Transcendental Kähler Cohomology Classes
Abstract
Associated with a smooth, $d$closed $(1, 1)$form $\alpha$ of possibly nonrational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was begun in the thesis of L. Laeng. We propose in this nonintegrable context a substitute for Hörmander's familiar $L^2$estimates of the $\bar\partial$equation of the integrable case that is based on analysing the spectra of the LaplaceBeltrami operators $\Delta_k"$ associated with $\bar\partial_k$. Global approximately holomorphic peak sections of $L_k$ are constructed as a counterpart to Tian's holomorphic peak sections of the integralclass case. Two applications are then obtained when $\alpha$ is strictly positive : a Kodairatype approximately holomorphic projective embedding theorem and a Tiantype almostisometry theorem for compact Kähler, possibly nonprojective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of $\alpha$ lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on $\alpha$.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 DOI:
 10.48550/arXiv.1201.0740
 arXiv:
 arXiv:1201.0740
 Bibcode:
 2012arXiv1201.0740P
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry
 EPrint:
 49 pages