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Staff Introduction

WENG, Lin / Professor

1) Between 1989-98, we developed a theory of relative Bott-Chern secondary characteristic classes, based on which we established an arithmetic Grothendieck-Riemann-Roch theorem for l.c.i. morphisms.

2) We also develop an Arakelov theory for surfaces with respect to singular metrics by establishing an arithmetic Deligne-Riemann-Roch isometry for them. Consequently, we study arithmetic aspect of the moduli spaces of punctured Riemann surfaces by introducing certain natural metrized line bundles related with Weil-Petersson metrics, Takhtajan-Zograf metrics. Intrinsic relations among them, some of which are open problems, are exposed as well. The difficulty here is that classical approach on determinant metric does not work.


3) We introduce genuine non-abelain L functions for global fields, based on a new cohomology, stability and Langlands' theory of Eisenstein series, and expose the relation between these non-abelian Ls and what we call the Arthur periods. Basic properties such as meromorphic continuation and functional equation(s) are established as well. In particular we show that the rank two non-abelian zetas for number fields satisfy the Riemann Hypothesis.


4) We develop a Program on what we call Geometric Arithmetic, in which an approach to non-abelian Class Field Theory using stability and an approach to the Riemann Hypothesis using intersection, together with a study on non-abelian L functions, are included.


5) We initiated an Arakelov approach to the study of what we call Kobayashi-Hitchin correspondence for manifolds aiming at establishing the equivalence between intersection stability and existence of KE metrics. I spent several years in discussion with Mabuchi. These almost weekly discussions prove to be quite crucial to problems involved. I have no formal publication in it. But one can trace them from some papers of Mabuchi.


6) Other works such as metrized version of projective flatness of certain bundles and degenerations of Riemann surfaces are of some importance to the related fields.


Most of the works listed above can be found either at xxx.lanl.gov or at MathSciNet. 

Keywords Algebraic and/or Arithmetic and/or Complex Geometry, Number Theory
Faculty , Department Faculty of Mathematics , Department of Mathematics