Richard Hain
This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal treatement of Teichmuller space and higher genus moduli spaces. The point of view is generally topological and analytical.
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Crossing the transcendental divide: from Schottky groups to algebraic curves
本文构造亏格 $g\geq 3$ 且带对称性与 $M$-curve 实结构的非超椭圆 Schottky Riemann 面，并数值计算其对应代数曲线与 Riemann 矩阵，以沟通解析与代数表示。
Crossing the transcendental divide: from translation surfaces to algebraic curves
本文利用离散 Riemann 曲面理论提出算法，从可方格化的平移曲面逼近 Jacobian 簇，并在 L 形多边形与 Jenkins-Strebel 代表上验证，结合 Riemann theta 函数给出至亏格 5 的数值实验和猜想。
As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts, and differ from each other. In this article we review some of constructions in order to provide materials for future research on "secondary geometry" of the moduli space ${\mathbb M}_g$.
The Hessian of a general cubic surface is a nodal quartic surface, hence its desingularisation is a K3 surface. We determine the transcendental lattice of the Hessian K3 surface for various cubic surfaces (with nodes and/or Eckardt points for example). Classical invariant theory shows that the moduli space of cubic surfaces is a weighted projective space. We describe the singular locus and some other subvarieties of the moduli space.
On the transcendental lattices of Hyperkähler manifolds
本研究引入由K3型霍奇结构诱导的超凯勒流形概念，针对已知形变类型给出判断其与K3/阿贝尔曲面层模空间双有理等价的格论准则，并揭示O'Grady型超凯勒流形的特殊行为。
We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.
We survey Mirzakhani's work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non-experts.
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.
We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface.
We give a short and "classical" proof of Borcherds' theorem that the moduli space of Enriques surfaces is quasi-affine. The use of the Borcherds' product is replaced in our proof by an application of the Grothendieck-Riemann-Roch theorem.
Moduli spaces of untwisted wild Riemann surfaces
本研究面向任意复约化结构群，完成纯未扭化情形野生黎曼曲面模叠的构造，同时定义了对应的纯野生映射类群。
We classify the finite groups associated to the singularities of the moduli space of $n/ge5$ unordered points on the Riemann sphere. We also realize the classification by an algorithm.
We compute the stringy E function of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E functions of the GIT quotients of linear representations.
In 1962 E. H. Rauch established the existence of points in the moduli space of Riemann surfaces not having a neighbourhood homeomorphic to a ball. These points are called here topologically singular. We give a different proof of the results of Rauch and also determine the topologically singular and non-singular points in the branch locus of some equisymmetric families of Riemann surfaces.
On non-normal subvarieties of the moduli space of Riemann surfaces
本文研究由有限群作用确定的紧 Riemann 曲面模空间的不可约子簇，探讨判定其中哪些为非正规子簇的一般问题。
We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional loca…
The moduli space of multi-monopoles on a Riemann surface
本研究针对紧黎曼曲面上含N个旋量的Seiberg-Witten方程解的模空间，完成其欧拉示性数计算并推导有理同调，为3-流形计数不变量及代数曲线几何研究提供支撑。
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational con…
In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by a graph defined by its (hyperbolic) pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratificat…
A holomorphic chain on a compact Riemann surface is a tuple of vector bundles together with homomorphisms between them. We show that the moduli space of holomorphic chains of rank one is identified with a fiber product of projective space bundles. We compute the Euler characteristic of the moduli space. The stability of chains involves real vector parameters. We also show that the variation of parameters corresponds to the characters of Gm.
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。
Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $φ:M\to X({\Bbb C})$ be a holomorphic map. For every positive real number $T$, let $A_U(T)$ be the cardinality of the set of $z\in U$ such that $φ(z)\in X(K)$ and $h_L(φ(z))\leq T$. After a revisitation of the proof of the sub exponential bound for $A_U(T)$, obtained by Bombieri and Pila , we show that there are intervals of $T$'s as big as we want for which $A_U(T)$ is upper bounded by a polynom…
A measure on the moduli space of super Riemann surfaces with Ramond punctures
本文利用超Mumford同构与超周期映射，在带Ramond奇点的超黎曼曲面模空间上构造了一个测度。
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.
Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory. In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces.
Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies
本研究证明双曲曲面模空间$M_g$中含足够多短测地线的曲面轨迹是柄体映射类群的分类空间，由此推出$M_g$的最高权上同调可单射到柄体映射类群的上同调中。
We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz' bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenera…
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…
Refinements of the Yang-Mills stratifications of spaces of connections over a compact Riemann surface are investigated. The motivation for this study was the search for a complete set of relations between the standard generators for the cohomology of the moduli spaces of stable holomorphic bundles of rank n and degree d when n and d are coprime and n>2.
Finite-dimensional construction of self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces
本文通过有限维全纯辛约简，在闭黎曼曲面上构建自对偶性及相关模空间的分层全纯辛结构。
We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.
We compute some Gromov-Witten invariants of the moduli space of odd degree rank two stable vector bundles over a Riemann surface of any genus. Next we find the first correction term for the quantum product of this moduli space and hence get the two leading terms of the relations satisfied by the natural generators of its quantum cohomology. Finally, we use this information to get a full description of the quantum cohomology of the moduli space when the genus of the Riemann surface is 3.
Inspired by his vanishing results of tautological classes and by Harer's computation of the virtual cohomological dimension of the mapping class group, Looijenga conjectured that the moduli space of smooth Riemann surfaces admits a stratification by affine subsets with a certain number of layers. Similarly, Roth and Vakil extended the conjecture to the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most k rational components. As a consequence of Lefschetz's theorem, Roth-Vakil's conjecture would also imply that the pre…
We construct a moduli scheme for semistable pre-$\D$-modules with prescribed singularities and numerical data on a smooth projective variety. These pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level structure'. We also construct a moduli scheme for perverse sheaves on the variety with prescribed singularities and other numerical data, and represent the de Rham functor (which gives the Riemann-Hilbert correspondence) by an analytic morphism between the two moduli schemes.
Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(π_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic for…
Let $X$ be any compact connected Riemann surface of genus $g \geq 3$. For any $r\geq 2$, let $M_X$ denote the moduli space of holomorphic $SL(r,C)$-connections over $X$. It is known that the biholomorphism class of the complex variety $M_X$ is independent of the complex structure of $X$. If $g=3$, then we assume that $r\geq 3$. We prove that the isomorphism class of the variety $M_X$ determines the Riemann surface $X$ uniquely up to isomorphism. A similar result is proved for the moduli space of holomorphic $GL(r,C)$-connections on $X$. We also show that the Torelli theorem remains valid for t…
The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli.
Local structure of the Teichmüller and the Riemann moduli stacks
本文介绍了D. Rydh关于Artin栈局部结构解析版本的问题，并通过在Teichmüller模空间和Riemann模空间上进行验证提供了支持证据。
Enumerative geometry via the moduli space of super Riemann surfaces
本文通过代数几何方法证明超黎曼面模空间体积递推关系等价于KdV tau函数，类比Mirzakhani的Kontsevich-Witten定理证明方法。
In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
Let $(C,\bt)$ ($\bt=(t_1,...,t_n)$) be an $n$-pointed smooth projective curve of genus $g$ and take an element $\blambda=(λ^{(i)}_j)\in\C^{nr}$ such that $-\sum_{i,j}λ^{(i)}_j=d\in\mathbf{Z}$. For a weight $\balpha$, let $M_C^{\balpha}(\bt,\blambda)$ be the moduli space of $\balpha$-stable $(\bt,\blambda)$-parabolic connections on $C$ and let $RP_r(C,\bt)_{\ba}$ be the moduli space of representations of the fundamental group $π_1(C\setminus\{t_1,...,t_n\},*)$ with the local monodromy data $\ba$ for a certain $\ba\in\C^{nr}$. Then we prove that the morphism $\RH:M_C^{\balpha}(\bt,\blambda)\righ…
We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology, and to compute their stable rational cohomology and their integral Picard groups. Using these methods we give a complete description of their integral Picard groups for genus at least 9 in terms of geometrically defined generators, and determine the relations between them.
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
The moduli space of Riemann surfaces of genus $g\geq 2$ is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$. This expectation is verified in low genus and supported by Harer's computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this paper we prove that such dimension is at most $2g-2$. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeau…
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta funct…
We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.
In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent.
One interpretation of Torelli's Theorem, which asserts that a compact Riemann Surface $X$ of genus $g > 1$ is determined by the $g(g+1)/2$ entries of the period matrix, is that the period matrix is a message about $X$. Since this message depends on only $3g-3$ moduli, it is sparse, or at least approximately so, in the sense of information theory. Thus, methods from information theory may be useful in reconstructing the period matrix, and hence the Riemann surface, from a small subset of the periods. The results here show that, with high probability, any set of $3g-3$ periods form moduli for th…
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.
For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(π_1(X), U(p,1))/U(p,1)$ is the moduli space of flat $\U(p,1)$-connections on $X$. There is an integer invariant, $τ$, the Toledo invariant associated with each element in $\Hom(π_1(X), U(p,1))/U(p,1)$. If $q = 1$, then $-2(g-1) \le τ\le 2(g-1)$. This paper shows that $\Hom(π_1(X), U(p,1))/U(p,1)$ has one connected component corresponding to each $τ\in 2Z$ with $-2(g-1) \le τ\le 2(g-1)$. Therefore the total number of connected components is $2(g-1) + 1$.
We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface $Σ$ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton Floer cohomology of the three manifold $Σ\times S^1$. (Note: There is some overlap with the previous paper: alg-geom/9711013).
We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector sp…
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group.
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…
The aim of this paper is to study the structure of the higher-dimensional Teichmüller and Riemann moduli spaces, viewed as stacks over the category of complex manifolds. We first show that the space of complex operators on a smooth manifold admits a foliation transversely modeled on a translation groupoid, a concept that we define here. We then show how to construct explicitly a holonomy groupoid for such a structure and show that in this case its objects and morphisms form a finite-dimensional analytic space and its source and target maps are smooth morphisms. This holonomy data encodes how t…
The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
本文通过轨形Mukai配对导出新的轨形HRR公式，用于计算K3曲面上有限群作用下稳定层等变模空间的维数，并证明这些模空间是形变等价于点Hilbert概型的不可约辛流形。
We study a pair of dual operads which arise in the study of moduli spaces of pointed genus 0 curves (this duality is similar to that between commutative and Lie algebras). These operads are both quadratic, and even Koszul, and arise in the theory of quantum cohomology.
The bad locus in the moduli of super Riemann surfaces with Ramond punctures parametrizes those super Riemann surfaces that have more than the expected number of independent closed holomorphic 1-forms. There is a super period map that depends on certain discrete choices. For each such choice, the period map blows up along a divisor that contains the bad locus. Our main result is that away from the bad locus, at least one of these period maps remains finite. In other words, we identify the bad locus as the intersection of the blowup divisors. The proof abstracts the situation into a question in …
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