# Subelliptic Spinℂ Dirac operators, I

Abstract Let X be a compact Kähler manifold with strictly pseudoconvex boundary, Y. In this setting, the Spinℂ Dirac operator is canonically identified with ∂¯b+∂¯∗b:∞(X;Λ0,e)→∞(X;Λ0,o). We consider modifications of the classical ∂¯b-Neumann conditions that define Fredholm problems for the Spinℂ 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 Spinℂ 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 Spinℂ Dirac operators on the components. This is a subelliptic analogue of Bojarski’s formula in the elliptic case.

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### 数学年刊（Annals of Mathematics）

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