Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action

For a complex manifold $\Sigma$ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index on $\Sigma$. By applying the method of 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 integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. We generalize $\mathbb{C}^{\ast }$-action to complex reductive 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-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. In the perspective of the equivariant algebraic cobordism theory $\Omega _{\ast }^{\mathbb{C}^{\ast }}(\Sigma ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.