Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jih-Hsin Cheng, Chin-Yu Hsiao, I-Hsun Tsai

TL;DR

The paper develops a transversal heat-kernel framework for open complex manifolds with a holomorphic $ ext{C}^{ imes}$-action, defining the $m$-th Fourier–Dolbeault cohomology $H_m^{q}(oldsymbol{ abla},oldsymbol{ abla})$ and the corresponding index. A local Hirzebruch–Riemann–Roch density $HRR_m$ is constructed via transversal heat-kernel asymptotics, yielding a local index formula that unifies the $m$-indices across all $m$ and recovers Kawasaki’s HRR for orbifolds through integrals over $oldsymbol{ abla}$ and singular strata. The work extends to complex reductive group actions, explores two compatible $ ext{C}^{ imes}$-actions with a mirror-type invariant theory, and develops a robust transversal spin$^{c}$ Dirac operator calculus. Central technical achievements include a carefully designed non-$ ext{C}^{ imes}$-invariant yet $ ext{S}^{1}$-invariant metric $G_{a,m}$, a transversal Hodge theory with modified Sobolev norms, and a constructive approximation to the transversal heat kernel with precise diagonal asymptotics and Lefschetz-type fixed-point contributions. These results yield a local index density, a global index formula, and provide a bridge to equivariant cobordism and potential physical applications.

Abstract

For a complex manifold $Σ$ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ 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 $\mathbb{C}^{\ast }$-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 $Ω_{\ast }^{\mathbb{C}^{\ast }}(Σ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.

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

TL;DR

The paper develops a transversal heat-kernel framework for open complex manifolds with a holomorphic -action, defining the -th Fourier–Dolbeault cohomology and the corresponding index. A local Hirzebruch–Riemann–Roch density is constructed via transversal heat-kernel asymptotics, yielding a local index formula that unifies the -indices across all and recovers Kawasaki’s HRR for orbifolds through integrals over and singular strata. The work extends to complex reductive group actions, explores two compatible -actions with a mirror-type invariant theory, and develops a robust transversal spin Dirac operator calculus. Central technical achievements include a carefully designed non--invariant yet -invariant metric , a transversal Hodge theory with modified Sobolev norms, and a constructive approximation to the transversal heat kernel with precise diagonal asymptotics and Lefschetz-type fixed-point contributions. These results yield a local index density, a global index formula, and provide a bridge to equivariant cobordism and potential physical applications.

Abstract

For a complex manifold with -action, we define the -th Fourier-Dolbeault cohomology group and consider the -index on . By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the -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 -action to complex reductive Lie group -action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all -invariant holomorphic -forms. Finally, in the case of two compatible holomorphic -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 local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.

Paper Structure

This paper contains 15 sections, 101 theorems, 533 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries on complex manifolds with $\mathbf{C}^{\ast }$-action
  3. A Hermitian metric on complex manifolds with $\mathbb{C}^{\ast }$-action
  4. A Hodge theory for $\square _{\Sigma ,m}^{(q)}$
  5. Transversally spin$^{c}$ Dirac operators
  6. Approximation of the transversal heat kernel $e^{-t\tilde{\square}_{m}^{c\pm }}$
  7. Asymptotic expansion of the transversal heat kernel
  8. Local $m$-index formula
  9. Part I of the local index formula ($k=0$ in (\ref{['7-46-1']}))
  10. Part II of the local $m$-index formula ($k\geq 1$ in ( \ref{['7-46-1']})) via Lefschetz type formulas.
  11. The local index formula completed
  12. Comparison with Duistermaat's formula for the Kähler case, Part I: from real to complex
  13. Comparison with Duistermaat's orbifold version of the index theorem, Part II: integrals over fixed point orbifolds
  14. Nonextendability of open group action; meromorphic action
  15. Complex manifolds with two holomorphic $\mathbb{C}^{\ast }$-actions

Key Result

Theorem 1.1

(proved in Subsection Subs-8-3) With the notations above, suppose that $\Sigma$ is an $n$-dimensional (open) complex manifold with a holomorphic, proper $\mathbb{C}^{\ast }$-action $\sigma (\rho e^{i\theta }).$ Assume that the $R^{+}$ part $\sigma (\rho )$ is globally free, the $S^{1}$ part $\sigma$ $ii)$ the following index is well defined and satisfies

Theorems & Definitions (204)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 194 more