2012/01/03 by Popovici, Dan
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1201.0740
Associated with a smooth, d-closed (1, 1)-form α of possibly non-rational De Rham cohomology class on a compact complex manifold X is a sequence of asymptotically holomorphic complex line bundles Lk on X equipped with (0, 1)-connections ∂k for which ∂k2≠ 0. Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for Hörmander's familiar L2-estimates of the ∂-equation of the integrable case that is based on analysing the spectra of the Laplace-Beltrami operators Δk" associated with ∂k. Global approximately holomorphic peak sections of Lk are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when α is strictly positive : a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact Kähler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of α lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on α.