Benjamin Bakker (UIC) Baily—Borel compactifications of period images and the b-semiampleness conjecture
Building on previous work of Satake and Baily, Baily and Borel proved in 1966 that arithmetic locally symmetric varieties admit canonical projective compactifications whose graded rings of functions are given by automorphic forms. Such varieties include moduli spaces of abelian varieties, and have rich algebraic and arithmetic geometry. Griffiths suggested in 1970 that the same might be true for the image of any period map, which would provide canonical compactifications of many moduli spaces, including for instance Calabi--Yau varieties. In joint work with S. Filipazzi, M. Mauri, and J. Tsimerman, we confirm Griffiths' suggestion, and prove that the image of any period map admits a canonical functorial compactification. We also show how the same techniques yield a resolution to an important conjecture in birational geometry, the b-semiampleness conjecture of Prokhorov—Shokurov. Both proofs crucially use o-minimal GAGA, and the latter application additionally uses results of Ambro and Kollár on the geometry of minimal lc centers.
Nathan Chen (Harvard) Characterizing varieties via birational automorphisms.
Cantat and Regeta--Urech--van Santen showed that rationality can be characterized via the Cremona group. In this talk, we will study more instances of how the birational automorphism group can be used to recover a variety. This is joint work with L. Esser, A. Regeta, C. Urech, and I. van Santen.
Philip Engel (UIC) Matroids and the integral Hodge conjecture for abelian varieties.
We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman and Stefan Schreieder.
Ludmil Katzarkov (Miami/Sofia) New birational Invariants.
In this talk we will give the origins of theory of atoms. Many examples will be considered. Future applications will be discussed.
Samuel Johnston (Imperial/MIT) The log-orbifold correspondence in intrinsic mirror symmetry.
The study of moduli spaces of maps from curves to simple normal crossings pairs has played a prominent role in modern enumerative geometry, with applications including flexible degeneration schemes for classical curve counting problems and mirror symmetry. The two primary foundations for this theory are logarithmic and orbifold Gromov-Witten theory. The different natures of these two theories endow them with complementary strengths and weaknesses. Recent work of Battistella-Nabijou-Ranganathan and upcoming joint work with Crumplin provides a correspondence relating these theories in genus 0 and arbitrary genus respectively. I will recall the constructions of these relative curve counting theories, illustrate the above correspondence, and apply it to yield the relations necessary for the construction of mirrors to log Calabi-Yau pairs, as well as relations related to periods of the mirror families.
Lisa Marquand (NYU) Birational Cubic fourfolds with birational Fano varieties of lines.
Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. However, not much is known about the birational geometry of the potentially non-rational cubics. In this talk, we will discuss conjectures for how to extract information on the birational geometry of a cubic fourfold using both the derived category, and hyperkahler manifolds. We will give new examples of pairs of birational cubic fourfolds that are Fourier-Mukai partners, as well as having birational Fano varieties of lines, a previously unknown phenomenon. All of our examples are (potentially) non-rational cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei.
Davesh Maulik (MIT) Intrinsic cohomology for families of compactified Jacobians.
The universal Jacobian over the moduli space M_g of smooth curves admits natural compactifications over the Deligne--Mumford compactification of M_g, known as universal fine compactified Jacobians. The geometry of these models depends on the degree and a choice of stability condition. In this talk, we show that although the cohomology and Chow rings vary with these choices, they all admit a common degeneration (what we call the intrinsic cohomology ring) which is independent of the choice of compactification. Its construction is based on a recent connection between Fourier-Mukai transforms and perverse filtrations. Time permitting, we will discuss a conjectural strengthening in the presence of a Lagrangian fibration structure. This is joint work with Younghan Bae, Junliang Shen, and Qizheng Yin.
Michael Temkin (Hebrew U., Jerusalem) New techniques in resolution of singularities.
Since Hironaka's famous resolution of singularities in characteristic zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However, one point remained quite mysterious: despite different descriptions of the basic resolution algorithm, it was essentially unique. Was it a necessity or a drawback of the fact that all subsequent methods relied on Hironaka's ideas essentially?
The situation changed in the last decade, when logarithmic, weighted and foliated analogues and generalizations were discovered in works of Abramovich-Temkin-Wlodarzcyk, McQuillan, Quek, Abramovich-Temkin-Wlodarzcyk-Belotto and others. At this stage we can already try to figure out general ideas and principles shared by all these methods and the picture is quite surprising -- it seems that each method is quite determined by its basic setting consisting of the class of geometric objects and basic blowings up one works with. In particular, the classical method is probably the only natural resolution (via principalization) method obtained by blowing up smooth centers in the ambient manifold.
In my talk I'll describe the settings and the methods on a very general level and will add some details about the classical algorithm, the logarithmic one and the simplest dream (or weighted) method, which has no memory and improves the singularity invariant by each weighted blowing up.