Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

Sao Paulo Journal of Mathematical Sciences. 2024. Vol. 18. P. 540–557.

Америк Е. Ю., Вербицкий М. С.

Let M be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and a compact Lagrangian submanifold. We construct a deformation to the normal cone, showing that a neighbourhood of X can be deformed to its neighbourhood in. This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to, and show that a certain neighbourhood of X is symplectically biholomorphic to a neighbourhood of the zero section of its cotangent bundle. This gives a holomorphic version of the Weinstein’s normal neighbourhood theorem.

Ключевые слова: normal form, Lagrangian submanifold, Hyperkähler manifold, Holomorphically symplectic manifold, 53C26

ПУБЛИКАЦИЯ ПОДГОТОВЛЕНА ПО РЕЗУЛЬТАТАМ ПРОЕКТА: Коммутативная, некоммутативная и мотивная алгебраическая геометрия и геометрия специальных многообразий (2024)