The purpose of this important paper is to construct a 1-parameter deformation of classical Hodge theory. For nonzero values of the parameter $b$, the corresponding Laplacian is a second order hypoelliptic operator on the cotangent bundle, which is in general non-selfadjoint. As $b\to 0$, one recovers classical Hodge theory. As $b\to\infty$, the deformed Laplacian converges to the generator of the geodesic flow. The construction is obtained using th superconnection formalism of {\it D. Quillen} [Topology 24, 89--95 (1985;

Zbl 0569.58030)].
The paper is organized as follows. In Section 1 the finite dimensional Hodge theory for finite dimensional complexes equipped with Hermitian forms of arbitrary signature, is developed. One of the main results of this theory is that, in general, the harmonic forms produce a subcomplex out of which one can extract the cohomology. It may even happen that the corresponding Laplacian is identically zero, which makes the general theory differ dramatically from standard Hodge theory. The reason for inclusion of this section is that the generalized Laplacians to be considered will precisely be associated to nonpositive Hermitian forms on the de Rham complex. Let $X$ be a Riemannian manifold, and let $F$ be a flat Hermitian sector bundle on $X$. In Section 2 one constructs the adjoint of the de Rham operator $d^{T^*X}$ with respect to a natural sesquilinear form on the de Rham complex of $T^*X$. One proves that this operator is also the adjoint of $d^{T^*X}$ with respect to a Hermitian form on the de Rham complex, which makes the theory of Section 1 applicable. In Section 3 one gives a Weitzenböck formula for the deformed Laplacian and one shows that Hörmander’s theorem can be applied to this operator. This new Laplacian is related to Mathai and Quillen’s formulas for Thom forms and one shows that it interpolates between the standard Laplacian and the geodesic flow. Also the case of $S_1$ receives special attention.
In section 4 one applies the above constructions in the context of families, the modal construction being an earlier work [{\it J.-M. Bismut, J. Lott}, J. Am. Math. Soc. 8, No. 2, 291--363 (1995;

Zbl 0837.58028)]. Special attention is given to the proof of Weitzenböck’s formulas in various forms. The motivation is connected to future applications to local index theory.
Finally we can remark that the “hypoelliptic Laplacian” introduced here looks like what {\it B. Helffer} and {\it F. Nier} [Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lect. Notes Math. 1862, Berlin: Springer (2005;

Zbl 1072.35006)] call the Fokker-Planck operator with a partial diffusion only in the momentum variable.