Answer to some question by Fujita on VHS.
 
Fabrizio Catanese (University of Bayreuth)



In the first part of my talk, which shall describe joint work with Michael Dettweiler, I shall provide details for a theorem announced by Fujita 34 years ago. THM. If one has a family fibred over a curve B, then the direct image V of the relative dualizing sheaf is the direct sum of an ample vector bundle A and of a unitary flat vector bundle W. I shall then describe a counterexample to a question raised by Fujita 31 years ago. Question: Is V semi ample ? In view of the previous theorem, the question is whether the flat bundle W corresponds to a finite representation of the fundamental group. While the answer is yes (Deligne) if we have a summand of W of rank one, or if the base has genus at most one, we show examples, based on hypergeometric integrals, where we get a representation of infinite order.



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