Algebraic Geometry Seminar Spring 2014
The seminar meets on Fridays at 2:25 pm in Van Vleck B231.
The schedule for the previous semester is here.
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Spring 2014 Schedule
|January 31||Marci Hablicsek (local)||Twisted derived intersections and twisted de Rham complexes|
|February 21||Rares Rasdeaconu (Vanderbilt)||Counting real rational curves on K3 surfaces||Maxim|
|April 4||Kevin Tucker (University of Illinois at Chicago)||Comparing multiplier ideals to test ideals on numerically QQ-Gorenstein varieties||Daniel|
|April 18||Manuel Gonzalez Villa (CIMAT, Mexico)||Recursions for motivic iterated vanishing cycles for quasi-ordinary surface singularities||Maxim|
|April 25||Charles Doran (University of Alberta)||From Elliptic Surfaces to Calabi-Yau Moduli via Euler Transform||Song|
Given a smooth variety and two smooth closed subvarieties, derived algebraic geometry assigns to this data a differential graded scheme, the derived intersection. Equipping the ambient space with an Azumaya algebra, we obtain the notion of twisted derived intersections. In order to compare the twisted and the "untwisted" derived intersections, we assume that the Azumaya algebra is split along the two subvarieties. For such twisted intersection problem, we associate a natural line bundle on the derived intersection, which measures the difference between the two derived intersections. We give a criterion for the triviality of this line bundle. As an application, we prove a special case of the Barannikov-Kontsevich theorem, and we give a decomposition theorem for the hypercohomology spaces of the twisted de Rham complexes. The work is joint with Dima Arinkin and Andrei Caldararu.
Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. I will discuss the case of counting real rational curves on K3 surfaces equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface, the real version of a result due to Gottsche.
The talk is based on a joint work with V. Kharlamov.
In this talk, I will focus on the connection between two important measures of singularities: multiplier ideals in characteristic zero and test ideals in positive characteristic. While their relationship is well understood in many cases (e.g. hypersurface or finite quotient singularities), it remains conjectural for non-QQ-Gorenstein varieties (such as the cone over the Segre embedding of PP^1 x PP^2 in PP^5). I will discuss positive recent progress on this conjecture for so-called numerically QQ-Gorenstein varieties (which include all normal surface singularities). This is joint work with T. de Fernex, R. Do Campo, and S. Takagi.
Manuel Gonzalez Villa
Kennedy and McEwan have proposed a geometrical scheme, inspired in the seminal work of Iomdin and Steenbrink, to investigate the Milnor fibration of quasi-ordinary surface singularities and the horizontal and vertical fibrations of their transversal sections. Their formulas for the (A´Campo) monodromy zeta function show a recursive relation between the invariant of the singularity and those of other two related singularities (a truncation and a derived object).
In our current joint project we analyze this framework with the help of the motivic invariants of Denef and Loeser. In our talk we report on our progress on recursive formulas for the motivic iterated vanishing cycles, introduced by Guibert, Loeser and Merle, in the case of quasi-ordinary surface singularities.
The talk is based on a joint work with Mirel Caibar, Gary Kennedy and Lee McEwan.
Starting from extremal elliptic surfaces, we construct a large class of one-parameter families of K3-surface fibered Calabi-Yau threefolds together with an explicit description of their periods. By a quadratic twist we construct convenient models for moduli spaces of lattice polarized K3 surfaces of high Picard-rank such that their multi-parameter K3 periods can be computed explicitly. Restricting to convenient sub-loci and carrying out another quadratic twist one obtains the desired one-parameter families of Calabi-Yau threefolds. The period computation makes essential use of a generalization of the classical Euler transform for the hypergeometric function. Time permitting, I will explain how our method also allows for the construction of more general Calabi-Yau threefolds with rational deformation spaces with more than 3 points removed. In some of those cases, we find interesting connections of the holomorphic period with the generating functions for the Apery numbers. This is joint work with Andreas Malmendier.