## Subelliptic $$\mathrm{Spin}_{\mathbb C}$$ Dirac operators. I.(English)Zbl 1154.32016

The Dolbeault operator $$\eth=\overline\partial+ \overline\partial{}^*$$ on a compact complex manifold $$X_+$$ with strictly pseudoconvex boundary $$Y$$ has an infinite-dimensional kernel. The elliptic Atiyah-Patodi-Singer boundary conditions provide a finite-dimensional kernel, but do not reflect the holomorphic structure of $$X_+$$. Instead, the author considers modified $$\overline\partial$$-Neumann boundary conditions and obtains a subelliptic operator.
Let $${\mathcal S}$$ be a generalized Szegő projector as defined by the author and R. Melrose in [Math. Res. Lett. 5, No. 3, 363–381 (1998; Zbl 0929.58012)]. A set of boundary conditions $${\mathcal R}_+$$, possibly twisted with a holomorphic vector bundle $$E\to X_\pm$$, consists of $${\mathcal S}$$ for the operator $$\overline\partial$$ in degree $$0$$, no conditions for $$\overline\partial$$ in higher degrees, and the $$\overline\partial$$-Neumann conditions for $$\overline\partial{}^*$$, which are modified by $$\text{Id}-{\mathcal S}$$ in degree $$1$$. Boundary conditions $${\mathcal R}_-$$ for a strictly pseudoconcave compact complex manifold $$X_-$$ are defined similarly. Boundary conditions of the type $$\text{Id}-{\mathcal R}_\mp$$ on $$X_\pm$$ with respect to the conjugate Szegő projector are dual to the above with respect to the Hodge star operator.
Subelliptic estimates proved in part II of this series [Ann. Math. (2) 166, No. 3, 723–777 (2007; Zbl 1154.32017)] guarantee that all four boundary value problems lead to associated Hodge decompositions. For the classical Szegő projector, the harmonic forms of $$\eth_{\mathcal R_\pm}$$ and $$\eth_{\text{Id}-{\mathcal R}_\mp}$$ are identified and compared with those of the classical $$\overline\partial$$-Neumann problem considered by G. B. Folland and J. J. Kohn in [The Neumann problem for the Cauchy Riemann complex. Princeton, N.J.: Princeton University Press and University of Tokyo Press (1972; Zbl 0247.35093)]. The author proves an Agranovich-Dynin type formula, by which the difference of the indices of two boundary value problems associated to two different generalized Szegő projectors is the relative index of these projectors on $$Y$$.
The author proves a modification of the long exact sequence of A. Andreotti and C. D. Hill in [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26, 325–363 (1972; Zbl 0256.32007)] that relates the kernels of $$\eth_{\mathcal R_+}$$ and $$\eth_{\text{Id}-\mathcal R_-}$$ to the Kohn-Rossi cohomology of $$Y$$. If a compact complex manifold $$X$$ is split along a strictly pseudoconvex hypersurface $$Y$$, the author obtains a gluing formula for the holomorphic Euler characteristic.

### MSC:

 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 58J32 Boundary value problems on manifolds 32W25 Pseudodifferential operators in several complex variables

### Citations:

Zbl 0929.58012; Zbl 0247.35093; Zbl 0256.32007; Zbl 1154.32017
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