HYPERBOLICITY AND SPECIALNESS OF SYMMETRIC POWERS

Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n 2 by Arapura and Archava, we study their analytic and algebraic hyperbolicity properties. First, we show that some (or equivalently any) Xmis rationally connected (resp. special) if and only if so is X (except when the core of X is a curve in the case of specialness). Then we construct dense entire curves in (sufficiently high) 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 hypersurfaces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension 5 n - 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, we use a metric approach to study symmetric powers of ball quotients.