We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and relate edges to jets. Using the polytopes, we also exhibit vector bundles that are ample but not globally generated, and vector bundles that are ample and globally generated but not very ample.

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ABSTRACT. Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y , all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold.The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Géométriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bogomolov-Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana's theory is included for the reader's convenience.

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y • , and suppose the family is nona simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D. Viehweg and Zuo have shown that for some m > 0, the m th symmetric power of this sheaf admits many sections. More precisely, the m th symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this "Viehweg-Zuo sheaf" comes from the coarse moduli space associated to the given family, at least generically.As an immediate corollary, if Y • is a surface, we see that the non-isotriviality assumption implies that Y • cannot be special in the sense of Campana. CONTENTS 1. Introduction and statement of main result 1 2. Review of Viehweg-Zuo's proof of Theorem 1.1 3 3. Proof of Theorem 1.4 5 4. Application of Theorem 1.4 to families over special surfaces 8 References 9

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