On the symplectic structure deformations related to the Monge–Ampère equation on the Kähler manifoldP2(ℂ)

Abstract

UDC 517.9 We analyze the cohomology structure of the fundamental two-form deformation related to a modified Monge–Ampère type on the complex Kähler manifoldP2(ℂ). Based on the Levi-Civita connection and the related vector-field deformation of the fundamental two-form, we construct a hierarchy of bilinear symmetric forms on the tangent bundle of the K\"{a}hler manifoldP2(ℂ),that generate Hermitian metrics on it and corresponding solutions to the Monge–Ampère-type equation. The classical fundamental two-form construction on the complex Kähler manifoldP2(ℂ)is generalized and the related metric deformations are discussed.