On the symplectic structure deformations related to the Monge–Ampère equation on the Kähler manifold $P_{2}(\mathbb{C})$

A. A.  Balinsky; A. K.  Prykarpatski; P. Ya.  Pukach; M. I.  Vovk

doi:10.37863/umzh.v75i1.7320

A. A. Balinsky Math. Inst. Cardiff Univ., Great Britain
A. K. Prykarpatski Cracow Univ. Technology, Poland and Inst. Appl. Math. and Fundam. Sci., Lviv Polytechn. Nat. Univ., Ukraine
P. Ya. Pukach Inst. Appl. Math. and Fundam. Sci., Lviv Polytechn. Nat. Univ., Ukraine
M. I. Vovk Inst. Appl. Math. and Fundam. Sci., Lviv Polytechn. Nat. Univ., Ukraine

DOI: https://doi.org/10.37863/umzh.v75i1.7320

Keywords: Kähler manifold, symplectic deformation, Levi–Civita connection, metric deformation, Monge–Ampère equation

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 manifold $P_{2}(\mathbb{C}).$ 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 manifold $P_{2}(\mathbb{C}),$ 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 manifold $P_{2}(\mathbb{C})$ is generalized and the related metric deformations are discussed.

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