On the moduli space of real Enriques surfaces

We introduce a new invariant, Pontryagin-Viro form, of real algebraic surfaces. We evaluate it for real Enriques surfaces with non-negative minimal Euler characteristic of the components of the real part and prove that, when combined with the known topological invariants, it distinguishes the deformation types of such surfaces.     

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms ${M or_P (\mathbb{G}, M(n, \xi))}$ and the moduli space of stable bundles over ${X \times \mathbb{G}}$ , where ${\mathbb{G}}$ is the Grassmannian ${\mathbb{G}(n - r, \mathbb{C}^n)}$ . Moreover, we give sufficient conditions for ${M or_{2ns}(\mathbb{P}^1, M(n, \xi))}$ to be non-empty, when s ≥ 1.     

Geometric aspects of the moduli space of riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kähler metrics were introduced on the moduli space and Teichmüller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kähler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincaré type growth. Furthermore, the Kähler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.

     

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.     

The Chow group of the moduli space of marked cubic surfaces

Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow-ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space. As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann–Roch theorem for the moduli space. Dedicated to the memory of our friend Fabio BardelliMathematics Subject Classification (2000) 14J15     

New results on the geometry of the moduli space of Riemann surfaces

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces,including the Weil-Petersson metric,the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric.We prove the dual Nakano negativity of the Weil-Petersson metric.As applications of these results we deduce certain important results about the L~2-cohomology groups of the logarithmic tangent bundle over the compactifled moduli spaces.     

The moduli space of Hessian quartic surfaces and automorphic forms

We shall show the existence of 15 automorphic forms of weight 8 on the moduli space of marked Hessian quartic surfaces of cubic surfaces. These automorphic forms can be interpreted in terms of the coefficients of the Sylvester form of a general cubic surface.     

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of \mathbbZ/2{\mathbb{Z}/2} by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Isotrivially fibred isles in the moduli space of surfaces of general type

We complement Catanese's results on isotrivially fibred surfaces by completely describing the components containing an isotrivial surface with monodromy group . We also give an example for deformation equivalent isotrivial surfaces with different monodromy group.