buondì,
vi segnalo il seminario di geometria di domani. a presto, Bruno
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Tuesday, November 11, 2014
Aula seminari, ore 14:00
Title: On the Dolbeault cohomological dimension of the moduli space of Riemann surfaces
Speaker: Gabriele Mondello
Abstract:
The moduli space M_g of Riemann surfaces of genus g is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension (i.e. the highest k such that H^{0,k}(M_g,E) does not vanish for some holomorphic vector bundle E on M_g). The conjecturally optimal bound is g-2, which is verified for g=2,3,4,5. In this talk, I will show that such dimension is at most 2g-2. The key point is to show that the Dolbeault cohomological dimension of each stratum of the Hodge bundle is at most g (still non-optimal bound). In order to do that, I produce an exhaustion function, whose complex Hessian has controlled index: in the construction of such a function basic geometric properties of translation surfaces come into play.