Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi’s conjecture for del Pezzo surfaces for the existence of Kahler–Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results This is a joint work with Yanir Rubinstein.
For G a finite subgroup of SL(n,CC) (esp. for n=2 or n=3), one seeks a crepant resolution Y ⟶ X = CC^n/G of the quotient space CC^n/G. The McKay correspondence predicts close relations between the equivariant geometry of G action on CC^n and the geometry of the resolution Y.
In practical cases, especially for Abelian groups A, this question leads to monomial calculations and a wide variety of combinatorial problems, some of which can be expressed pictorially in an entertaining way. The words in the title indicate some of the topics that I intend to cover by drawing pictures and waving my arms:
The first half of this talk will be a survey of the theory of stable pairs, and its relationship to Gromov-Witten theory via the MNOP conjecture. Secondly I will outline an application -- a proof of the "KKV formula", expressing the "reduced" Gromov-Witten invariants of K3 surfaces in all classes and genera in terms of modular forms. This is joint work with Rahul Pandharipande.
Spherical twists are certain autoequivalences of the derived category D(X) of an algebraic variety X. Each such twist is defined by a functor D(Z) ⟶ D(X), where Z is some other variety. In geometrical case, this functor is defined by a subvariety of X flatly fibered over Z.
In this talk I will first give a short introduction to these twists, recalling ther definition and construction. Then I will explain sufficient criteria for several such twists to define a categorical braid group action on D(X). I will give these first in general form, which applies to the twists defined by any abstract functors. Then I will give a geometrical interpretation for the case when the twists are defined by fibered subvarieties of X as above. This is joint work with Rina Anno (UPitt).
After introducing the basic notions of the theory of Higgs bundles and the Hitchin fibration we will move on to study some concrete geometric realizations of their moduli spaces. We will see that elliptic curves (and in particular those with complex multiplication) give rise to algebraic surfaces whose Hilbert schemes are moduli spaces of (parabolic) Higgs bundles. The categorical McKay correspondence allows us to establish the autoduality conjecture for the derived categories of the constructed examples.
Floppable rational curves in 3-folds exhibit a range of interesting behaviours, according to their normal bundle, ADE type, and other discrete invariants. I will discuss joint work with Michael Wemyss (arXiv:1309.0698) which uses noncommutative deformation theory to investigate the associated geometry and homological algebra. In particular, we give a new description of the symmetries of the derived category which naturally correspond to such curves, and new calculations of their deformations.