Marked Riemann Surface Research Articles

On the stability of extensions of tangent sheaves on Kähler–Einstein Fano/Calabi–Yau pairs

Let S be a smooth projective variety and $$\Delta $$ a simple normal crossing $${\mathbb {Q}}$$ -divisor with coefficients in (0, 1]. For any ample $${\mathbb {Q}}$$ -line bundle L over S, we denote by $$\mathscr {E}(L)$$ the extension sheaf of the orbifold tangent sheaf $$T_S(-\log (\Delta ))$$ by the structure sheaf $$\mathcal {O}_S$$ with the extension class $$c_1(L)$$ . We prove the following two results: These results generalize Tian’s result where $$-K_S$$ is ample and $$\Delta =\emptyset $$ . We give two applications of these results. The first is to study a question by Borbon–Spotti about the relationship between local Euler numbers and normalized volumes of log canonical surface singularities. We prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked Riemann surface. In particular we complete the computation of Langer’s local Euler numbers for any line arrangements in $${\mathbb {C}}^2$$ . The second application is to derive Miyaoka–Yau-type inequalities on K-semistable log-smooth Fano pairs and Calabi–Yau pairs, which generalize some Chern-number inequalities proved by Song–Wang.

Connected components of strata of Abelian differentials over Teichmüller space

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups of strata inside of the mapping class group. The main techniques of proof are mod r winding numbers and a mapping class group–theoretic analogue of the Euclidean algorithm.

T-branes and G2 backgrounds

Compactification of M- / string theory on manifolds with $G_2$ structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure "hidden" in T-branes.

Hyperelliptic d-Tangential Covers and d× d-Matrix KdV Elliptic Solitons

Abstract More than $40$ years ago I. Krichever developed the Theory of (vector) Baker–Akhiezer functions and devised a criterion for a $d$-marked compact Riemann surface to provide $d\times d$-matrix solutions to the KdV equation. Later on he also found a criterion for a $d$-marked curve to provide $d\times d$-matrix solutions to the Kadomtsev-Petviashvili (KP) equation, doubly periodic with respect to $x$, the 1st KP flow. In particular, when both criteria apply, one should obtain $d\times d$-matrix KdV elliptic solitons. It seems, however, that the latter issue has been completely neglected until very recently (cf. [10] where the $d=2$ case is treated). In this article we fix a complex elliptic curve $X=\mathbb{C}/\Lambda$, corresponding to a lattice $\Lambda \subset \mathbb{C}$, and define so-called hyperelliptic $d$-tangential covers as $d$-marked covers of $X$ satisfying a geometric condition inside their Jacobians. They satisfy Krichever’s criteria and give rise, therefore, to families of $d\times d$-matrix KdV elliptic solitons. We also construct an anticanonical rational surface ${\mathcal S}$ naturally attached to $X$, with a Picard group of rank $10$. It turns out that the former covers of $X$ correspond to rational irreducible curves in suitable divisor classes of ${\mathcal S}$. We thus reduce their construction to proving that the associated Severi Varieties (of rational irreducible nodal curves) are not empty. The final key to the problem consists in finding rational reducible nodal curves in the latter divisor classes that can be deformed to irreducible ones, according to A. Tannenbaum’s criterion (see [5]). At last we deduce, for any $d\geq 2$, infinite families (of arbitrary high genus and degree) of hyperelliptic $d$-tangential covers, giving rise to $d\times d$-matrix KdV elliptic solitons.

Hyperelliptic graphs and the period mapping on outer space

The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively. The map Phi is a free group analog of the classical period mapping that sends a marked Riemann surface to its Jacobian. In this paper, we analyze the fibers of Phi in T_n, showing that they are aspherical, pi_1-injective subspaces. Metric graphs admitting a 'hyperelliptic involution' play an important role in the structure of Phi, leading us to define the hyperelliptic Torelli group, ST(n) < Out(F_n). We obtain generators for ST(n), and apply them to show that the connected components of the locus of 'hyperelliptic' graphs in T_n become simply-connected when certain degenerate graphs at infinity are added.

Derived invariants for surface algebras

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler [11]. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential (Q,W) associated with the triangulation.Fixing a set ϵ of generators of the fundamental group of the surface π1(S), we associate to any cut d a weight wϵ(d)∈Z2g+b, where g is the genus of S and b the number of boundary components. The main result of the paper asserts that the derived equivalence class of the surface algebra is determined by the corresponding weight wϵ(d) up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤2, and any surface algebras coming from the same surface (S,M) are cluster equivalent, in the sense of [2]. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2].Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in [6] is determined by the weight.

Singularity categories of gentle algebras

We determine the singularity category of an arbitrary finite dimensional gentle algebra $\Lambda$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $\Lambda$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.

Tau function and Chern–Simons invariant

We define a Chern–Simons invariant for Schottky hyperbolic 3-manifolds of infinite volume. We then prove an expression relating the Bergman tau function on a fiber space over the Teichmüller space to the lifting of the function F defined by Zograf on Teichmüller space, and a holomorphic function on this space which we introduce. If the point in this space corresponds to a marked Riemann surface X, then this function is constructed from the renormalized volume and our Chern–Simons invariant for the bounding 3-manifold of X given by Schottky uniformization, together with a regularized Polyakov integral. We also obtain a relation between the Chern–Simons invariant and the eta invariant of the bounding 3-manifold, with defect given by the phase of the Bergman tau function of X.

Coloured quivers for rigid objects and partial triangulations: the unpunctured case

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi–Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalized cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and we give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi–Yau case. We further show that Iyama–Yoshino reduction can be interpreted as cutting along an arc in the surface.

Historical development of Teichmüller theory

Originally, the expression “Teichmuller theory” referred to the theory that Oswald Teichmuller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmuller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmuller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmuller theory and the development of its early major ideas.

Behavior of geodesic-length functions on Teichmüller space

Let T be the Teichmuller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichmuller space of marked Riemann surfaces with nodes, (Abi77, Ber74). Pro- vided with the WP metric T is a complete CAT(0) metric space, (DW03, Wol03, Yam04). An invariant of a marked hyperbolic struc- ture is the lengthof the geodesic in a free homotopy class. A basic feature of Teichmuller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop the understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include ex- pansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bound- ing configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a dis- crete group of isometries and bounding sums of the inverse square exponential-distance.

Group-valued implosion and parabolic structures

The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian K -manifolds, where K is a simply connected compact Lie group. The imploded cross-section of the double K × K turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian K -manifolds, such as the "spinning 2 n -sphere" for K = SU ( n ). Secondly we construct a universal ("master") moduli space of parabolic bundles with structure group K over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of K and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for K = SU (2)), the master moduli space turns out to be a complex algebraic variety.

The supermoduli space of matrix string theory

We study matrix string scattering amplitudes and matrix string instantons on a marked Riemann surface in the limit of a vanishing string coupling constant. We give an explicit parameterization of the moduli space of such instantons. We also give a description of the set of fermionic supermoduli. The integration over the supermoduli leads to the inclusion of picture changing operators at the interaction points. Finally we investigate the large N limit of the measure on the instanton moduli space and show its convergence to the Weil-Petersson measure on the moduli space of marked Riemann surfaces.

Schottky uniformization and the symplectic structure of the cotangent bundle of a Teichmüller space

Using the Schottky uniformization of a marked Riemann surface, the space of equivalence classes of complex projective structures on a compact oriented C ∞ surface gets identified, in an obvious fashion, with the total space of the holomorphic cotangent bundle of the corresponding Teichmüller space. This identification is proved to be compatible with the natural symplectic structures on these two spaces. If we consider the other identification of the same two spaces obtained using the Bers’ construction of universal Riemann surface, then the compatibility of the symplectic structures was established by Kawai [S. Kawai, Math. Ann. 305 (1996) 161–182].

New parameters for Fuchsian groups of genus $2$

We give a new real-analytic embedding of the Teichmiiller space of closed Riemann surfaces of genus 2 into R6. The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in PSL(2, ), where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to define the image of the embedding; the four matrices corresponding to a point in this image generate a fuchsian group representing a closed Riemann surface of genus 2. In this note we introduce a new, canonical, real-analytic embedding of the Teichmiiller space T2, of closed Riemann surfaces of genus 2, onto an explicitly defined region R C R6. The embedding is defined in terms of the underlying hyperbolic geometry; in particular, the parameters are elementary functions of lengths of simple closed geodesics, and angles and distances between simple closed geodesics. We start with a specific marked Riemann surface So, and a specific set of normalized (non-standard) generators, ao, bo, co, do E PSL(2, R), for the fuchsian group Go representing So. Then we can realize a point in T2 as a set of appropriately normalized generators a, b, c, d E PSL(2, I) for the fuchsian group G representing a deformation S of So. We write the entries in the generators, a,... , d, as elementary functions of eight parameters, all defined in terms of the underlying hyperbolic geometry, and we write down explicit formulae expressing two of these parameters as functions of the other six. Three of our six parameters are necessarily positive; we give two additional inequalities to obtain necessary and sufficient conditions for the group G C PSL(2, R), generated by a,... ,d, to be an appropriately normalized quasiconformal deformation of Go. There is a related embedding in [3], where the parameters are fixed points of hyperbolic elements of G. As in [3], we identify the Teichmiiller space with DY), the identity component of the space of discrete faithful representations of 7rl(So) into PSL(2, ) modulo conjugation. The main differences between these two embeddings is that in [3] the four matrices do not have unit determinant and the parameters there are defined in terms of fixed points of elements of the group; here our matrices do have unit determinant, and the parameters are defined in terms of the underlying hyperbolic geometry. Received by the editors October 20, 1997 and, in revised form, February 20, 1998. 1991 Mathematics Subject Classification. Primary 30F10; Secondary 32G15. Research supported in part by NSF Grant DMS 9500557. (?1999 American Mathematical Society

Non-differentiability of the Teichmuller cometric in infinite-dimensional Teichmuller spaces

The author shows the non-differentiability of the Teichmuller cometric for infinite-dimensional Teichmuller spaces. Let Y be a Riemann surface, the Teichm uller space of Y denoted by T(Y). For[X, f]∈T(Y), where [X. f] is an equivalent class of marked Riemann surfaces (X. f), let

On the natural line bundle on the moduli space of stable parabolic bundles

We construct the natural holomorphic line bundle on the moduli space of stable parabolic bundles on a compact marked Riemann surface, which is the prequantum line bundle for the Chern-Simons gauge theory. The fusion rule in the Chern-Simons gauge theory can be viewed as the existence condition of this line bundle.

ON UNIFORMIZATION OF RIEMANN SURFACES AND THE WEIL-PETERSSON METRIC ON TEICHMÜLLER AND SCHOTTKY SPACES

A potential is constructed for the Weil-Petersson metric on the Teichm?ller space of marked Riemann surfaces of genus 1$ SRC=http://ej.iop.org/images/0025-5734/60/2/A03/tex_sm_3170_img2.gif/> in terms of the density of the Poincar? metric on the region of discontinuity of the corresponding normalized marked Schottky group. It is proved that the difference between the projective connections corresponding to the Fuchsian uniformization and the Schottky uniformization for a marked Riemann surface of genus 1$ SRC=http://ej.iop.org/images/0025-5734/60/2/A03/tex_sm_3170_img2.gif/> is the -derivative of this potential, and the Weil-Petersson symplectic form on Teichm?ller space is the -derivative of the Fuchsian projective connection. The results establish how the accessory parameters of the Fuchsian uniformization and the Schottky uniformization of a Riemann surface are connected with the geometries of Teichm?ller space and Schottky space.Bibliography: 31 titles.

ON LIOUVILLE'S EQUATION, ACCESSORY PARAMETERS, AND THE GEOMETRY OF TEICHMÜLLER SPACE FOR RIEMANN SURFACES OF GENUS 0

For the Weil-Petersson metric on the Teichmuller space of marked Riemann surfaces of genus 0 with punctures, a potential is constructed in terms of the density of the hyperbolic metric on the corresponding surface (i.e., in terms of a solution of Liouville's equation). It is shown that this potential is a generating function of the accessory parameters of the Fuchsian uniformization of the corresponding Riemann surface. Also, a connection is established between the accessory parameters and the Eichler integrals of Fuchsian groups.Bibliography: 18 titles.

Moduli of marked Riemann surfaces

Moduli of marked Riemann surfaces