Subelliptic Spin_C Dirac operators, I

TL;DR

This paper explores subelliptic boundary conditions for Spin_C Dirac operators on complex manifolds, deriving index formulas and boundary index relations, extending classical elliptic results to subelliptic settings.

Contribution

It introduces subelliptic boundary conditions for Spin_C Dirac operators and derives index formulas, including a subelliptic analogue of Bojarski's formula and boundary index relations.

Findings

Derived subelliptic estimates for boundary value problems.

Expressed holomorphic Euler characteristic as index of Spin_C-Dirac operator.

Established an analogue of the Agranovich-Dynin formula for subelliptic cases.

Abstract

We consider modifications of the classical dbar-Neumann conditions that define Fredholm problems for the Spin_C Dirac operator. In part II, we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a Spin_C-Dirac operator with a subelliptic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface, then we obtain formulae for the holomorphic Euler characteristic of X as sums of indices of Spin_C-Dirac operators on the components. This is a subelliptic analogue of Bojarski's formula in the elliptic case.