We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-$g$ Riemann surface, where $g$ is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-$g$ Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand-Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explictly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).
Given an n-tuple of positive real numbers, Konno defines an algebraic variety called a hyperpolygon space, a hyperkahler analogue of the Kahler variety parametrizing spacial polygons with fixed edge lengths. The ordinary polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, the hyperpolygon space is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural action of the nonzero complex numbers, and the union of the precompact orbits is called the core. We study the com…
Given an $n$-tuple of positive real numbers $α$ we consider the hyperpolygon space $X(α)$, the hyperkähler quotient analogue to the Kähler moduli space of polygons in $\mathbb{R}^3$. We prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This isomorphism allows us to prove that hyperpolygon spaces $X(α)$ undergo an elementary transformation in the sense of Mukai as $α$ crosses a wall in the space of its admissible…
We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $Σ$ and consider a natural involution, studying its fixed point locus when $Σ$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space.
We study the hyperkähler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperkähler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.
We introduce an unfolded moduli space of connections, which is an algebraic relative moduli space of connections on complex smooth projective curves, whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli space of unramified irregular singular connections. On the moduli space of unramified irregular singular connections, there is a subbundle of the tangent bundle defining the generalized isomonodromic deformation produced by the Jimbo-Miwa-Ueno theory. On an analytic open subset of the unfolded moduli space of connections, we construct a non-c…
We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polarized Calabi-Yau manifolds. We also give an estimate of the complex Hessian of the BCOV torsion using the relation. After establishing a degenerate version of the Schwarz Lemma of Yau, we prove that the complex Hessian of the BCOV torsion is bounded by the Poincaré metric.
Generalized Hypergeometric Functions for Degree k Hypersurface in CP^{N-1} and Intersection Numbers of Moduli Space of Quasimaps from CP^{1} with Two Marked Points to CP^{N-1}
本文将 CP^{N-1} 中 k 次超曲面的镜像计算所用广义超几何函数实现为带两个标记点的拟映射模空间相交数的生成函数，建立了超曲面镜像对称与拟映射计数之间的具体联系。
A ribbon is a first-order thickening of a non-singular curve. Motivated by a question of Eisenbud and Green, we show that a compactification of the moduli space of line bundles on a ribbon is given by the moduli space of semi-stable sheaves. We then describe the geometry of this space, determining the irreducible components, the connected components, and the smooth locus.
In this paper, we study generalized line bundles over $C_n$, a primitive multiple curve of arbitrary multiplicity $n$, where $n$ is a positive integer. In particular, we give a structure theorem for them and we characterize their semistability in terms of $n-1$ integral invariants, the indices. These results are used to describe the irreducible components that contain stable generalized line bundles in the Simpson moduli space of semistable sheaves of generalized rank $n$ and fixed generalized degree on $C_n$. We compute also the dimension of the Zariski tangent space to this moduli space in a…
We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we intro…
We construct the moduli space of finite dimensional representations of generalized quivers for arbitrary connected complex reductive groups using Geometric Invariant Theory as well as Symplectic reduction methods. We explicit characterize stability and instability for generalized quivers in terms of Jordan-Hölder and Harder Narasimhan objects, reproducing well-known results for classical case of quiver representations. We define and study the Hesselink and Morse stratifications on the parameter space for representations, and bootstrap them to an inductive formula for the equivariant Poincaré P…
In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the modu…
This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abe…
As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and only if $S$ admits an anticonformal automorphism. In the case that the field of moduli of $S$ is ${\mathbb R}$, then $S$ can be defined over the field of moduli if and only if $S$ admits an anticonformal involution. It may happen that the field of modu…
We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric i…
We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions…
We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic eightfold $Z$, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that $Z$ is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bri…
It is known that generalized deformation in the sense of Hitchin-Gaultieri is a geometric realization of the degree-2 component of Kontsevich-Barannikov's homological approach to extended deformation. Through extended deformation, one associates a Frobenius structure to the extended moduli space. In this notes, we prove that on primary Kodaira manifolds the restriction of the Frobenius structure on the degree-2 component of the extended moduli space is trivial. It generalizes the author's past observation on Kodaira surface.
Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi-Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period …
We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincare polynomial in singular cohomology of these moduli spaces.
The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are identified using subalgebras of a deformed Jacobian ring.
In this paper I present a new geometric approach to the factorization rule for generalised theta functions. Let $X$ be an irreducible projective nodal curve with one singularity and let $Y$ be its normalization. Recently I have constructed the moduli stack $GVB(X)$ of rank $n$ Gieseker vector bundles on $X$ and have shown that its normalization is a locally trivial fibration over the moduli stack $VB(Y)$ of vector bundles on $Y$, where the fibre is a canonical compactification of $Gl_n$. In this paper I prove a canonical direct sum decomposition of the space of global sections of a power of th…
In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fundamental classes defined in the two categories. We prove this statement in an interesting case, namely the moduli problem associated with the symplectic factorization problem which yields the complete toric varieties. Therefore, our main objects are the moduli spaces of "twisted"…
This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or $r$-spin curves, are pairs $(X,L)$ with $X$ a smooth curve and $L$ a line bundle whose r-th tensor power is isomorphic to the canonical bundle of $X.$ These are a natural generalization of $2$-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over $\Bbb{Z}[1/r],$ all using torsion-free sheaves, are constructed. All three yield algebraic stacks…
The goal is to verify the Hodge conjecture (and some related conjectures) for certain moduli spaces. It is shown that the (generalized) Hodge conjecture holds for the projective moduli spaces of vector bundles over an abelian or K3 surface or a curve if it holds for all powers of the surface or curve. In the latter case, the curve can be replaced by its Jacobian. As a consequence the generalized Hodge conjecture holds for these moduli spaces when the curve is very general. A similar result is proved for the Hilbert schemes of points over an arbitrary surface.
In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study what are called the Verlinde bundles in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.
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.
The theory of generalized matric Massey products has been applied for some time to $A$-modules $M$, $A$ a $k$-algebra. The main application is to compute the local formal moduli $\hat{H}_M$, isomorphic to the local ring of the moduli of $A$-modules. This theory is also generalized to $\mathcal{O}_X$-modules $\mathcal{M}$, $X$ a $k$- scheme. In these notes we consider the definition of generalized Massey products and the relation algebra in any obstruction situation (a differential graded $k$-algebra with certain properties), and prove that this theory applies to the case of graded $R$-modules,…
By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve (arising in the theory of elliptic special functions). The common feature of the relaxations is that they can be identified with moduli spaces of sheaves on rational surfaces. Not only does this make various natural questions become purely geometric (rigid equations correspond to -2-curves), it also establishes a number of nontrivial correspondences between dif…
We provide a description of the moduli space of framed autodual instanton bundles on projective space, focusing on the particular cases of symplectic and orthogonal instantons. Our description will use the generalized ADHM equations which define framed instanton sheaves.
This lecture note is devoted to prove that the space of vacua is isomorphic with the space of generalized theta functions (joint work with M.S. Narasimhan, and A. Ramanathan)
We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of a semiclassical limit of the r-th Gelfand-Dickey (or KdV_r) hierarchy. Additionally, we prove that tensor products of the Frobenius manifolds associated to such hierarchies admit a geometric interpretation in terms of moduli spaces of higher spin structures. We also elaborate upon the analogy to Gromov-Witten invariants of a smooth, projective variety.
This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,Θ_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.
We prove moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are globally $F$-regular type.
Let $A$ be a sheaf of Azumaya algebras over a Noetherian base $S$. In this paper we describe using generalized Severi-Brauer varieties, a quasi-projective moduli space parametrizing sheaves of ètale subalgebras of $A$. In the case that $S$ is the spectrum of a field, we study the geometry of this moduli space, and show that in certain cases it is a rational variety.
We investigate stable holomorphic vector bundles on a compact complex Kähler manifold and more generally on an orbifold that is equipped with a Kähler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of coherent sheaves. Such an extension will be called a Weil-Petersson current. When the orbifold is of Hodge type, t…
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by Selberg-Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instant…
Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:Δ-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $Δ$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (i.e. there are no entire cu…
We introduce coupled Seiberg-Witten equations, and we prove, using a generalized vortex equation, that, for Kaehler surfaces, the moduli space of solutions of these equations can be identified with a moduli space of holomorphic stable pairs. In the rank 1 case, one recovers Witten's result identifying the space of irreducible monopoles with a moduli space of divisors. As application, we give a short proof of the fact that a rational surface cannot be diffeomorphic to a minimal surface of general type.
We describe the moduli stack of Gushel-Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of Lagrangian data; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel-Mukai varieties and construct some complete nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approa…
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. And then we obtain an algorithm expressing intersection numbers $<τ_{n,m}\prod_{i=0}^{r-1}τ_{0,i}^{k_i}>_g$ via correlation functions of primaries.
We give an introduction to the compactification of the moduli space of surfaces of general type introduced by Kollár and Shepherd-Barron and generalized to the case of surfaces with a divisor by Alexeev. The construction is an application of Mori's minimal model program for 3-folds. We review the example of the projective plane with a curve of degree d > 3. We explain a connection between the geometry of the boundary of the compactification of the moduli space and the classification of vector bundles on the surface in the case H^{2,0}=H^1=0.
Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $CP^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $CP^1$ with Two Marked Points to $CP^{N-1}$
