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Register a new userExtending rationally connected fibrations from ample subvarieties
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Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.
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Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.
We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.
We prove that a degeneration rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $epsilon$-CY type form a birationally bounded family for $epsilon>0$. Moreover, we show that the set of $epsilon$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
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.
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