2013 / June Volume 8 No.2
Hypoellipticity and Vanishing Theorems
| Published Date |
2013 / June
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|---|---|
| Title | Hypoellipticity and Vanishing Theorems |
| Author | |
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| Download | |
| Pagination | 231-258 |
| Abstract | 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.
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| AMS Subject Classification |
58C4, 32L20, 32V05, 58J10
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| Received |
2013-04-20
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| Accepted |
2013-04-21
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