Subelliptic $\mathrm{Spin}_{\mathbb{C}}$ Dirac operators, I

Abstract

Let $X$ be a compact Kähler manifold with strictly pseudoconvex boundary, $Y.$ In this setting, the Spin${}_{\mathbb{C}}$ Dirac operator is canonically identified with $\bar{\partial}_b+\bar{\partial}_b^*:{\mathcal C}^{\infty}(X;\Lambda^{0,e})\rightarrow {\mathcal C}^{\infty} (X;\Lambda^{0,o}).$ We consider modifications of the classical $\bar{\partial}_b$-Neumann conditions that define Fredholm problems for the Spin${}_{\mathbb{C}}$ Dirac operator. In Part 2, [7], 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${}_{\mathbb{C}}$ 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 formulæ for the holomorphic Euler characteristic of $X$ as sums of indices of Spin${}_{\mathbb{C}}$ Dirac operators on the components. This is a subelliptic analogue of Bojarski’s formula in the elliptic case.