# Boston Algebraic Geometry Day

### October 6, 2024 at Harvard University

organized by Dawei Chen, Aaron Landesman, Mihnea Popa

Speakers: Lena Ji, Radu Laza, Davesh Maulik, Mircea Mustațǎ

Location: Harvard University Science Center 507

Parking: If you need a parking pass, please contact Aaron Landesman (landesman@math.harvard.edu) by September 28, a week before the event, and tell him your full name and email address.

Schedule:

9:30-10 Check-in and refreshments

10-11 Davesh Maulik: D-equivalence conjecture for varieties of K3^[n]-type

11-11:30 Coffee break

11:30-12:30 Lena Ji: Symmetries of Fano varieties

12:30-2:30 Lunch (not provided)

2:30-3:30 Radu Laza: The Core of Calabi-Yau degenerations

3:30-4 Coffee break

4-5 Mircea Mustațǎ: The minimal exponent of hypersurface singularities

Abstracts:

Davesh Maulik (MIT): D-equivalence conjecture for varieties of K3^[n]-type

The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.

Lena Ji (University of Illinois Urbana-Champaign): Symmetries of Fano varieties

Prokhorov and Shramov proved that the boundedness of Fano varieties (which Birkar later proved) implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. In particular, in each dimension n, there is an upper bound on the size of semisimple groups (i.e. those with no nontrivial normal abelian subgroups) acting on n-dimensional complex Fano varieties. In this talk, we study the action by a particular semisimple group—the symmetric group—and investigate the geometric consequences of such an action. This work is joint with Louis Esser and Joaquín Moraga.

Radu Laza (Stony Brook University): The Core of Calabi-Yau degenerations

It is problem of high interest to construct meaningful compactifications for moduli spaces of algebraic varieties. For varieties of general type and Fano type, the situation is well understood via KSBA theory and K-stability theory respectively. In the remaining K-trivial case, one understands the “classical case” (i.e. abelian varieties, K3 surfaces, and hyper-Kaehler manifolds) via period maps and Hodge theory. Thus, the only remaining primitive case for constructing compact moduli spaces is that of strict Calabi-Yau’s of dimension at least 3.

In this talk, I conjecture the existence of a minimal compactification for the moduli of Calabi-Yau varieties, analogous to the Baily-Borel compactification in the classical case. I will then explain that such a compactification exists at least as a compact stratified topological space, with quasi-projective strata. The key concepts here are the Type and Core of a Calabi-Yau degeneration. The Type is an integer invariant that measures the depth of the degeneration, while the Core is a pure Hodge structure of Calabi-Yau type, which should be understood as the Hodge theoretic information that is visible in any reasonable model of the degeneration.

Mircea Mustațǎ (University of Michigan): The minimal exponent of hypersurface singularities

I will give an introduction to the minimal exponent, an invariant of singularities introduced by Morihiko Saito and which refines the log canonical threshold. I will discuss different characterizations, some basic properties, and a few open problems. Time permitting, I will present some work in progress with Qianyu Chen in the direction of a birational description of this invariant.

We gratefully acknowledge the support of the NSF and the host institution.

We are dedicated to upholding the principles and practices of equal opportunity, affirmative action, and nondiscrimination. Our commitment extends to fostering an inclusive environment for all BAGD participants. Harvard University's related policies are applicable to all BAGD participants. If there are any concerns or requests for assistance related to discrimination or harassment, please reach out to one of the organizers.