Analyzing the L2∂̄ -cohomology

Abstract

For the bundle-valued differential forms on complex manifolds, a method of solving ∂̄ -equations with a control of L2 norm is discussed. Basic results are existence theorems for such solutions under curvature conditions. They are variants of Kodaira’s cohomology vanishing theorem on compact Kähler manifolds, and formulated as vanishing theorems with L2 conditions. Some of these L2 vanishing theorems are generalized to finite-dimensionality theorems under the assumptions on the bundle-convexity. Besides applications to holomorphic functions, extensions of the Hodge theory to noncompact manifolds will also be discussed.