Abstract:

Inspired by the LG/CY correspondence, we study the local index theory of the Schrodinger operator associated to a singularity defined on C-n by a quasi-homogeneous polynomial f. Under some mild assumption to f, we show that the small time heat kernel expansion of the corresponding Schrodinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integration. The heat kernel expansion provides the spectral invariants of f. Furthermore, we can define the torsion type invariants associated to a homogeneous singularity. The spectral invariants provide another way to classify the singularity.

(Link: https://arxiv.org/pdf/1603.06530.pdf) 

Papers Published：
[1] H. Fan, and H. Fang, Torsion type invariants of singularities, Vietnam Journal of Mathematics (celebrate Juergen Jost's 65 birthday), Published online,50 pages, 2021.
[2] H. Fan, T. Lan and Z. Yang, LG/CY Correspondence Between tt*Geometries, Comm. Math. Research, Vol.37,No.3,297-349, 2021.

        Prof. Fan Huijun