2024/12/15 by Jih-Hsin Cheng, Cheng, Jih-Hsin, Chin-Yu Hsiao +3
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2412.11037
For a complex manifold Σ with ℂ∗ -action, we define the m-th ℂ∗ Fourier-Dolbeault cohomology group and consider the m-index on Σ. 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 ℂ∗ -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 ℂ∗ -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 ℂ∗ local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory Ω∗ ^ℂ∗ (Σ), a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.