On a compact complex manifold the index of the Spin_C Dirac operator computes the holomorphic Euler characteristic. On manifolds with boundary this operator has an infinite dimensional null space. We describe subelliptic boundary condtions that define Fredholm problems for strictly pseudoconvex and pseudoconcave complex manifolds. We give generalizations of the Agranovich-Dynin and Bojarski formulae and gluing formulae for holomorphic Euler characteristics.