### $\mathbb{Q}$-Fano threefolds of large Fano index. I.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

In this paper we classify rank two Fano bundles $\mathcal{E}$ on Fano manifolds satisfying ${H}^{2}(X,\mathbb{Z})\cong {H}^{4}(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(\mathcal{E}\right)$, that allows us to obtain the cohomological invariants of $X$ and $\mathcal{E}$. 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.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in ${c}_{1}\left(\phantom{\rule{-0.166667em}{0ex}}L\right)$ and its automorphism group $\mathrm{Aut}(V\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}},\phantom{\rule{-0.166667em}{0ex}}L)$ 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 $\mathrm{Aut}(V,L)$ is not discrete.

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.

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 ${\mathbb{P}}^{3}$ 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...

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 ${H}_{4}(X,\mathbf{C})$. Let $f:Y\to X$ 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 $X$.

O’Grady showed that certain special sextics in ${\mathbb{P}}^{5}$ 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.