2013 / June

Hypoellipticity and Vanishing Theorems

Gerardo Mendoza

Spectral theory, hypoellipticity, CR manifolds, cohomology, vanishing theorems, Spectral theory, hypoellipticity, CR manifolds, cohomology, vanishing theorems

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231-258

Let $-i\mathcal{L}_\mathcal{T}$ (essentially Lie derivative with respect to $\mathcal{T}$, a smooth nowhere zero real vector field) and $P$ be commuting differential operators, respectively of orders $1$ and $m\geq 1$, the latter formally normal, both acting on sections of a vector bundle over a closed manifold. It is shown that if $P+(-i\mathcal{L}_\mathcal{T})^m$ is elliptic then the restriction of $-i\mathcal{L}_\mathcal{T}$ to $\mathscr{D}\subset \ker P\subset L^2$ ($\mathscr{D}$ is carefully specified) yields a selfadjoint operator $-i\mathcal{L}_\mathcal{T}|_\mathscr{D}:\mathscr{D}\subset\ker P\to \ker P$ with compact resolvent. It is also shown that, in the presence of an additional hypothesis on microlocal hypoellipticity of $P$, $-i\mathcal{L}_\mathcal{T}|_\mathscr{D}$ is semi-bounded. These results are applied to CR manifolds on which $\mathcal{T}$ acts as an infinitesimal CR transformation which are then shown to yield versions of Kodaira's vanishing theorem.

58C4, 32L20, 32V05, 58J10

2013-04-20

2013-04-21