14 oct 2019 -- 14:00   [open in google calendar]

Scuola Normale Superiore, Aula Mancini

Abstract.

For a complex manifold with $\mathbb{C}^{\ast }$ action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index. By applying the method of \textit{transversal} heat kernel asymptotics, we obtain a local index formula for the $m$-index. We can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our single integral over a (smooth) complex manifold. We generalize $\mathbb{C}^{\ast }$ action to complex semisimple Lie group $G$ action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all $G$ invariant holomorphic $p$-forms. Finally, in the case of two compatible holomorphic $\mathbb{C}^{\ast }$ actions, a mirror-symmetry type isomorphism is found between two linear spaces of holomorphic forms, and the Euler characteristic associated with these spaces can be computed by our $\mathbb{C}^{\ast }$ local index formula on the total space. This is joint work with Chin-Yu Hsiao and I-Hsun Tsai.