O. García-Prada, P. B. Gothen, V. Muñoz
We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.
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Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincare polynomials depend on the system of weights of the parabolic bundle.
We determined the first two Betti numbers of the moduli of rank two stable sheaves on an arbitrary algebraic surface
Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obtained by Liu and Schaffhauser.
Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
本文研究光滑射影表面的一维斜稳定层模空间的交贝蒼地数，证明在特定度数范围内与点希尔伯特方案的稳定贝蒼地数一致，并在表面具有 nef 反可以能除数时证明这些交贝蒼地数在每个固定度数中稳定化。
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$, then the $2n$th Betti number of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.
Betti numbers of the moduli space of Higgs bundles over a real curve
本文基于Mellit、Fedorov-Soibelman-Soibelman和Schiffmann的动机公式，推导出实曲线上的稳定Higgs丛模空间M_r^d的Z_2-Betti数公式，并证明虚拟的Z_2庞加莱多项式是实数上的动机度量。
Local Constancy of the Betti Automorphic Category over the Moduli of Curves
本文采用纯微局部方法证明了曲线模空间上Betti自守范畴的局部常性猜想，该结果不依赖新近确立的Betti几何朗兰兹猜想，还验证了Ben-Zvi与Nadler关于Betti自守范畴仅依赖曲线亏格的猜想。
Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps
本文研究了带有 lagrangian 纤维的不可约 holomorphically symplectic 流形的抛物自同构，证明了相关 Betti 映射具有最大秩，从而得出诱导平移为有限阶的纤维集合是稠密的。
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.
This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on the projective plane P2 and its blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up. These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for P2 follow from those for the blow-up by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions.
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
Let $M(d,χ)$ with $(d,χ)=1$ be the moduli space of semistable sheaves on $\mathbb{P}^2$ supported on curves of degree $d$ and with Euler characteristic $χ$. The cohomology ring $H^*(M(d,χ),\mathbb{Z})$ of $M(d,χ)$ is isomorphic to its Chow ring $A^*(M(d,χ))$ by Markman's result. W. Pi and J. Shen have described a minimal generating set of $A^*(M(d,χ))$ consisting of $3d-7$ generators, which they also showed to have no relation in $A^{\geq d-2}(M(d,χ))$. We compute the two Betti numbers $b_{2(d-1)}$ and $b_{2d}$ of $M(d,χ)$ and as a corollary we show that the generators given by Pi-Shen have no…
We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers i…
We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.
In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.
Using the $L^2$-norm of the Higgs field as a Morse function, we count the number of connected components of the moduli space of parabolic $U(p,q)$-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. This space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in $U(p,q)$, with fixed compact holonomy classes around the marked points. We apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.
We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.
In this paper we use Weil conjectures (Deligne's theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of Harder-Narasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.
We calculate the Betti numbers of the coarse moduli space of stable maps of genus 0 to projective space, using a generalization of the Legendre transform.
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…
In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.
The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic stable bundles of rank 2 and also prove the existence of components with smooth points.
On the Chow group of Moduli of parabolic connections
本文研究亏格至少为3的紧Riemann曲面上有理一般权重抛物联络的模空间，确定底层抛物丛稳定情形的周群，并讨论其有理性与有理连通性。
Simpson, in 1994, introduced the notion of $Λ$-modules and constructed the corresponding moduli space, where $Λ$ is a sheaf of rings of differential operators. Higgs bundles, connections and $λ$-connections (as defined by Delgine) are particular cases of $Λ$-modules. In this article the concept of parabolic $Λ$-modules over a curve is introduced and their moduli space is built. As an application, we construct the parabolic Hodge moduli space parameterizing parabolic $λ$-connections.
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.
In this paper, we treat moduli spaces of parabolic connections. We take étale coverings of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation for each étale morphism.
We prove that, given the isomorphism class of the parabolic Deligne-Hitchin moduli space over a smooth projective curve, we can recover the isomorphism class of the curve and the parabolic points.
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.
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.
Density of integral points in the Betti moduli of quasi-projective varieties
本文证明了在参数化拟射影簇上具有固定代数整数迹的 $SL_2$-局部系统的特征簇中整点潜在稠密，通过化简至黎曼曲面情形并构造在映射类群下 Zariski 稠密轨道的整点完成证明。
We define a moduli space of stable regular singular parabolic connections of spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.
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 generalize the construction of moduli spaces of parabolic sheaves given by Maruyama and Yokogawa in [MY92] to the case of a projective fine saturated log scheme with a fixed global chart. Furthermore we construct moduli spaces of parabolic sheaves without fixing the weights.
Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
该研究构造了黎曼曲面上带标架的对数联络与抛物联络的模空间，证明其具有自然代数辛结构，并给出抛物联络模空间正则函数代数超越次数的上界。
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 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.
For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kaehler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein-Maass series. The cuspidal defect is explicitly expressed through curvat…
In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on $\mathbb{P}^1$ and rank 2 (parabolic) Higgs bundle on $\mathbb{P}^1$. Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
We prove the SYZ (Strominger-Yau-Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface.The SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface by Hausel and Thaddeus.
This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli…
Moduli spaces of parabolic bundles over $\mathbb{P}^1$ with five marked points
本文研究五标记点 $\mathbb{P}^1$ 上秩2、次数1的抛物对数平坦丛与Higgs丛的模空间/模叠，分析其叶状结构与分层，并在非特殊权重系统下证实Simpson猜想。
For every set of parabolic weights, we construct a Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles on a curve. It is based on the notion of parabolic slope, introduced by Mehta and Seshadri. We also prove that the stratification is schematic, that each stratum is complete, and establish an analogue of Behrend's conjecture for parabolic vector bundles. A comparison with recent $Θ$-stratification approaches is discussed.
Derived category of moduli of parabolic bundles on $\mathbb{P}^1$
该研究针对$\u2119^1$上秩2抛物丛模空间的凝聚层有界导出范畴的结构问题，提出了一项相关猜想，为代数几何领域该方向的研究提供了新的探索方向。
In this article we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.
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.
Like the Higgs bundles on a Riemann surface who played an important role in the study of representation of the fundamental group of the surface, the parabolic Higgs bundles play also their importance in the study of the fundamental group but of punctured surface. In this paper, we shall give an algorithm to calculate the (virtual) motive (i.e in a suitable Grothendieck ring) of the moduli spaces of parabolic Higgs bundles of fixed rank, fixed degree and fixed parabolic structure, using localization with respect to the circle action.
Hitchin maps and parabolic Hitchin maps on the moduli spaces of Hitchin sheaves on nodal curves
本文研究了Hitchin映射在有理曲线上的模空间及其抛物情形，证明了抛物Poincaré sheaf的稳定性。
Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
本文通过微局部几何方法，验证了弱几何 P=W 猜想对双球面上带一个不规则奇点的秩 n 野特征簇成立，其对偶边界复形同伦等价于 $d-1$ 维球面。
Moduli of parabolic sheaves and filtered Kronecker modules
本文利用滤过Kronecker模的模空间构造了纯抛物线层的函子模空间，证明了极大维纯抛物线层模空间的射影性，并得到了模栈层面的态射。
This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D), pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.
We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces o…
We determine the Brauer group and Picard group of the moduli space $U^{' s}_{L,par}$ of stable parabolic vector bundles of rank $r$ with determinant $L$ on a real curve $Y$ of arithmetic genus $g \,\ge\, 2$ with at most nodes as singularities.
Brauer group of moduli of stable parabolic $\text{SL}(r,\mathbb{C})$ and $\text{PGL}(r,\mathbb{C})$-connections and Higgs bundles over a curve
本文研究了稳定抛物SL(C)连接和Higgs包的模空间的Brauer群，并建立了稳定抛物PGL(C)连接模堆与粗模空间光滑部分的Brauer群相等性。
Generators for the cohomology of the moduli space of irregular parabolic Higgs bundles
该研究证明非正则抛物Higgs丛模空间上同调环纯部分的生成元构成，并将类似结果推广到正则完全旗情形下的抛物与强抛物Higgs丛模空间。
Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands corr…
Moduli of parabolic bundles on an elliptic curve
本文研究秩3、行列式平凡且带一个标记点的半稳定抛物丛在椭圆曲线上的模空间的几何性质，证明其为有理的，给出几何描述、自同构群并证明托雷利型结果。
Brauer group of moduli stack of stable parabolic $\textnormal{PGL}(r)$-bundles over a curve
本文研究曲线上稳定抛物型 PGL(r)-丛模叠的 Brauer 群，证明其与粗模空间光滑轨迹的 Brauer 群一致，并在一般拟抛物类型与权重条件下完成计算。
We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of the complexity of the actions of the corresponding groups of bundle automorphisms on stable parabolic structures. For the given choices of parabolic weights, $\mathscr{N}$ consists entirely of isomorphism classes of strictly stable parabolic bundles whose underlying Birkhoff-Grot…
Brauer group of moduli stacks of parabolic principal bundles over a curve
本文研究了抛物主丛模堆的 Brauer 群，证明其与粗模堆的光滑部分的 Brauer 群相同，并显示了 quasi-抛物主丛的 Brauer 群为零。
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