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

• Date: 2023
• Main Authors: Balinsky, A. A., Prykarpatski, A. K., Pukach, P. Ya., Vovk, M. I.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2023
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7320
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.