Hyperbolicity and Specialness of Symmetric Powers

TL;DR

This paper investigates the hyperbolic and special properties of symmetric powers of complex projective varieties, establishing criteria for hyperbolicity, specialness, and general type, with applications to hypersurfaces and ball quotients.

Contribution

It provides new criteria linking the hyperbolic properties of symmetric powers to the geometry of the original variety, including dense entire curves and hyperbolicity results.

Findings

Symmetric power is special iff the original variety is special, except when the core is a curve.

Constructs dense entire curves in symmetric powers of K3 surfaces and products of curves.

Establishes criteria for pseudo-hyperbolicity and Kobayashi hyperbolicity of symmetric powers.

Abstract

Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n $\ge$ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension $\le$ n -- 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.