Flat λ\lambda-Connections, Mochizuki Correspondence and Twistor Spaces

In this paper, we first collect some basic results for $\lambda$-flat bundles, and then get an estimate for the norm of $\lambda$-flat sections, which leads to some vanishing theorem. Mochizuki correspondence provides a homeomorphism between the moduli space of (poly-)stable $\lambda$-flat bundles and that of (poly-)stable Higgs bundles, and provides a dynamical system on the later moduli space (the Dolbeault moduli space). We investigate such dynamical system, in particular, we discuss the corresponding first variation and asymptotic behavior. We generalize the Deligne's twistor construction for any element $\gamma$ of the outer automorphism group of the fundamental group of Riemann surface to obtain the $\gamma$-twistor space, and we apply the twistor theory to study a Lagrangian submanifold of the de Rham moduli space. As an application, we prove a Torelli-type theorem for the twistor spaces, and meanwhile, we prove that the oper stratum in the oper stratification of the de Rham moduli space is the unique closed stratum of minimal dimension, which partially confirms a conjecture by Simpson.Comment: Simpson pointed out a mistake on the Moishezon property for the twistor space in the last version, we delete it and add a section on the study of oper stratification of the de Rham moduli space as an applicatio

The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized K\"ahler Manifolds

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.Comment: To appear in The Journal of Geometric Analysi

Asymptotics in randomized urn models

This paper studies a very general urn model stimulated by designs in clinical trials, where the number of balls of different types added to the urn at trial n depends on a random outcome directed by the composition at trials 1,2,...,n-1. Patient treatments are allocated according to types of balls. We establish the strong consistency and asymptotic normality for both the urn composition and the patient allocation under general assumptions on random generating matrices which determine how balls are added to the urn. Also we obtain explicit forms of the asymptotic variance-covariance matrices of both the urn composition and the patient allocation. The conditions on the nonhomogeneity of generating matrices are mild and widely satisfied in applications. Several applications are also discussed.Comment: Published at http://dx.doi.org/10.1214/105051604000000774 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Non-Archimedean meromorphic solutions of functional equations

In this paper, we discuss meromorphic solutions of functional equations over non-Archimedean fields, and prove analogues of the Clunie lemma, Malmquist-type theorem and Mokhon'ko theorem