Transcendental K\"ahler Cohomology Classes

Abstract

Associated with a smooth, $d$-closed $(1, 1)$-form $\alpha$ of possibly non-rational 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 non-integrable context a substitute for H\"ormander's familiar $L^2$-estimates of the $\bar\partial$-equation of the integrable case that is based on analysing the spectra of the Laplace-Beltrami 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 integral-class case. Two applications are then obtained when $\alpha$ is strictly positive : a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact K\"ahler, possibly non-projective, 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 $\alpha$.