Title: Finite order birational automorphisms of Fano hypersurfaces.
Abstract: The birational automorphism group is a natural birational invariant associated to an algebraic variety. In this talk, we study the specialization homomorphism for the birational automorphism group. As an application, building on work of Kollár and work of Chen–Stapleton, we show that a very general n-dimensional complex hypersurface X of degree = 5⌈(n+3)/6⌉ has no finite order birational automorphisms. This work is joint with N. Chen and D. Stapleton.
Title: Higher index Fano varieties with finitely many birational automorphisms.
Abstract: Given a projective variety X, its birational automorphism group -- denoted Bir(X) -- is an important birational invariant. The easiest way to study Bir(X) is to study how it acts on the space of global differential i-forms. E.g. when X is general type, the action on the pluricanonical forms show that Bir(X) is finite. So studying Bir(X) is most difficult and interesting when X is Fano, in which case the groups can be very large: e.g. if X is projective space or a cubic hypersurface. In the case X is a smooth Fano hypersurface of index 1 over C, the Noether-Fano method was used by many authors (Fano-Segre-Iskovskikh-Manin-Pukhlikov-Cort-Cheltsov-De Fernex-Ein-Mustata-Zhuang) to prove that X is birationally super-rigid, in particular Bir(X) is finite. Pukhlikov proved a similar result for index 2 hypersurfaces. To our knowledge, there are no known results in index 3 and higher. Our main result is to give examples of Fano varieties of arbitrarily large index -- at the price of working in characteristic p and allowing singularities. Our main observation is that the forms that Kollár produced on p-cyclic covers in characteristic p give rise to a natural Bir(X)-equivariant ample line bundle on these varieties.
Title: Higher index Fano varieties with finitely many birational automorphisms.
Abstract: Holomorphic forms are an important birational invariant for studying the geometry of a variety. In characteristic 0, Fano varieties do not have any holomorphic forms. Surprisingly, Kollár showed that in characteristic p>0 certain singular Fano hypersurfaces admit many global (n-1)-forms, and he combined this observation with a specialization method to prove nonrationality of many Fano hypersurfaces. We use similar methods to study their possible rational endomorphisms. This is joint work with David Stapleton.
Title: Rationality of conic bundle threefolds over non-closed fields.
Abstract: To show that a variety is not rational, one must exhibit some birational invariant that is trivial for projective space and nontrivial for the given variety. The intermediate Jacobian is one such obstruction to rationality in dimension three, first introduced over the complex numbers by Clemens–Griffiths to show that cubic threefolds are irrational. Hassett–Tschinkel and Benoist–Wittenberg recently refined this obstruction over non-closed fields using torsors over the intermediate Jacobian. In joint work with L. Ji, S. Sankar, B. Viray, and I. Vogt, we identify these torsors for conic bundle threefolds, and we give applications to rationality over non-closed fields.
Title: Finite quotients of abelian varieties with a Calabi-Yau resolution.
Abstract: Let G be a group acting freely in codimension 1 on an abelian variety A. Terminalizations (and, if any, crepant resolutions) of the singular quotient A/G are K-trivial varieties which, depending on A and G, may have various types of Beauville-Bogomolov decomposition. In particular, they may be symplectic or not, simply-connected or not... Assuming that G acts freely in codimension 2 tremendously reduces the possible outcomes, but this strong assumption is still reasonable enough that there are a few examples illustrating it, namely two smooth Calabi-Yau threefolds arising as crepant resolutions of cyclic quotients of abelian varieties. These examples, which give the full picture in dimension 3, are due to Oguiso.
In this talk, I will present some results toward a higher dimensional counterpart to Oguiso's classification. In particular, using some results in McKay correspondence and elementary group theory, I will explain why there are no simply-connected crepant resolutions for finite quotients of abelian fourfolds by groups acting freely in codimension 2.
Title: Fundamental groups of algebraic singularities.
Abstract: In this talk, we will discuss about the fundamental group of the link of algebraic singularities. We will review some results due to Kollár and Kollár-Kapovich regarding the fundamental groups of algebraic singularities, rational singularities, and Cohen-Macaulay singularities. Then, we will focus on singularities of the minimal model program (MMP). We will present a structural result about the fundamental group of klt type singularities. Finally, we will discuss some recent developments on the fundamental group of log canonical singularities.
Title: Fundamental groups of orbifolds with nef anti-canonical bundle.
Abstract: For compact complex varieties, a natural way to understand their topologies is to compute their fundamental groups. For smooth Fano varieties, Kobayashi proved in 1960s that their fundamental groups are trivial. Recently Lukas Braun generalised this result to klt case by showing that klt Fano varieties have finite fundamental groups. For a smooth variety X with nef anti-canonical bundle, Demaily-Peternell-Schneider constructed a sequence of Kähler metrics on X whose negative part tend to 0 by using Aubin-Yau theorem. With a simple volume estimate, they proved that π_{1}(X) has subexponential growth. Later Mihai Păun adapted DPS’s proof by applying a geometric Margulis lemma of Cheeger-Colding to show that π_{1}(X) is virtually nilpotent. In this talk, we show how to adapt Păun's proof to orbifold case to prove that the fundamental groups of orbifolds with nef anti-canonical bundle are virtually nilpotent. This work is part of my PhD thesis under the direction of Benoît Claudon and Andreas Höring.
Abstract: Kawamata proved boundedness for Fano 3-folds in 1992: there are only finitely
many deformation families of (Q-factorial terminal Picard rank 1) Fano 3-folds.
The proof puts bounds on the invariants that appear in the (anticanonical) Hilbert series,
showing that there are only finitely many such series, and then finiteness of the
Hilbert scheme concludes.
The Graded Ring Database (GRDB) makes those bounds explicit by listing
all the series that satisfy Kawamata’s bounds. Of course for any given series
in the database there is no guarantee that any Fano actually has that as its
Hilbert series, and if there is one then there is no formula for how many deformation
families might realise it. So this is not a method of counting the number of
deformation families.
Nevertheless, it does provide an extremely useful map of all Fano 3-folds
(far more generally than the Mori-theoretic hypotheses above), and, more than that,
it provides a strategy for proving their existence in many cases, as I will explain.
Title: Regularizations of positive entropy pseudo-automorphisms.
Abstract:Given a birational automorphism of an algebraic variety it is natural to ask whether there exists a projective birational model of the variety on which the induced automorphism is regular. The question when one can construct a regularization model of an automorphism is quite hard and the answer is known only for curves and surfaces. In my talk I am going to recall brilliant theorems which show the relation between the dynamics of birational automorphisms of surfaces and their regularization properties. Then I am going to speak about birational automorphisms of threefolds and the difficulties that arise when we try to find out if they are regularizable.
Title: Log minimal model program for Kahler 3-folds.
Abstract: The Minimal Model Program for Kahler varieties (which are not necessarily projective) aims to mimic the same recipe as the projective case, i.e. given a compact Kahler variety X with mild singularities, after a finitely many K_X-negative elementary contractions, can we find a variety X’ such that X - - -> X’ is bimeromorphic and either K_X’ is nef or there is a Mori fiber space g:X’—> Z? In 2015-16, in a series of papers, Campana, Höring and Peternell established that this is possible when X has dimension 3 and terminal singularities. There are some fundamental differences in the way Kahler MMP works compared to the projective MMP, for example, there is no Base-point free theorem in the Kahler setting (which is fundamental for proving the contraction of a negative extremal ray in the projective case), the Mori cone \overline{NE}(X) may have too few curves, etc. In their papers Campana, Höring and Peternell introduced various new tools and techniques to handle this problems. In my talk I will explain what are those new tools and how to extend their results for DLT pair (X, B), \dim X=3 in full generality, this is a joint work with Christopher Hacon. If time permits, then I will talk about some more recent results on Kahler 4-folds and generalized pairs in the Kahler settings in dimension 3.
Title: Deformations of mildly singular Calabi-Yau varieties.
Abstract: The well-known Bogomolov-Tian-Todorov theorem says that the deformations of Calabi-Yau manifolds are unobstructed. The unobstructedness of deformations continues to hold Calabi-Yau varieties with ordinary nodal singularities (Kawamata, Ran, Tian), but surprisingly the smoothability of such varieties is subject to topological constrains. These obstructions to existence of smoothings are linear in dimension 3 (Friedman), and non-linear in higher dimensions (Rollenske-Thomas).
In this talk, I will give vast generalizations to both the unobstructedness of deformations for mildly singular Calabi-Yau varieties, and to constraints on the existence of smoothings for certain classes of singular Calabi-Yau varieties. Additionally, I will establish the proper context of this type of results: the Hodge theory of degenerations with constrained singularities, specifically higher rational and higher Du Bois singularities.
This is joint work with Robert Friedman.
Abstract: Geometric invariant theory (GIT) is the theory of producing quotients of projective varieties, as heavily used in the construction of moduli spaces. The construction depends on a choice of ample line bundle, and "variation of GIT" explains how the GIT quotient depends on the choice of ample line bundle (one obtains finitely many birational models of the GIT quotient). This is the basic motivation for wall-crossing in algebraic geometry. I will explain a variant of this story, where one instead asks how the GIT quotient depends on the variety itself. Allowing birational models of the variety, the main result will be that one "fully captures" the birational geometry of the GIT quotient, in contrast with variation of GIT. This will be made precise using an analogue of Riemann-Zariski spaces, and is joint work with Rémi Reboulet.
Title: Higher rational and higher Du Bois singularities.
Abstract: For any effective Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many applications in algebraic geometry and commutative algebra. In this talk, I will discuss the construction of a new family of ideal sheaves associated to (X,D), which recovers the usual multiplier ideals. This family of ideals is closely related to, but different from, the theory of Hodge ideals developed by Popa and Mustata. We establish their local and global properties systematically. Among inputs, we prove a vanishing theorem for ''twisted complex Hodge modules'', whose corollary is a Kawamata-Viehweg type log Akizuki-Nakano vanishing theorem; on the other hand, we establish a Shokurov-Koll\'ar connectedness principle type result for zero loci of higher multiplier ideals. If time permits, we will discuss some application to conjectures of Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties and the Riemann-Schottky problem. This is joint work in progress with Christian Schnell.
Abstract: Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin components); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work with Brian Lehmann and Eric Riedl.
Title: Centres of Noncommutative Crepant Resolutions are Kawamata Log Terminal.
Abstract: This is joint work with Takehiko Yasuda. Stafford and Van den Bergh have shown that centres of Noncommutative crepant resolutions have at most rational singularities. We use their techniques to extend their result, showing that they have at worst log terminal singularities. We also show that a homologically homogeneous algebra has a centre and ramification divisor which is Kawamata log terminal.
Title: Studying Dynamics of Higher Dimensional Varieties via the Minimal Model Program.
Abstract: In this talk, we aim to understand the behavior of surjective morphisms on normal projective varieties with mild singularities under composition. This is a difficult problem in general. To get a feeling for the difficulty note that a complete understanding of surjective morphisms of rank two projective bundles over projective spaces would yield a resolution of Hartshorne's conjecture.
To address this problem, we propose an approach that utilizes the minimal model program. We focus on studying some dynamical property D. For instance, the property that f has a dense set of pre-periodic points. To determine if f has the property D, we employ the following strategy. Suppose that X has an extremal contraction p:X->Y.
1. If p:X->Y is a divisorial contraction, construct a morphism g:Y->Y such that f o p = p o g. Then prove that f has the property D if and only if g has the property D.
2. If p:X->Y is a small extremal contraction, we let X^ be the associated flip. The morphism f:X->X gives a dominant rational map f^:X^-->X^. Show that f^ extends to a morphism f^:X->X and show that f has the property D if and only if f^ has the property D.
3. If p:X->Y is a fibering contraction, construct a morphism g:Y->Y such that f o p = p o g. Prove that f has the property D if and only if g has the property D.
We discuss the challenges that arise when employing these strategies and situations where the difficulties can be overcome. For example, we will describe how to employ these strategies to study the pre-periodic points of surjective morphisms of toric varieties as well as applications to the Medvedev-Scanlon conjecture and Dynamical Manin-Mumford conjecture.
Title: Higher du Bois and higher rational singularities for LCI varieties.
Abstract: For complex algebraic varieties, there are two important ``cohomological" classes of singularities, called du Bois singularities and rational singularities. These are related by the work of Kovács: rational implies du Bois. This talk will define some natural generalizations of these classes of singularities, called higher du Bois singularities and higher rational singularities, in the case of a local complete intersection complex variety. I will explain the relation to the minimal exponent, which is a refinement of the log canonical threshold, and also relate these notions to the structure of the local cohomology of a complete intersection subvariety of a smooth variety. This work is joint with Qianyu Chen, Mircea Mustata and Sebastián Olano.