Geometry in İstanbul

Abstract: I will survey some recent results on the classification of singular Fano threefolds, focusing, on the case of Fano threefolds with terminal singularities. Some new results will be presented.

Abstract

Abstract: There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a Kaehler-Einstein metric or not. This is a joint work with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess and Nivedita Viswanathan.

Abstract: (joint work with Kharlamov) We present real enumerative invariants not sensitive to changing the real structure on del Pezzo surfaces of degrees 1,2 and 3. These invariants are obtained by signed summation of Welschinger invariants for a fixed anticanonical degree. The signs come from certain canonical Pin-structures on the real loci of these surfaces.

Abstract: We give a survey on conjectures that relate Hodge structures of Fano varieties and mixed Hodge structures of log Calabi--Yau varieties related to them with some filtrations and Hodge structures constructed from Landau--Ginzburg models that are mirrors to the Fano varieties.

Abstract: A non-degenerate square matrix with positive integer entries defines an invertible polynomial – a polynomial with as many terms as the number of variables. To study the monodromy of a hypersurface defined by an invertible polynomial, we propose to consider its dehomogenization through torus action so that it become Delsarte polynomial. We will be concerned with a generalized version of Dubrovin’s conjecture on the Stokes matrix associated to the Delsarte polynomial. Firstly, we remark the equivalence between oscillating integral for Delsarte polynomial (Landau-Ginzburg model) and quantum cohomology of a weighted projective space P. Secondly,we verify a correspondence between the Stokes matrix of oscillating integral and the Gram matrix of the full exceptional collection on P. By means of the Laplace transform, the oscillating integral turns out a period integral of the Delsarte polynomial hypersurface that can be understood as Pochhammer hypergeometric function. We apply the theory of hypergeometric functions to calculate the monodromy of the period integrals then the Stokes matrix of their inverse Laplace transform.

Abstract: In a previous work with M.Bhupal, we proved that each minimal symplectic filling of any oriented lens space, viewed as the singularity link of some cyclic quotient singularity and equipped with its canonical contact structure, can be obtained from the minimal resolution of the singularity by a sequence of symplectic rational blowdowns along linear plumbing graphs. Here we give a dramatically simpler visual presentation of our rational blowdown algorithm in terms of the triangulations of a convex polygon. As a consequence, we are able to organize the symplectic deformation equivalence classes of all minimal symplectic fillings of any given lens space equipped with its canonical contact structure, as a graded, directed, rooted, and connected graph, where the root is the minimal resolution of the corresponding cyclic quotient singularity and each directed edge is a symplectic rational blowdown along an explicit linear plumbing graph. Moreover, we provide an upper bound for the rational blowdown depth of each minimal symplectic filling. (This is a joint work with M. Bhupal)

Abstract: We present a new version of tropical-to-complex correspondence theorem. Given a regular tropical curve C in R^N passing through a collection of points P, we realize each component of C-P by a compact holomorphic curve with smooth boundary on the corresponding fiber tori in C*^N. In other words, the holomorphic curve is cut to individual pieces whose amoebas have pinching points at P. The regularity condition holds automatically if N=2, and the curve is immersed. This version of the tropical correspondence theorem is motivated by a question of Cheuk Yu Mak and Ilia Zharkov.