Tag - Algebraic geometry

Jihun Park: Sasaki-Einstein 5-manifolds

7th May, 2022

A numerous number of closed simply connected Sasaki-Einstein manifolds, in particular 5-manifolds, have been found based on the method introduced by Shoshichi Kobayashi, and developed by Charles Boyer, Krzysztof Galicki, and János Kollár. In this talk, we briefly explain how to find closed simply connected 5-manifolds that allow Sasaki-Einstein metrics. By using methods in K-stability, we newly verify that several closed simply connected 5-manifolds allow Sasaki-Einstein structures. We then list closed simply connected 5-manifolds that are known so far to admit Sasaki-Einstein structures.

Miles Reid: Godeaux surfaces in mixed chacteristic

7th May, 2022

It is known that Godeaux's construction of surfaces with χ = 1, K2 = 1 as the quotient of a quintic surface by an action of the cyclic group of order 5 can be modified to work in chacteristic 5, with any of the possible group schemes of order 5. These cases can all be put together into a single deformation family in mixed characteristic, and a similar construction also produces non-singular Calabi-Yau 3-folds with polarisation of degree A3 = 1 and Pic0 containing any of the group schemes ℤ/5 or μ5 or α5. For more information, see this website.

Joaquín Moraga: Coregularity of Fano varieties

6th May, 2022

In this talk, we will introduce the coregularity of Fano varieties.

This invariant measures how large of a dual complex can we find among log Calabi-Yau structures on a Fano variety. The coregularity relates to log canonical thresholds, existence of complements, and the index of log Calabi-Yau pairs. In this talk, we will discuss some recent results about this invariant and other future directions. The results of this talk are joint work with Fernando Figueroa, Stefano Filipazzi, Mirko Mauri, and Junyao Peng.

Angela Gibney: Vector bundles on the moduli space of curves from representations of VOAs

6th May, 2022

Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This gives rise to a functor taking n-tuples of A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is strongly rational (in which case A(V) is finite and semi-simple), much is known about these sheaves, including that they are coherent and satisfy a factorization property. Factorization ultimately allows one to show the sheaves are vector bundles with Chern classes in the tautological ring. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is strongly finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Krashen and Damiolini, extending work with Damiolini and Tarasca.

Yuchen Liu: ACC for local volumes and boundedness of singularities

5th May, 2022

Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, Kähler-Einstein geometry, and K-stability. Recently, Chi Li invented a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. In this talk, we will explore the relation between local volumes and certain boundedness condition of singularities related to the existence of ε-plt blow-ups. As a main result, we show that the set of local volumes of klt singularities is discrete away from zero (resp. satisfies ACC) if the coefficient set is finite (resp. satisfies DCC) and the ambient spaces are analytically bounded.

Yusuke Nakamura: Inversion of adjunction for quotient singularities

5th May, 2022

In this talk, we will discuss the minimal log discrepancies of quotient singularities. I will show that the PIA (precise inversion of adjunction) conjecture holds for quotient singularities. The main tool of this talk involves the theory of the arc space of a quotient singularity established by Denef and Loeser. I will also explain some technical difficulties when dealing with non-linear group actions.