Lunt Hall, 105
10.00am - 11.00am
| Sébastien Boucksom (Jussieu)|
Degenerations of Calabi-Yau manifolds and non-Archimedean geometry - Abstract
I will present joint work in progress with Mattias Jonsson. Kontsevich and Soibelman have given a conjectural description of the Gromov-Hausdorff limit of a maximally degenerate family of polarized Calabi-Yau manifolds in terms of the Berkovich space attached to the degeneration. Motivated by this, Mustata, Nicaise and Xu recently studied the essential skeleton of this Berkovich space, which is a natural realization of the dual complex of a minimal model of the degeneration. We show that the volume form induced by a holomorphic form of top degree on a fiber converges, in some appropriate sense, to an explicit Lebesgue type measure on the essential skeleton.
11.30am - 12.30pm
| Andreas Höring (Université de Nice)|
Rational curves on compact Kähler manifolds - Abstract
Rational curves play a crucial role in the classification of projective manifolds via the minimal model program (MMP). The construction of these curves on projective manifolds or low-dimensional Kähler manifolds is based on deformation theory of curves and, in the projective case, reduction to positive characteristic. In this talk I will present a new strategy to construct rational curves via a "subadjunction formula for adjoint cohomology classes". This is joint work with Junyan Cao.
2.30pm - 3.30pm
| Jian Song (Rutgers University)|
Analytic base point free theorem - Abstract
The abundance conjecture predicts that if the canonical bundle of a projective manifold is nef, then it is semi-ample. A special case is proved by Kawamata for big and nef canonical bundles. We give an analytic proof of Kawamata's theorem using the Ricci flow, L2 theory and degeneration of Riemannian manifolds. We further construct unique Kähler-Einstein metrics with a global Riemannian structure on canonical models. We will also discuss a more general analytic base point free theorem.
4.00pm - 5.00pm
| Tristan Collins (Harvard University)|
Restricted volumes for big (1,1) classes on Kähler manifolds - Abstract
In this talk I will discuss work in progress with Valentino Tosatti. I will introduce a transcendental definition of the restricted volume of a big (1,1) class, and discuss the conjectural relation between the retricted volume and the non-Kähler locus.
Questions? e-mail to: emphasisGA@math.northwestern.edu