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Register a new userSections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations
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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.
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In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formula for a BG-manifold $M$, showing that there exists a symmetric form q on the second cohomology such that for any $win H^2(M)$ one has $int_M w^{2n}=q(w,w)^n$. This form is a non-Kahler version of the Beauville-Bogomolov-Fujiki form known in hyperkahler geometry.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.
Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let $K$ be its Kahler cone, which is an open, convex subset in the space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature $(1,n)$ obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in the space of signature $(1,n)$ is a disconnected union of two convex cones. The positive cone is the component which contains the Kahler cone. We say that the Kahler cone is round if it is equal to the positive cone. The manifolds with round Kahler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmuller space. We prove thay any maximal holonomy hyperkahler manifold with $b_2 > 4$ has a deformation with round Kahler cone and the Picard lattice of signature (1,1), admitting two non-collinear integer isotropic classes. This is used to show that all known examples of hyperkahler manifolds admit a deformation with two transversal Lagrangian fibrations, and the Kobayashi metric vanishes unless the Picard rank is maximal.
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.
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
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