LOGARITHMIC HEAT PROJECTIVE OPERATORS

Abstract

The sheaf of differential operators on log schemes is defined and studied. Then logarithmic differential operators on compactified Jacobians of singular curves are studied. In particular, a logarithmic heat equation for theta functions is produced geometrically.

Key Words:

• Heat operators
• Logarithmic
• Moduli spaces

Mathematics Subject Classification:

• 14F05
• 14F10
• 14D20

ACKNOWLEDGMENTS

This work was done during my stay at FB 6 Mathematik of Universität Essen. I would like to express my hearty thanks to Prof. H. Esnault and Prof. E. Viehweg for their hospitality and encouragement. I have benefited from stimulating discussions with them, which helped to prove Lemma 3.3 and Definition 1.3. Discussions with H. Clemens, I-Hsun Tsai, and Kang Zuo, and the emails of Z. Ran, were very helpful. I thank them very much.