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A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Mosers isotopy theorem for families of C-symplectic structures and list several applications of this result. We prove that the degenerate twistorial deformation associated to a holomorphic Lagrangian fibration is locally trivial over the base of this fibration. This is used to extend several theorems about Lagrangian fibrations, known for projective hyperkahler manifolds, to the non-projective case. We also exhibit new examples of non-compact complex manifolds with infinitely many pairwise non-birational algebraic compactifications.
We formulate and prove the analogue of Mosers stability theorem for locally conformally symplectic structures. As special cases we recover some results previously proved by Banyaga.
In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists language, i.e. a submanifold which is either c
We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irr
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple uni