The Pseudo-Index of Horospherical Fano Varieties

Abstract

We prove a conjecture of L. Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties. Mathematics Subject Classification. 14J45 14L30 52B20Keywords. Horospherical varieties, Picard number, pseudo-index, Fano varieties.Let X be a normal, complex, projective algebraic variety of dimension d. Assume that X is Fano, namely the anticanonical divisor −K X is Cartier and ample. The pseudo-index ι X is the positive integer defined byThe aim of this paper is to prove the following result. Theorem 1. Let X be a Q-factorial horospherical Fano variety of dimension d, Picard number ρ X and pseudo-index ι X . Then Moreover, equality holds if and only if X is isomorphic toThis inequality has been conjectured by L. Bonavero, C. Casagrande, O. Debarre and S. Druel for all smooth Fano varieties [1]. Moreover, they have proved Theorem 1 in the case of Fano varieties of dimension 3 and 4, all flag varieties, and also for some particular toric varieties. More generally, this result has been also proved by C. Casagrande for all Q-factorial toric Fano varieties [5]. Theorem 1 generalizes these results to the family of 1

The Mukai conjecture for Fano quiver moduli

“…In [2], the moduli spaces M st d (Q) of stable quiver representations [9] which are Fano varieties are identified, forming a rather special (for example, always being rational [11] and of algebraic cohomology [6]), but arbitrarily high-dimensional, class of Fano varieties. In this note, we verify the Mukai conjecture for Fano quiver moduli spaces (in the spirit of such a verification for toric varieties [1], horospherical varieties [8] and symmetric varieties [3]), under a reasonable genericity assumption on the dimension vector d, to be discussed below.…”

On the classification of normal G-varieties with spherical orbits

“…The calculation is expressed in terms of the ramification indices of the quotient map by F . In particular, this formula is a first step toward the classification of Fano varieties in this setting as it was studied in some particular cases in [Bat94,Pas08,Pas10,Sus14,GH17]. We believe that the combinatorial description developed in this article can be useful for describing the deformation theory of spherical varieties as in [AB05,AB06].…”

Real Structures on Horospherical Varieties

“…A presentation of the theory of horospherical varieties, and their relation to Fano varieties, can be found in [Pas06,Pas08]. They form a fertile ground to tackle many problems in algebraic geometry: the Mukai conjecture [Pas10], the (log) minimal model program [Pas15a,Pas18], the stringy invariants [BM13,LPR], the quantum cohomology [GPPS]... However, as far as we know, the theory of equivariant real structures on horospherical varieties has never been systematically studied.…”