector bundles and moduli of tame stacks
本文利用充足向量丛概念将诺特环上的驯顺叠嵌入加权射影叠的推广中，并构造了带充足向量丛的驯顺叠的代数模叠及驯顺轨道曲线的代数叠。
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.
In this paper, we define $Λ$-quot-functors on Deligne-Mumford stacks. We prove that the $Λ$-quot-functor is representable by an algebraic space. Then, we construct the moduli space of $Λ$-modules on a projective Deligne-Mumford stack. We prove that this moduli space is a quasi-projective scheme.
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…
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.
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.
In this paper we study four families of moduli problems which give rise to two dimensional examples of the Hitchin map. Using a few Fourier-Mukai transforms on the corresponding spectral curves, we give isomorphisms between these moduli problems.
We construct five families of two-dimensional moduli spaces of parabolic Higgs bundles (respectively local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve. We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively local systems), confirming a conjecture of Boalch in these cases and extending a result of Gorsky--Nekrasov--Rubtsov. Using the McKay correspondence we establish the autoduality conjecture for the derived categories of the moduli spaces …
Here we calculate the Chern classes of ${\bar {\mathcal M}}_{g,n}$, the moduli stack of stable n-pointed curves. In particular, we prove that such classes lie in the tautological ring.
This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperkähler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite …
We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori's program are also moduli spaces of parabolic vector bundles with certain parabolic weights.
A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $α$. In this paper we study wall crossing for the moduli spaces of $α$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential rol…
Hausel and Rodriguez-Villegas conjectured that the intersection form on the moduli space of stable PGL_n-Higgs bundles on a curve vanishes if the degree is coprime to n. In this note we prove this conjecture. Along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of Higgs bundles.
Let $G$ be a semisimple linear algebraic group over the field $\mathbb C$, and let $C$ be an irreducible smooth complex projective curve of genus at least three. We compute the Brauer group of the smooth locus of the moduli space of semistable principal $G$--bundles over $C$. We also compute the Brauer group of the moduli stack of principal $G$--bundles over $C$.
We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.
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