An analogue of Demailly’s inequality for strictly pseudoconvex CR manifolds. (English) Zbl 0714.58053

Getzler, Ezra

J. Differ. Geom. 29, No. 2, 231-244 (1989).

Let L be a Heisenberg line bundle on a compact \((2n+1)\)-dimensional Heisenberg manifold M. The author gives an expression involving local data for the limit as \(m\to \infty\), of \((4\pi t/m)^{n+1}\cdot trace(\exp (-t\square /m))\) on \(\Omega^{0,q}(M;L^ m).\)
In defining Heisenberg structures, the author states that the most important examples of Heisenberg manifolds are strictly pseudo-convex hypersurfaces in \({\mathbb{C}}^{n+1}\). The restriction of a holomorphic vector bundle with Hermitian connection on \({\mathbb{C}}^{n+1}\) to the hypersurface is then an example of a Heisenberg bundle on the hypersurface. A Heisenberg manifold has a Cauchy-Riemann operator that generalizes the operator \({\bar \partial}_ b\), and this Cauchy-Riemann operator can be defined on \(\Omega^{0,*}(M;E)\) when E is a Heisenberg bundle. The Laplacian associated to the Cauchy-Riemann operator is denoted \(\square.\)
As in an earlier paper [Commun. Math. Phys. 92, No.2, 163-178 (1983; Zbl 0543.58026)], the author proves his theorem by developing an extended pseudodifferential operator calculus. This time the basic calculus is that developed for CR manifolds by R. Beals, P. C. Greiner and N. K. Standon [J. Differ. Geom. 20, No.2, 343-387 (1984; Zbl 0553.58029)] from the algebra of left-invariant operators on the Heisenberg group.
A preprint by J.-M. Bismut suggested the author’s analogue of the result of J.-P. Demailly [Ann. Inst. Fourier 35, No.4, 189-229 (1985; Zbl 0565.58017)]. For a complex line bundle on a compact complex manifold M with \(\dim_ CM=n\), Demailly calculated the asymptotic dimension of the \({\bar \partial}\)-cohomology of \(L^ m\) as \(m\to \infty\). Observing that the dimension of the \({\bar \partial}\)-cohomology was bounded by the trace of the \({\bar \partial}\)-heat operator, Bismut recovered Demailly’s result by using methods from probability theory to develop an expression for the limit, as \(m\to \infty\), of \((2\pi /m)^ n\) times the trace of the appropriate \({\bar \partial}\)-heat operator.

MSC:

58J35	Heat and other parabolic equation methods for PDEs on manifolds
32V40	Real submanifolds in complex manifolds
58J40	Pseudodifferential and Fourier integral operators on manifolds
35K05	Heat equation
32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators

Keywords:

Demailly’s inequality; heat equation; strictly pseudo-convex hypersurfaces in \({\mathbb{C}}^{n+1}\); Heisenberg manifold; pseudodifferential operator calculus; CR manifolds

Citations:

Zbl 0564.58032; Zbl 0575.58013; Zbl 0543.58026; Zbl 0553.58029; Zbl 0565.58017