Some remarks on the Dolbeault-Kostant complex (Q1099856)

The Dolbeault-Kostant complex is a sort of de Rham cohomological complex. There are two differences: instead of being constructed from the tangent bundle of some manifold, it is costructed on a vector subbundle defined by a constant rank integrable distribution, i.e. a foliation. Instead of differential forms we consider sheaves of germs of differential forms on the base manifold. This permits one to handle situations in the process of geometrical quantization where the Hilbert space representation is trivial. It was extended by \textit{H. R. Fischer} and \textit{F. L. Williams} [Trans. Am. Math. Soc. 252, 163-195 (1979; Zbl 0423.14008)] to the general case of a manifold (no longer symplectic) with a complex foliation on it. The author of this paper studies eigenvalue properties of the Laplace operator for this general cohomology complex. Among other things, properties of harmonic forms, foliated versions of the Hodge theorem and asymptotic heat expansions are studied. It is also shown that the Euler characteristic is finite, and a formula is given to compute it in terms of the spectrum of the Laplacians.