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Bimeromorphic geometry of LCK manifolds
Proceedings of the American Mathematical Society. 2024. Vol. 152. No. 2. P. 701 – 707.
Verbitsky M., Ornea L.
A locally conformally Kähler (LCK) manifold is a complex manifold M which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of M. We prove that any bimeromorphic map M′→M is in fact holomorphic; in other words, M has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.
Keywords: minimal modelLocally conformally Kählerglobal Kähler potentialbimeromorphismnormal variety
Publication based on the results of:
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A complex manifold X is called "LCK manifolds with potential" if it can be realized as a complex submanifold of a Hopf manifold. Let Y its $\Z$-covering, considered as a complex submanifold in Cn∖0. We prove that Y is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on Y is independent from the choice of X. ...
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An LCK manifold is a complex manifold (M,I) equipped with a Hermitian form ω and a closed 1-form θ, called the Lee form, such that dω=θ∧ω. An LCK manifold with potential is an LCK manifold with a positive Kähler potential on its universal cover, such that the deck group multiplies the Kähler potential by a constant. A Lee class of an ...
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A Hopf manifold is a quotient of Cn \0 by the cyclic group generated by a holomorphic contraction. Hopf manifolds are diffeomorphic to S1 × S2n−1 and hence do not admit Kähler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kähler (LCK) metrics. In this paper, we prove that the Hopf ...
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An LCK manifold with potential is a quotient M of a Kähler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that a compact M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove ...
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Let M be a complex manifold and L an oriented real line bundle on M equipped with a flat connection. A “locally conformally Kähler” (LCK) form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian of a function called LCK potential. We ...
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