Program - December 20, 2016 (Aula C, second floor)
You can download the poster and the program.
10:30-11:20
Carmelo di Natale
Hodge Theory and Deformations of Affine Cones of Subcanonical Projective Varieties
Abstract:
We investigate the relation between the Hodge theory of a smooth subcanonical n-dimensional projective variety X and the deformation theory of the affine cone A_X over X. We start by identifying H^{n-1,1}_{prim}(X) as a distinguished graded component of the module of first order deformations of A_X, and later on we show how to identify the whole primitive cohomology of X as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over X. In the particular case of a projective smooth hypersurface X we recover Griffiths isomorphism between the primitive cohomology of X and certain distinguished graded components of the Milnor algebra of a polynomial defining X. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds. This is a joint work with Enrico Fatighenti and Domenico Fiorenza.
11:30-12:20
Olaf Schnürer
DG enhanced six functor formalism and applications
Abstract:
We explain that Grothendieck-Verdier-Spaltensteinss six functor formalism for derived categories of sheaves on topological spaces can be lifted to dg enhancements, as soon as we work with ringed spaces over a base field. Some applications of this dg enhanced formalism are given.
14:00-14:50
Riccardo Moschetti
Twisted derived categories in the case of cubic fourfolds containing a plane
Abstract:
Kuznetsov proved that a component of the derived category of a generic cubic fourfold containing a plane is equivalent to the twisted derived category of a certain K3 surface. The original purpose of his work was to formulate a conjecture concerning the rationality of cubic fourfolds; then it has become clear that twisted derived categories play a central role in the study of the moduli space of cubic fourfolds. The whole theory is very rich from both the geometrical point of view and the side of derived categories and it allows several new directions which are worth further analysis.
I will talk about some applications of the study of the derived category of a non generic cubic fourfold containing a plane.
15:00-15:50
Diletta Martinelli
Rational curves on fibered Calabi-Yau manifolds
Abstract:
Calabi-Yau manifolds are of interest in both algebraic geometry and theoretical physics. In particular the problem of determining whether Calabi-Yau manifolds do contain rational curves has big relevance in string theory. Moreover, a folklore conjecture
in algebraic geometry predicts the existence of rational curves on every Calabi-Yau manifolds. There are several positive answers in dimension three but very little is known in higher dimension. I will talk about a joint work with Simone Diverio and Claudio Fontanari where we prove the existence of rational curves on Calabi-Yau manifolds of any dimension that admit an elliptic fibration. If time permits, I will show how to use this result to produce rational curves on Calabi-Yau manifolds that admit a fibration onto a curve whose fibers are abelian varieties.