Polarized Parallel Transport and Uniruled Divisors on Deformations of Generalized Kummer Varieties

Abstract

Abstract. In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.

Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles

“…These three parts of the proof are correct and will be presented here essentially as in [CP14]. The main difference is that, after the appearance of [OSY18] we realised that the examples we provided in (c) did not (and actually could not, because of [OSY18, Corollary A.3]) cover all the connected components of M. The same type of considerations hold in the generalized Kummer case, treated in [MP17] and amended in [MP].…”

Rational curves on O'Grady's tenfolds

“…The existence of ample uniruled divisors on ihs manifolds of Beauville's deformation type has been studied by Charles, Mongardi and Pacienza [CMP19] in the K 3 [n] case, and by Mongardi and Pacienza [MP17], [MP19] in the K n (A) case. In both cases, the authors have proved that for any polarized ihs manifold out of finitely many connected components of the moduli space of polarized ihs manifolds of the respective Beauville's deformation type, there exists a multiple of the ample divisor that is linearly equivalent to a sum of uniruled divisors.…”

“…We find explicit examples of polarized ihs manifolds (X, h) ∈ M pol OG 10 such that a multiple of the polarization h has a representative that is uniruled, and then conclude that the same holds true for any element in the connected component of M pol OG 10 containing (X, h), thanks to a result of deformation of rational curves ruling a divisor on an ihs manifold. This strategy was already used in [CMP19] and [MP17] for the K 3 [n] and K n (A) case.…”

Deformations of rational curves on primitive symplectic varieties and applications

“…Ample uniruled divisors on irreducible symplectic manifolds of K3 [n] , Kum n or OG10 deformation types are investigated respectively in [CMP19, MP18,Ber21]. The OG6 deformation type is the object of an ongoing project by Bertini, Grossi, and Onorati.…”

“…The proof of the theorem above uses crucially the analogous existence results proved in[CMP19,MP18] together with a rational map constructed in [PR18, Lemma 3.19] from a smooth moduli space of sheaves M u (S, H) (respectively K u (S, H)), where u is primitive, onto M v (S, H) (respectively onto K v (S, H)), see (2.6), and another important result contained in[PR18] (cf. Theorem 1.19 therein).…”