Évènements

We introduce the canonical model singularities (CMS) criterion for bigness of the cotangent bundle for surfaces. The CMS-criterion for bigness involves invariants for canonical singularities that we describe and we give formulas for A_n singularities. The CMS-criterion leads to conjectures and some answers about the geography and the possible ratios $c_1^2/c_2$ of surfaces with big cotangent bundle. Two cases are naturally separated: regular and irregular surfaces. For regular surfaces we apply the CMS-criterion to show the existence of deformations of hypersurfaces in $\mathbb{P}^3$ with big cotangent bundle for degree $d\ge 8$ and give an example of the regular surface with big  cotangent bundle with ratio close to 1/5. For irregular surfaces we show that there are examples with ratio as close to 1/5 as possible. If time permits, we talk about ratios  below 1/5.

I will introduce a method to compute efficiently and with arbitrary precision a basis of harmonic functions with singularities in a Riemann surface of any genus. This basis is a powerful tool to approximate harmonic functions with spectral speed, using the method of particular solutions: we give a full characterization of the speed of convergence in any genus, depending on the singularities of the harmonic extension.