Abstract
Let M be a holomorphically symplectic complex manifold, not necessarily compact or quasiprojective, and \(X \subset M\) 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 \(T^* X\). This is used to study Lagrangian submanifolds which can be bimeromorphically contracted to a point. We prove that such submanifolds are biholomorphic to \({\mathbb {C}}P^n\), 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.
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Notes
To give an example, let M be holomorphically Lagrangian fibered over a disc. The general fiber of this fibration is a complex torus, and one sees from the Leray spectral sequence that \(H^1({\mathcal {O}}_M)\) does not vanish.
This part has been added after the first version of this paper has been written, and is suggested by the subsequent discussions with C. Shramov and Yu. Prokhorov.
I.e. such that all of its members passing through a general point are irreducible. Informally, X should not covered by rational curves splitting off some curves in this family.
See [7], Theorem 4.2. Since [7] is known to be not always accurate, we prefer to rely on [16], who proves in Proposition 3.1 that the base \(H_x\) of a family of minimal rational curves passing through the general point x and covering the variety X is \({\mathbb {P}}^{n-1}\), and in the following Sect. 3.2 computes that the total space \(U_x\) of this family is \({\mathbb {P}}(\mathcal{O}_{{\mathbb {P}}^{n-1}}\oplus \mathcal{O}_{{\mathbb {P}}^{n-1}}(-1))\), so that the evaluation map contracts the exceptional section onto \({\mathbb {P}}^n\).
See also [1] for a simple argument using twistor spaces, which should be generalizable to some extent.
This is what \(\mu _{\alpha , max}(TX)>0\) means by definition.
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Acknowledgements
We are grateful to Frédéric Campana, Andreas Höring, Dmitry Kaledin, Yuri Prokhorov, Costya Shramov and Andrey Soldatenkov for their insightful discussions, and to Arnaud Beauville for his valuable email communication.
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Communicated by Ivan Cheltsov.
In memory of Sasha Ananin, our dear friend and a colleague. We miss you so much.
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Ekaterina Amerik and Misha Verbitsky acknowledge support of HSE University basic research program; also partially supported by ANR (France) project FANOHK. Partially supported by FAPERJ E-26/202.912/2018 and CNPq - Process 310952/2021-2.
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Amerik, E., Verbitsky, M. Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds. São Paulo J. Math. Sci. (2024). https://doi.org/10.1007/s40863-024-00426-7
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DOI: https://doi.org/10.1007/s40863-024-00426-7