Fano fourfolds of K3 type

Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility of our methods, we construct in both situations many Calabi--Yau manifolds of dimension three and four, as well as a few Fano varieties, including some new Fano fourfolds.

We prove that certain Fano fourfolds of K3 type constructed by Fatighenti–Mongardi have a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.

In this paper, we study smooth complex projective 4-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperkahler manifolds in dimension ≥ 4 are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperkahler fourfold which is deformation equivalent to the Hilbert scheme S[2] of two points of a projective K3 surface S.

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper, we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, and hence has codimension two, all the anticanonical divisors are singular.

Let A A be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over A A form an abelian category HVec A \textrm {HVec}_A ; the Fourier-Mukai transform yields an equivalence of HVec A \textrm {HVec}_A with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of HVec A \textrm {HVec}_A with the category of finite-dimensional representations of a commutative affine group scheme (the “affine fundamental group” of A A ). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.

We are investigating homogeneous p-adic vector bundles on abelian varieties that are analytic tori. We show that for each homogeneous vector bundle on such a variety there exists an integer N > 0, such that the pullback of this vector bundle via the N-multiplication is attached to an integral representation of the topological fundamental group.

In this paper we relate two constructions of representations of semisimple Lie groups constructions that appear quite different at first glance. Homogeneous vector bundles are one source of representations: if a real semisimple Lie group Go acts on a vector bundle E --* M over a quotient space M=Go/Ho, then Go acts also on the space of sections C~(M, E), and on any subspace VcC~(M, E) defined by a Go-invariant system of differential equations. Ordinary induction, so-called cohomological induction and the construction of representations by "quantizat ion" all fit into the framework of homogeneous vector bundles. For any complex semisimple Lie algebra g, there is an equivalence of categories, due to Beilinson-Bernstein [1], between ~-modules on the one hand, and sheaves of ~-modules over the flag variety X of g on the other. In the context of real semisimple Lie groups this equivalence of categories associates infinitesimal representations to orbits in the flag variety of the complexified Lie algebra orbits not of the group Go itself, but of the complexification of maximal compact subgroup Ko c Go. We shall show that these Beilinson-Bernstein modules are naturally dual to modules attached to certain homogeneous vector bundles. In the special case of a compact group, both the Beilinson-Bernstein construction and the construction via homogeneous vector bundles reduce to the Borel-Weil-Bott theorem; our duality theorem is then a particular instance of Serre duality.

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset ∑ ⊂ ∏ of simple roots of G, and let E ϕ be a homogeneous vector bundle over the flag manifold M = G/P corresponding to a linear representationϕ of P. Using Bott’s theorem, we obtain sufficient conditions on ϕ in terms of the combinatorial structure of ∑ ⊂ ∏ for cohomology groups H q(M,ε ϕ) to be zero, where ε ϕ is the sheaf of holomorphic sections of E ϕ . In particular, we define two numbers d(P), ℓ(P) ∈ ℕ such that for any ϕ obtained by natural operations from a representation \(\tilde \varphi \) of dimension less than d(P) one has H q(M,ε ϕ) = 0 for 0 < q < ℓ(P). Applying this result to H 1(M,εϕϕ), we see that the vector bundle E ϕ, is rigid.Key wordsHomogeneous vector bundlecomplex flag manifoldcohomology of the sheaf of sectionsBott’s theoremroot systemDynkin diagramrigid vector bundle

Holomorphic vector bundles on homogeneous spaces

A classification of homogeneous operators in the Cowen–Douglas class

Let $G$ be a complex reductive linear algebraic group and $G_0 \subseteq G$ a real form. Suppose $P$ is a parabolic subgroup of $G$ and assume that $P$ has a Levi factor $L$ such that $G_0 \cap L = L_0$ is a real form of $L$ . Using the minimal globalization $V_{\min}$ of a finite length admissible representation for $L_0$ , one can define a homogeneous analytic vector bundle on the $G_0$ orbit $S$ of $P$ in the generalized flag manifold $Y = G/P$ . Let $A(P, V_{\min})$ denote the corresponding sheaf of polarized sections. In this article we analyze the $G_0$ representations obtained on the compactly supported sheaf cohomology groups $H^p_c(S,A(P, V_{\min}))$ .

In the present article, we combine some techniques in harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ( $$ \mathcal{D} $$ -modules), and reformulate the composition series and branching problems for objects in the Bernstein–Gelfand–Gelfand parabolic category $$ {\mathcal{O}}^{\mathfrak{p}} $$ geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold G / P for G = SL(n + 2, ℂ) and P the Heisenberg parabolic subgroup; and algebraic analysis gives a complete classification of $$ {\mathfrak{g}}_r^{\prime } $$ -singular vectors for all $$ {\mathfrak{g}}_r^{\prime }=\mathfrak{s}\mathfrak{l}\left(n-r+2,\mathbb{C}\right)\kern1em \subset \mathfrak{g}=\mathfrak{s}\mathfrak{l}\left(n+2,\mathbb{C}\right),n-r>21 $$ . A consequence of our results is that we classify SL(n − r + 2, ℂ) -covariant differential operators acting on homogeneous line bundles over the complexification of the odd-dimensional CR-sphere S 2n+1 and valued in homogeneous vector bundles over the complexification of the CR-subspheres S 2(n-r)+1.

Dolbeault Cohomological Realization of Zuckerman Modules Associated with Finite Rank Representations

On the cohomology of certain homogeneous vector bundles of [formula omitted] in characteristic zero

The goal of this paper is to give a cohomological characterization of F n , t {F_{n,t}} , where F n , t := Ker ⁡ ( ( n + t ; n ) O P n ( − t ) → O P n ) {F_{n,t}}: = \operatorname {Ker} ((n + t;n){\mathcal {O}_{{{\mathbf {P}}^n}}}( - t) \to {\mathcal {O}_{{{\mathbf {P}}^n}}}) .