On the isotriviality of families of projective manifolds over curves

TL;DR

This paper investigates the conditions under which families of projective manifolds over curves are isotrivial, establishing lower bounds on the number of singular fibers based on the genus and properties of the fibers.

Contribution

It proves that non-isotrivial families with certain fiber properties must have a minimum number of singular fibers, extending previous results and providing explicit bounds on related sheaf degrees.

Findings

Non-isotrivial families over genus 0 curves have at least 3 singular fibers.

For fibers of general type or with semi-ample canonical divisor, the number of singular fibers meets expected lower bounds.

Explicit bounds are given for the degree of the direct image of powers of the dualizing sheaf.

Abstract

Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-ample canonical divisor. The corresponding result has been obtained by Migliorini and Kovacs, for families of surfaces of general type and for families of canonically polarized manifolds, and by Oguiso-Viehweg for families of elliptic surfaces. As a byproduct we obtain explicit bounds for the degree of the direct image of powers of the dualizing sheaf, generalizing those obtained by Bedulev-Viehweg for families of surfaces of general type.