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

Abstract

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of K3 [n] -type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed n, we show that there are only finitely many polarization types of holomorphic symplectic variety of K3 [n] -type that do not contain such a uniruled divisor. As an application we provide a generalization of a result due to Beauville-Voisin on the Chow group of 0-cycles on such varieties.

Deformations of rational curves on primitive symplectic varieties and applications

“…The deformation theory of rational curves is one of the key tools to deduce general results from special cases. The main technical contribution of the present note extends to the most general singular, not necessarily projective setting a result established in [AV15,CMP19] in the smooth case.…”

“…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).…”

“…To see this we argue as in[CMP19, Corollary 4.7]. Since Y is projective and has Picard rank at least two, its Picard lattice is indefinite and contains primitive elements of arbitrarily positive BBF square.…”

Rational curves on O'Grady's tenfolds

“…As in the case of projective K 3 surfaces, the existence of rational curves ruling a divisor has a strong consequence on the 0-Chow group of the manifold. Namely, by the results of Charles, Mongardi and Pacienza in [CMP19] we deduce the following.…”

“…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.…”