Hodge Theory and Rationality

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Schedule

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  • Monday October 5th, 2020

    Decomposition of the Diagonal and Specialization – Lecture I

    Dr. Claire Voisin
    IMJ-PRG

    9:00am - 10:00am
    Via Zoom
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    Abstract: The decomposition of the diagonal is a powerful tool to study the stable irrationality of smooth projective varieties. I will discuss the specialization method, which rests on a certain specialization property of the decomposition of the diagonal, in the first lecture. The decomposition of the diagonal controls many other stable birational invariants and I will focus in the second lecture on invariants related to the Abel-Jacobi map and the coniveau.


    Equivariant Birational Types

    Dr. Yuri Tschinkel
    NYU & Simons Foundation

    10:15am - 11:15am
    Via Zoom
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    Abstract: I will explain new obstructions in equivariant birational geometry, introduced and studied in joint work with Kontsevich, Pestun, Kresch, and Hassett.


    Completions of Period Mappings

    Dr. Philip Griffiths
    IAS & IMSA

    2:00pm - 3:00pm
    Via Zoom
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    Abstract: We will define and discuss some properties of and results about the objects in the title.

    To view Dr. Philip Griffiths presentation, click here


    Rationality for Threefolds Over Non-Closed Fields

    Dr. Brendan Hassett
    Brown University

    3:30pm - 4:30pm
    Via Zoom
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    Abstract: We survey recent results on rationality questions for geometrically rational threefolds over non-closed fields due to Benoist-Wittenberg, Kuznetsov-Prokhorov, Tschinkel, etc. as well new approaches via symbol invariants.


    A Look at Fourier-Mukai Transforms and GIT in the Smooth and Singular Settings

    Dr. David Favero
    University of Alberta

    5:00pm - 6:00pm
    Via Zoom
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    Abstract: I will survey recent progress with various collaborators on constructing derived equivalences/semi-orthogonal decompositions from birational geometry.  More specifically, I will look at how partial compactifications of group actions lead to Fourier-Mukai functors which compare geometric invariant theory quotients.  Passing to the perspective of derived algebraic geometry, I will suggest that deriving these Fourier-Mukai functors could give a general comparison, even in the singular setting.

  • Tuesday October 6th, 2020

    Decomposition of the Diagonal and Abel-Jacobi Map – Lecture II

    Dr. Claire Voisin
    IMJ-PRG

    9:00am - 10:00am
    Via Zoom
    Click here to view video

    Abstract: The decomposition of the diagonal is a powerful tool to study the stable irrationality of smooth projective varieties. I will discuss the specialization method, which rests on a certain specialization property of the decomposition of the diagonal, in the first lecture. The decomposition of the diagonal controls many other stable birational invariants and I will focus in the second lecture on invariants related to the Abel-Jacobi map and the coniveau.


    Loci of Rational Varieties

    Dr. Brendan Hassett
    Brown University

    10:15am - 11:15am
    Via Zoom
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    Abstract: Given a family of smooth complex projective varieties, which members are rational? Stably rational? We discuss general structure theorems as well as guiding examples like cubic fourfolds.


    Birational Geometry and Milnor K-theory

    Dr. Alena Pirutka
    NYU Courant Institute of Mathematical Sciences

    2:00pm - 3:00pm
    Via Zoom
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    Abstract: In this talk, we will discuss results on reconstruction of function fields from Milnor K-theory. This is a joint work with A.Cadoret.


    Rationality, Unirationality and Infinite Transitivity

    Dr. Fedor Bogomolov
    NYU Courant Institute of Mathematical Sciences

    3:30pm - 4:30pm
    Via Zoom
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    Abstract: In my talk I want to touch several classical approaches to the proof of (stable) rationality and also relate the problem of unirationality and infinite transitivity.

     

  • Wednesday October 7th, 2020

    On One Question of Janos Kollar

    Dr. Ivan Cheltsov
    University of Edinburgh

    9:00am - 10:00am
    Via Zoom
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    Abstract: In 2008, Kollar studied how birational rigidity behaves when we extend the field of definition. He asked the following question: if X is a Fano variety over a field F such that X is birationally rigid over the algebraic closure of F, is X birationally rigid over F? Kollar mentioned that he thinks that this is very unlikely but he does not have a counter-example. His question can be restated for G-Fano varieties as follows: if X is a G-Fano variety for some algebraic group G in Aut(X) such that X is H-birationally rigid for some subgroup H in G, is X G-birationally rigid? I will explain why this question can have a negative answer and will explain how one can try to find a counter-example to it using toric geometry.


    A (Biased) Survey of Some Results Inspired by the (Stable) Rationality Problem for Cubic Hypersurfaces – Lecture I

    Dr. Christian Böhning
    University of Warwick

    10:15am - 11:15am
    Via Zoom
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    Abstract: We will give an overview of joint work with several co-authors: Asher Auel, Hans-Christian v. Bothmer, Michel van Garrel, Alena Pirutka, and Pawel Sosna. Particular emphasis will be placed on recent further developments of the degeneration method involving the p-torsion of the Brauer group in characteristic p or prelog Chow groups (and time permitting both).


    Factorization Centers in Dimension Two and the Grothendieck Ring of Varieties

    Dr. Evgeny Shinder
    University of Sheffield

    2:00pm - 3:00pm
    Via Zoom
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    Abstract: I explain the concept of factorization centers for birational isomorphisms. The main result is that for smooth projective surfaces over perfect fields these factorization centers for a birational isomorphism f: X->Y are independent of the choice of f and only depend on X and Y. The proof relies on the two dimensional MMP providing the minimal models for surfaces and the link decomposition for morphisms. I explain the relationship to the rationality problem for surfaces and to the structure of the Grothendieck ring of varieties. This is joint work with H.-Y. Lin and S. Zimmermann.


    Complex Surfaces with p_g=q=0 via Birational Geometry, Derived Category and Gauge theory

    Dr. Kyoung-Seog Lee
    University of Miami & IMSA

    3:30pm - 4:30pm
    Via Zoom
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    Abstract: Surfaces with p_g=q=0 play important roles in the theory of compact complex surfaces. Investigation of these surfaces has a long history and it is still a very active area of research. In the first part of this talk, I will review some parts of the history and recent developments of the theory. Then I will discuss several aspects of these surfaces via birational geometry, derived category and gauge theory.


    A Tropical Analog of the Hodge Conjecture for Smooth Algebraic Varieties Over Trivially Valued Fields

    Dr. Ryota Mikami
    Kyoto University

    5:00pm - 6:00pm
    Via Zoom
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    Abstract: Tropical geometry is a combinatorial shadow of algebraic geometry. We propose a tropical approach to problems on cycle class maps such as the Hodge conjecture. In this talk, I would like to explain a proof of a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields. The main ingredients are a theorem for general "cohomology theories", developed by many mathematicians, e.g., Quillen, a tropical analog of Milnor K-theory (introduced in this talk), and explicit calculations of tropical cohomology of the trivial line bundles by non-archimedean geometry.

  • Thursday October 8th, 2020

    Finite-Dimensional DG Algebras and their Properties

    Dr. Dmitri Orlov
    Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

    9:00am - 10:00am
    Via Zoom
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    Abstract: The talk will focus on finite-dimensional DG algebras and categories of perfect complexes over such DG algebras. These categories can be considered as proper derived noncommutative schemes. We are going to discuss basic properties of these noncommutative schemes and to establish a connection between such categories and DG categories with (semi)exceptional collections.


    Wall Finiteness Obstruction for DG Categories I & II

    Dr. Alexander Efimov
    Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

    10:15am - 11:15am
    Via Zoom
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    Abstract: The classical Wall finiteness obstruction theorem gives a criterion where by a finitely dominated CW complex is homotopy equivalent to a finite CW complex. I will explain that an analogue of this result holds for DG categories over a field.

    Namely, a homotopically finitely presented DG category is Morita equivalent to a finite cell DG category iff the class of its diagonal bimodule in $K_0$ is generated by the classes of (box) tensor products of left and right perfect modules. Moreover, finite cell DG categories are (up to Morita equivalence) exactly the quotients of proper DG categories with full exceptional collections, by a subcategory generated by a single object.

    As an application, we show that any smooth and proper phantom DG category can be embedded fully faithfully into a proper DG category with a full exceptional collection. The same holds for the derived category of the Barlow surface. This disproves two conjectures of Orlov (since the Barlow surface is not rational). We also obtain that for any smooth proper algebraic variety with a stratification by affine spaces, its derived category admits a fully faithful embedding as above.

    We will also sketch the proof of a similar result (Wall finiteness obstruction) for algebras over (colored) DG operads. In this case the class of the diagonal should be replaced by the class of the cotangent complex.


    Rigid, Not Infinitesimally Rigid Surfaces of General Type with Ample Canonical Bundle – Lecture II

    Dr. Christian Böhning
    University of Warwick

    2:00pm - 3:00pm
    Via Zoom
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    Abstract: In the talk I will report on work in progress, joint with Roberto Pignatelli and Hans-Christian von Bothmer, that concerns the construction of surfaces of general type with ample canonical bundle and Kuranishi space (and possibly also Gieseker moduli space) a non-reduced point. The main tools are configurations of lines and their incidence schemes as well as the theory of abelian covers due to Pardini and others.

    I apologize for the thematic remoteness from the main topic of the conference; however, nearly rational varieties help in the construction, and the topic is fun!


    Rationality and Exceptional Collections for Toric Varieties

    Dr. Matthew Ballard
    University of South Carolina 

    3:30pm - 4:30pm
    Via Zoom
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    Abstract: A conjecture of Orlov asks whether all varieties with full exceptional collections are rational. We will discuss two interpretations of this question over non-closed fields. The strict one, where End(E)=k is the definition of exceptional for X over k, holds for toric varieties over a general field. The looser one, End(E)/k is a separable extension of fields, does not guarantee the existence of a k-point much less rationality. The work discussed is joint with Alexander Duncan, Alicia Lamarche, and Patrick McFaddin.


    Holomorphic Bisectional Curvature and Applications to Deformations and Rigidity of Mixed Hodge Structure

    Dr. Gregory Pearlstein
    Texas A&M University

    4:45pm - 5:45pm
    Via Zoom
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    Abstract: The curvature properties of period domains play a fundamental role in Hodge theory.  In this talk, I will discuss recent work with Chris Peters on Hodge metrics arising in mixed Hodge theory, and applications to rigidity results for geometric classes of examples.

  • Friday October 9th, 2020

    Intermediate Jacobians of Gushel--Mukai Varieties

    Dr. Alexander Kuznetsov
    Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

    9:00am - 10:00am
    Via Zoom
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    Abstract: A Gushel--Mukai variety X is a smooth dimensionally transverse linear section of a 6-dimensional intersection of the cone CGr(2,5) over Gr(2,5) with a quadric. The nontrivial part of the derived category of X is of K3 type when \dim X is even and of a curve-Enriques type when \dim X is odd.

    In the talk I will present a description of intermediate Jacobians of GM threefolds and fivefolds as Albanese varieties of the associated double EPW surfaces and discuss birational and categorical counterparts of this phenomenon. This talk is based on joint results with Olivier Debarre and Alex Perry.


    Wall Finiteness Obstruction for DG Categories I & II

    Dr. Alexander Efimov
    Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

    10:15am - 11:15am
    Via Zoom
    Click here to view video

    Abstract: The classical Wall finiteness obstruction theorem gives a criterion where by finitely dominated CW complex is homotopy equivalent to a finite CW complex. I will explain that an analogue of this result holds for DG categories over a field.

    Namely, a homotopically finitely presented DG category is Morita equivalent to a finite cell DG category iff the class of its diagonal bimodule in $K_0$ is generated by the classes of (box) tensor products of left and right perfect modules. Moreover, finite cell DG categories are (up to Morita equivalence) exactly the quotients of proper DG categories with full exceptional collections, by a subcategory generated by a single object.

    As an application, we show that any smooth and proper phantom DG category can be embedded fully faithfully into a proper DG category with a full exceptional collection. The same holds for the derived category of the Barlow surface. This disproves two conjectures of Orlov (since the Barlow surface is not rational). We also obtain that for any smooth proper algebraic variety with a stratification by affine spaces, its derived category admits a fully faithful embedding as above.

    We will also sketch the proof of a similar result (Wall finiteness obstruction) for algebras over (colored) DG operads. In this case the class of the diagonal should be replaced by the class of the cotangent complex.


    On Rational Cubic Fourfolds Containing Veronese Surfaces

    Dr. Yu Wei Fan
    University of California at Berkeley

    2:00pm - 3:00pm
    Via Zoom
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    Abstract: Veronese surfaces in P^5 induce birational automorphisms on P^5. We show that these birational automorphisms give rise to a non-trivial birational involution on the space of cubic fourfolds containing a Veronese surface. Using this involution, we prove the rationality of certain cubic fourfolds that was not known previously. We also prove that if a cubic fourfold X contains a Veronese surface, then the only non-isomorphic cubic that shares the same K3 category with X is the image of X under the Cremona transformation. Joint work with Kuan-Wen Lai.


    NCHS and Rationality

    Dr. Ludmil Katzarkov
    University of Miami & IMSA

    3:30pm - 4:30pm
    Via Zoom
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    Abstract: In this talk we will discuss new approached to nonrationality based on Mirror Symmetry.