2023/09/20 by Popovici, Dan
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2309.11291
Given a complete Kähler manifold (X, ω) with finite second Betti number, a smooth complex hypersurface Y⊂ X and a smooth real d-closed (1, 1)-form α on X with arbitrary, possibly non-rational, De Rham cohomology class \α\ satisfying a certain assumption, we obtain extensions to X, with control of their L2-norms, of smooth sections of the canonical bundle of Y twisted by the restriction to Y of any C^∞ complex line bundle Lk in a sequence of asymptotically holomorphic line bundles whose first Chern classes approximate the positive integer multiples k\α\ of the original class. Besides a known non-integrable (0, 1)-connection ∂k on Lk, the proof uses two twisted Laplace-type elliptic differential operators that are introduced and investigated, leading to Bochner-Kodaira-Nakano-type (in-)equalities, a spectral gap result and an a priori L2-estimate. The main difference from the classical Ohsawa-Takegoshi extension theorem is that the objects need not be holomorphic, but only asymptotically holomorphic as k→∞. The possibility that ∂k does not square to 0 accounts for its lack of commutation with the Laplacian Δ''k it induces. We hope this study is a possible first step in a future attack on Siu's conjecture predicting the invariance of the plurigenera in Kähler families of compact complex manifolds.