Number Theory Day
Friday, 1.6.18, 00:00-01:00, Basel
TBA
Friday, 8.6.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Quantised dihedral angles and quantum dilogarithms
Monday, 11.6.18, 13:15-14:15, Hörsaal FRIAS, Albertstr. 19
I will describe a relation between quantum dilogarithms and\n3-dimensional hyperbolic geometry obtained by quantising the dihedral\nangles of an ideal hyperbolic tetrahedron with respect to the\nNeumann—Zagier symplectic structure. In this way, one constructs a\n(metaplectic) quantum operator \(Q\) realising the 3-3 Pachner move for\n4-dimensional triangulations. This realisation admits a natural\ngeneralisation to any self-dual locally compact abelian group, together\nwith a fixed gaussian exponential. The 5-term operator identity,\nsatisfied by a quantum dilogarithm over such a group, is equivalent to\nan integral identity involving the operator kernel of \(Q\).
TBA
Friday, 15.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Moduli spaces of sheaves on K3 surfaces and irreducible symplectic varieties
Friday, 22.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Irreducible symplectic manifolds are one of the three building blocks of compact K\b"ahler manifolds with numerically trivial canonicl bundle (together with abelian varieties and Calabi-Yau manifolds), thanks to the Beauville-Bogomolov decomposition theorem. A recent result of A. H\b"oring and T. Peternell has completed the extension of this decomposition theorem to singular projective varieties: irreducible symplectic varieties are the singular analogue of irreducible symplectic manifolds, and they are one of the building blocks of normal, projective varieties having canonical singularities and numerically trivial canonical bundle. In a recent joint work with A. Rapagnetta we prove that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to a generic polarization) are irreducible symplectic varieties, with the only excption of those isomorphic to symmetric products of K3 surfaces, and compute their Beauville form and Fujiki constant. Similar results are shown to hold for the Albanese fiber of moduli spaces of sheaves over Abelian surfaces.
A variety that cannot be dominated by one that lifts
Wednesday, 27.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
In the sixties, Serre constructed a smooth projective variety in\ncharacteristic p that cannot be lifted to characteristic 0. If a variety\ndoesn't lift, a natural question is whether some variety related to it does\nlift. We construct an example of a smooth projective variety that cannot be\nrationally dominated by a smooth projective variety that lifts.\n