Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications

Abstract

We prove a Second Main Theorem type inequality for any log-smooth projective pair (X, D) such that X \ D supports a complex polarized variation of Hodge structures. This can be viewed as a Nevanlinna theoretic analogue of the Arakelov inequalities for variations of Hodge structures due to Deligne, Peters and Jost-Zuo. As an application, we obtain in this context a criterion of hyperbolicity that we use to derive a vast generalization of a well-known hyperbolicity result of Nadel.The first ingredient of our proof is a Second Main Theorem type inequality for any log-smooth projective pair (X, D) such that X \ D supports a metric whose holomorphic sectional curvature is bounded from above by a negative constant. The second ingredient of our proof is an explicit bound on the holomorphic sectional curvature of the Griffiths-Schmid metric constructed from a variation of Hodge structures.As a byproduct of our approach, we also establish a Second Main Theorem type inequality for pairs (X, D) such that X \ D is hyperbolically embedded in X.

Second Main Theorem on the Moduli Spaces of Polarized Varieties

“…Our setting and approach are different however: they consider base spaces which carry polarized VHS while we consider base spaces which carry families of polarized varieties; they use the Griffiths-Schmid metric and the logarithmic derivative lemma while we use the Finsler (pseudo)metric and the tautological inequality. The common feature is that we both use certain ample line bundle A on X and obtain an inequality of the Nevanlinna characteristic function of A ( it is called the Arakelov-Nevanlinna inequality in [BB20], see Theorem 1.1 loc. cit.).…”

“…cit.). In [BB20] they used the Griffiths line bundle associated to the VHS, and we use the Viehweg line bundle A associated to the family.…”