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A classification theorem on Fano bundles

Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)

Annales de l’institut Fourier

In this paper we classify rank two Fano bundles on Fano manifolds satisfying H 2 ( X , ) H 4 ( X , ) . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization ( ) , that allows us to obtain the cohomological invariants of X and . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

An example of an asymptotically Chow unstable manifold with constant scalar curvature

Hajime Ono, Yuji Sano, Naoto Yotsutani (2012)

Annales de l’institut Fourier

Donaldson proved that if a polarized manifold ( V , L ) has constant scalar curvature Kähler metrics in c 1 ( L ) and its automorphism group Aut ( V , L ) is discrete, ( V , L ) is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where Aut ( V , L ) is not discrete.

Are rational curves determined by tangent vectors?

Stefan Kebekus, Sándor J. Kovács (2004)

Annales de l’institut Fourier

Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of X is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.

Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Gianni Ciolli (2004)

Bollettino dell'Unione Matematica Italiana

We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from P 3 or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing...

Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Jun-Muk Hwang (2007)

Annales de l’institut Fourier

Let X be a Fano manifold with b 2 = 1 different from the projective space such that any two surfaces in X have proportional fundamental classes in H 4 ( X , C ) . Let f : Y X be a surjective holomorphic map from a projective variety Y . We show that all deformations of f with Y and X fixed, come from automorphisms of X . The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of X .

Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

O’Grady showed that certain special sextics in 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

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