Moduli Space Of Pairs Research Articles

Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

This paper gives a construction for all minimal immersions $f$ of the Poincar\'{e} disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $\rho$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs $(\rho,f)$ is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for $PU(2,1)$, and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.

On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4\pi$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $\zeta$-regularized determinant $\operatorname{Det}' \Delta_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\operatorname{Det}' \Delta_F$.

Moduli spaces of meromorphic functions and determinant of the Laplacian

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $\Delta^{|df|^2}$ and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential $df$. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the $S$-matrix, $S(\lambda)$, of a surface with conical singularities as a function of the spectral parameter $\lambda$ and relate its behavior at $\lambda=0$ with the Schiffer projective connection on the Riemann surface $X$. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.

Brauer Group of Moduli Spaces of Pairs

We show that the Brauer group of the moduli space of stable pairs with fixed determinant over a curve is zero.

Stable spherical varieties and their moduli

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group $G$ and their flat equivariant degenerations. Given any projective space $\bP$ where $G$ acts linearly, we construct a moduli space for stable spherical varieties over $\bP$, that is, pairs $(X,f)$, where $X$ is a stable spherical variety and $f : X \to \bP$ is a finite equivariant morphism. This space is projective, and its irreducible components are rational. It generalizes the moduli space of pairs $(X,D)$, where $X$ is a stable toric variety and $D$ is an effective ample Cartier divisor on $X$ which contains no orbit. The equivariant automorphism group of $\bP$ acts on our moduli space; the spherical varieties over $\bP$ and their stable limits form only finitely many orbits. A variant of this moduli space gives another view to the compactifications of quotients of thin Schubert cells constructed by Kapranov and Lafforgue.

Torelli theorem for the moduli spaces of pairs

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,\phi) over X consists of an algebraic vector bundle E over X and a holomorphic section \phi of E. There is a concept of stability for pairs which depends on a real parameter \tau. Here we prove that the third cohomology groups of the moduli spaces of \tau-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank at least two and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H^1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori A θ {\mathbb A}_{\theta } admits a natural desingularization by the group E x t ( A θ , A θ ) Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta }) . Namely, we prove that the moduli space of pairs ( A θ , E x t ( A θ , A θ ) ) ({\mathbb A}_{\theta }, Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta })) is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.

Compact moduli of plane curves

We construct a compactification $\mathcal{M}$d of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ℙ2 as a surface-divisor pair (ℙ2,C), and we define $\mathcal{M}$d as a moduli space of pairs (X,D), where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack $\mathcal{M}$d is smooth and the degenerate surfaces X can be described explicitly.

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

Elliptic Sklyanin integrable systems for arbitrary reductive groups

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve E. We consider the moduli space of pairs (P,f), where P is a principal G-bundle on E, and f is a meromorphic section of the adjoint group bundle. We show: 1) The moduli space admits an algebraic Poisson structure. It is related to the poisson structures on loop groupoids, constructed by Etingof and Varchenko, using Felder's elliptic solutions of the Classical Dynamical Yang-Baxter Equation. 2) The symplectic leaves are finite dimensional. A symplectic leaf is determined by labeling, finitely many points of E, each by a dominant co-character of the maximal torus of G. 3) Each leaf is an algebraically completely integrable system.

Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

Recently, due largely to its importance in fermionic string theory, there has been much interest in the moduli spaces ~/gl-e] of Riemann surfaces of genus g with spin structure of Arf invariant e e Z/2Z. Algebraic geometers have long studied these spaces in their alternate guise as moduli spaces of pairs (algebraic curve, square root of the canonical bundle). For topologists these spaces are rational classifying spaces for spin mapping class groups. However, despite the fact that ~r162 is a finite cover of the ordinary moduli space J/4g, little is known about the topology of these spaces. In this paper we begin a study of the homology of ~'g[e] by proving that its homology groups are independent of g and e when g is adequately large (Theorem3.1). In a second paper [H4] we will compute nl(v//g[e]) and H2(./r thereby calculating the Picard group of Jlg[e]. Putting this all together we know approximately the same amount about the homology of Jt'g[e] as we do about that of d/g itself. The techniques used here are an extension of those of [H 2] which are in turn strongly related to those of [C; Q; V; W] and others. We begin by constructing several simplicial complexes from configurations of simple closed curves and properly imbedded arcs in a surface of genus g. The homology of the spin moduli space is identified with that of the spin mapping class group G, which acts on these complexes in a natural way. The Borel construction is then applied to obtain a spectral sequence which describes the homology of G in terms the homology of the stabilizers of the cells of these complexes. These turn out to be spin mapping class groups (in an extended sense) of smaller genus and the result is established inductively. The complexes are exactly the same as those of [H 2]; however, the spectral sequence arguments are more difficult because there are more orbits of cells under the action of G. Furthermore, in Sect. 4 we apply an entirely different and much simpler version of the argument of [H 2] to obtain stability in the case of a closed surface.

The moduli space of curves of genus three together with an odd theta-characteristic is rational

In this note we use a “normal form”, due to Sylvester, for the equation of a generic cubic surface in ℙ 3 (ℂ) to prove that %plane1D;4B0;= {moduli space of pairs ( S,P ) with S smooth cubic surface, P a point on S } is rational. We then prove that %plane1D;510; 3 oth = {moduli space of curves of genus three together with one odd theta-characteristic} is birational to %plane1D;4B0; and so rational.