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Register a new userWebs of Lagrangian Tori in Projective Symplectic Manifolds
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For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauvilles. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandts theory of subnormal subgroups.
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We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.
Let $(M,I, Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $pi:; M mapsto X$, and $eta$ a closed form of Hodge type (1,1)+(2,0) on $X$. We prove that $Omega:=Omega+pi^* eta$ is again a holomorphically symplectic form, for another complex structure $I$, which is uniquely determined by $Omega$. The corresponding deformation of complex structures is called degenerate twistorial deformation. The map $pi$ is holomorphic with respect to this new complex structure, and $X$ and the fibers of $pi$ retain the same complex structure as before. Let $s$ be a smooth section of of $pi$. We prove that there exists a degenerate twistorial deformation $(M,I, Omega)$ such that $s$ is a holomorphic section.
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.
In the framework of Special Bohr - Sommerfeld geometry it was established that an ample divisor in compact algebraic variety can define almost canonically certain real submanifold which is lagrangian with respect to the corresponding Kahler form. It is natural to call it lagrangian shadow; below we emphasize this correspondence and present some simple examples, old and new. In particular we show that for irreducible divisors from the linear system $vert - frac{1}{2} K_{F^3} vert$ on the full flag variety $F^3$ their lagrangian shadows are Gelfand - Zeytlin type lagrangian 3 - spheres.
In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.
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