In this paper we compute the asymptotics of the natural metric on the line bundle over the moduli space ℳ _g associated to the algebraic cycle C–C ^– in the jacobian Jac C of a smooth projective curve C of genus g ≥ 3. The asymptotics are related to the structure of the mapping class group of a genus g surface.

For a birational analogue of minimal elliptic surfaces f: X→Y, the singularities of the fibers allow us to define a log structure (Y,B _Y ) in codimension one on Y. Via base change, we have a log structure (Y′,B _Y′) in codimension one on Y′, for any birational model Y′ of Y. We show that these codimension one log structures glue to a unique log structure, defined on some birational model of Y (Shokurov's BP Conjecture). As applications, we obtain Inversion of Adjunction for the above mentioned fiber spaces, and the invariance of Shokurov's FGA-algebras under adjunction.

We extend the “miraculous cancellation” formulas of Alvarez–Gaumé, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. We apply our result to discuss intrinsic relations between the higher dimensional Rokhlin type congruences due to Ochanine, Finashin and Zhang. In particular, an analytic proof of the Finashin congruence is given.

We introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including ordinary Hamiltonian $G$-spaces, Lu's momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev–Malkin–Meinrenken. More precisely, we carry out the following program:

(1) We define and study properties of quasi-symplectic groupoids.

(2) We study the momentum map theory defined by a quasi-symplectic groupoid $\Gamma \rightrightarrows P$. In particular, we study the reduction theory and prove that $J^{-1} \mathcal {O})/ \Gamma$ is a symplectic manifold for any Hamiltonian $\Gamma$-space $(X \stackrel {J} \rightarrow P, \omega_X$ (even though $\omega_X \in \Omega^2 (X)$ may be degenerate), where $\mathcal{O} \in P$ is a groupoid orbit. More generally, we prove that the intertwiner space $(X_1 \times_{P} \overline{X_2}$ between two Hamiltonian $\Gamma$-spaces $X_1$ and $X_2$ is a symplectic manifold (whenever it is a smooth manifold).

(3) We study Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories. Moreover the intertwiner space $(X_1 \times_P \overline{X_2}$depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including the Alekseev–Ginzburg–Weinstein linearization theorem and the Alekseev–Malkin–Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group spaces.

For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so called theta line bundle on the family of moduli spaces. When the Kodaira–Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so called Hitchin's connection on the direct image of (powers of) theta line bundle is derived naturally from the identification theorem. A logarithmic extension of Hitchin's connection to certain singular stable curves is also presented in this paper.

This paper develops new tools for understanding surfaces with more than one end and infinite topology which are properly minimally embedded in Euclidean three-space. On such a surface, the set of ends forms a totally disconnected compact Hausdorff space, naturally ordered by the relative heights of the ends in space. One of our main results is that the middle ends of the surface have quadratic area growth, and are thus not limit ends. This implies that the surface can have at most two limit ends, which occur at the top and bottom of the ordering, and thus only a countable number of ends, which is a strong topological restriction. There are also restrictions on the asymptotic geometry and conformal structure of such a surface: for example, we prove that if the surface has exactly two limit ends, then it is recurrent (that is, almost all Brownian paths are dense in the surface), and in particular, any positive harmonic function on the surface is constant. These results have played an important role in several recent advances in the theory, including the uniqueness of the helicoid, the invariance of flux for a coordinate function on a properly immersed minimal surface, and the topological classification of properly embedded minimal surfaces of finite genus.