# $\mathbb R$-invariant CR structures, hypoellipticity, and Kodaira's vanishing theorem

Gerardo Mendoza (Temple University)
Wednesday, May 23, 2018 - 4:00pm to 5:00pm
Let $\overline {\mathcal K}$ be CR structure on a compact manifold $\mathcal N$, let $\mathcal T$ a nonvanishing real vector field that preserves $\overline{\mathcal K}$ and a Hermitian structure on $\overline {\mathcal K}$. The null spaces, $\mathcal H^q(\mathcal N;\overline {\mathcal K})$ of the resulting Laplacians of the CR complex have a decomposition into eigenspaces of $-i\mathcal L_{\mathcal T}$ (Lie derivative) which are finite-dimensional. All eigenvalues are real and form a discrete set without finite points of accumulation. Assuming non-degeneracy of the Levi form, only finitely many eigenvalues of $-i\mathcal L_{\mathcal T}$ lie on the positive (or negative, depending on $q$ and the signature of the Levi form) real axis. Finiteness of spectrum on one side or the other of $0$ is strongly related to Kodaira's vanishing theorem.