092-802-4490

Ph. D (University of Science and Technology of China)

Algebraic and/or Arithmetic and/or Complex Geometry, Number Theory

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.

Recently, we are studying zeta functions and general class field theory
using stability and Galois representations.

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

Research Interests

• Geometric Arithmetic
  keyword : Non-abelian Class Field Theory, Abelian and Non-Abelian Zeta Functions
  1999.06.
• Arithmetic Aspects of Moduli Spaces of Punctured Riemann Surfaces
  keyword : Weil-Petersson, Takhtajan-Zograf
  2000.03.
• Relative Bott-Chern Secondary Characteristic Classes and Arithmetic Grothendieck-Riemann-Roch Theorem for L.C.I. Morphisms
  keyword : Relative Bott-Chern Secondary Class, Grothendieck-Riemann-Roch Theorem
  1990.06.

Academic Activities

Educational Activities