##
**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.

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.

Reviewer: Sebastian Goette (Freiburg)

### MSC:

32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |

58J32 | Boundary value problems on manifolds |

32W25 | Pseudodifferential operators in several complex variables |