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 group of moduli spaces.
Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O'Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension $\geq4$ they do no admit projective symplectic resolutions.
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3-surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3-surface.
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…
A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from normal elliptic quintics. The natural projection to the dual projective 5-space parametrizing the hyperplane sections is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally, is equal) to the quasi-symplectic one, induced by the Yoneda pairing on the moduli space.
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.
We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.
Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed lo…
We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.
Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE ---> L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_{Sp}(L) is generated by automorphisms sending E to ExM, where M is a 2-torsion line bundle, and automorphisms induced by automorphisms of X.
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
We describe here a degeneration of the symplectic desingularization of the moduli spaces of topologically trivial $GL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$-Higgs bundles over a hyperelliptic curve, into O'Grady's ten and six dimensional exceptional examples of irreducible holomorphic symplectic manifolds. Most of this note is a survey of work on these degenerations by Donagi-Ein-Lazarsfeld, de Cataldo-Maulik-Shen and Felissetti-Mauri, although in certain cases we provide details that were missing the previous articles.
In a recent paper \cite{3}, a semi-stable degeneration of moduli space of Higgs bundles on a curve has been constructed. In this paper, we show that there is a relative log-symplectic form on this degeneration, whose restriction to the generic fibre is the classical symplectic form discovered by Hitchin. We compute the Poisson ranks at every point and describe the symplectic foliation on the closed fibre. We also show that the closed fibre, which is a variety with normal crossing singularities, acquires a structure of an algebraically completely integrable system.
We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.
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.
A symplectic bundle over an algebraic curve has a natural invariant $\sLag$ determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on $\sLag$ which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by $\sLag$ on moduli spaces of symplectic bundles, and get a full picture for the case of rank four.
We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles.
Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in general and A. Apostolov counted the number of their components. We extend Apostolov's results to the case of manifolds deformation equivalent to generalised Kummer varieties.
Let X be a complex projective curve which is smooth and irreducible of genus 2. The moduli space M_2 of semistable symplectic vector bundles of rank 4 over X is a variety of dimension 10. After assembling some results on vector bundles of rank 2 and odd degree over X, we construct a generically finite cover of M_2 by a family of 5-dimensional projective spaces, and outline some applications.
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.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
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 …
A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$. This invariant $t$ defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on $t(V)$, which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In part…
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.
The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $σ$ of order $n\geq 2$ is one dimensional exactly when $φ(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,σ)$ of the irreducible components of maximal dimension of such moduli spaces. In particular we show that there is a unique one-dimensional component for $n=20,22, 24$, three irreducible components for $n=15$ and two components in the remaining cases.
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.
Several situations are known when a holomorphic 2-form on a moduli space of sheaves over some base S is induced by a holomorphic 2-form on S. Moreover, the closedness of the 2-form on the base implies the closedness on the moduli space, which provides a stock of symplectic structures on moduli spaces (Mukai, Kobayashi, O'Grady, Tyurin, Huybrechts--Lehn). A parallel theory was developed for bivector fields and Poisson structures (Bottacin). However, there exist symplectic moduli spaces of sheaves over bases that have no holomorphic forms at all. A well-known example (Beauville--Donagi) is the f…
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 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.
Symplectic instanton vector bundles on the projective space $\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r,\ n\equiv r({\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.
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).
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 $X$.
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 $X$.
Let $X$ be a projective K3 surface with generic polarization $\cO_X(1)$ and let $M_c=M(2,0,c)$ be the moduli space of semistable torsion-free sheaves on $X$ of rank 2, with Chern classes $c_1=0$ and $c_2=c$. When $c=2n\ge 4$ is even, $M_c$ is a singular projective variety. We show that there is no symplectic desingularization of $M_{2n}$ if $\frac{n a_n}{2n-3}$ is not an integer where $a_n$ is the Euler number of the Hilbert scheme $X^{[n]}$ of $n$ points in $X$.
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.
In this article we study the Gieseker-Maruyama moduli spaces $\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We construct two new infinite series $Σ_0$ and $Σ_1$ of irreducible components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the seri…
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
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 $(X,D)$ and $(X',D')$ be two compact Riemann surfaces of genus $g \geq 4$ with the set of marked points $D \subset X$ and $D' \subset X'$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ and $\mathcal{N}'_{\textnormal{Sp}}(2m,α,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X'$ respectively, with rank $2m$ and fixed parabolic structure $α$, with the symplectic form taking values in $L$. We prove that if $\mathcal{N}_{\textnormal{Sp}}(2m,α,L)$ is isomorphic to $\mathcal{N}'_{\textnormal{Sp}}(2…
