We discuss recent progress on Shokurov’s ACC conjecture for mlds in dimension 3. Partially joint works with Jingjun Han, Yujie Luo, and Liudan Xiao.
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.
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.
I will survive some recent results on the study of the birational geometry of foliations over complex projective varieties, focusing, in particular, on the case of algebraically integrable foliations.
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.
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.
We study a singularity of a pair, consisting of a smooth variety and a multi-ideal, defined over a field of positive characteristic. Singularities of a variety defined over a field of characteristic 0 are extensively studied by making use of many good properties which are not available for the positive characteristic case. So we cannot expect the same results in positive characteristic by the same method as in characterisitc 0. In my talk I will propose another way to prove the results which are known in characteristic 0. The way is to construct a bridge between positive characteristic and characteristic 0 by means of inversion of modulo p-reduction and discussion of arc spaces. I will introduce our expected conjecture and its consequences. Then, I will show my recent results towards the conjecture. Actually, for every 'multi-ideal' on a smooth variety over the base field of positive characteristic, there exists its lifting 'multi-fractional ideal' on a smooth variety over characteristic 0 with the same mld (minimal log discrepancy). By this bridge, we can induce some results on mld for positive characteristic from those for characteristic 0.
I will report on a joint work (in progress) with Hamid Abban, Erroxe Etxabarri-Alberdi, Dongchen Jiao, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Elena Denisova, and Theo Papazachariou about the K-moduli of smooth Fano threefolds in deformation families 2-22, 2-24, 2-25, 3-12, 3-13, 4-13 (see this website for the description of these families). These are all 1-dimensional families of smooth Fano threefolds for which K-moduli exist. We know all smooth K-polystable Fano threefolds in these families. I will explain how to find their K-polystable singular limits.
