Schedule for: 23w5131 - Explicit Moduli Problems in Higher Dimensions

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We discuss the KSBA moduli space of surfaces of general type, focusing first on general techniques to construct and study singular surfaces and their moduli. To illustrate one such technique, we showcase a delightful example (one of many discovered by my undergraduate students in a summer project) that builds a singular surface from a smooth elliptic surface. In the second part of the talk, we describe joint work with S. Rollenske on moduli spaces of Horikawa surfaces. Classically, certain moduli spaces of Horikawa surfaces consist of two disconnected components. We show that the closures of these two components intersect in a divisor parametrizing semi-smooth surfaces. More generally, we describe new, generically non-reduced, irreducible components of increasing dimensions. ****Note from the organizers: this talk is a two-part introduction to the moduli space of surfaces of general type. There will be a break during the talk from 10 - 10:20 for coffee.****

We discuss the KSBA moduli space of surfaces of general type, focusing first on general techniques to construct and study singular surfaces and their moduli. To illustrate one such technique, we showcase a delightful example (one of many discovered by my undergraduate students in a summer project) that builds a singular surface from a smooth elliptic surface. In the second part of the talk, we describe joint work with S. Rollenske on moduli spaces of Horikawa surfaces. Classically, certain moduli spaces of Horikawa surfaces consist of two disconnected components. We show that the closures of these two components intersect in a divisor parametrizing semi-smooth surfaces. More generally, we describe new, generically non-reduced, irreducible components of increasing dimensions. ****Note from the organizers: this talk is a two-part introduction to the moduli space of surfaces of general type. There will be a break during the talk from 10 - 10:20 for coffee.****

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The moduli theory of Fano varieties has various pathologies that do not appear for canonical polarized varieties. To address these issues, a current solution is to only parameterize Fano varieties that are K-stable, which is a notion introduced by differential geometers to characterize KE metrics on Fano varieties. In this talk, I will explain the definition of K-stability, how it produces a moduli theory for Fano varieties analogous to the KSBA theory, and end with the computation of explicit examples. ****Note from the organizers: this talk is a two-part introduction to K-stability and K-moduli. There will be a break during the talk from 3 - 3:20 for coffee.****

The moduli theory of Fano varieties has various pathologies that do not appear for canonical polarized varieties. To address these issues, a current solution is to only parameterize Fano varieties that are K-stable, which is a notion introduced by differential geometers to characterize KE metrics on Fano varieties. In this talk, I will explain the definition of K-stability, how it produces a moduli theory for Fano varieties analogous to the KSBA theory, and end with the computation of explicit examples. ****Note from the organizers: this talk is a two-part introduction to K-stability and K-moduli. There will be a break during the talk from 3 - 3:20 for coffee.****

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. The method presented here can also be applied to the moduli space of stable map spaces from semi-log-canonical surfaces to projective varieties.

A rational elliptic surface of index two is a rational surface that comes equipped with a (relatively minimal) genus one fibration that has exactly one multiple fiber of multiplicity two. In this talk I will describe how one can construct a moduli space for these surfaces when the choice of a bisection is part of the classification problem. This is based on joint work with Rick Miranda.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Smooth Fano hypersurfaces are all conjectured to be K-polystable. This is verified for a large class of hypersurfaces, and a next step is to describe their K-moduli space. This is known for surfaces as well as cubic 3-folds and 4-folds. In particular, it is known that the K-moduli coincides with the GIT-moduli for cubic hypersurfaces up to dimension 4, and conjectured in higher dimensions. The next natural class to be considered is the K-moduli of quartic 3-folds. In this talk I present some recent results on the K-moduli of these objects. This is joint work with Ivan Cheltsov, Andrea Petracci, Alexander Kasprzyk, and Yuchen Liu.

It is well known that Calabi-Yau manifolds have good deformation theory, which is controlled by Hodge theory. By work of Friedman, Namikawa, M. Gross, Kawamata, Steenbrink and others, some of these results have been extended to Calabi-Yau threefolds with canonical singularities. In this talk, I will report on further extensions in two directions: in dimension 3, we sharpen and clarify some of the existing results, and, secondly, we obtain some higher dimensional analogues. I will also briefly explain the related case of Fano varieties, where stronger results hold. One surprising aspect of our study is the role played by higher du Bois and higher rational singularities, notions that were recently introduced by Mustata, Popa, Saito and their collaborators. This is joint work with Robert Friedman.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I want to explain how toric geometry can help in finding explicit K-polystable Fano varieties which give one of the following two behaviours: i) singular points on K-moduli of Fano varieties (joint work with Anne-Sophie Kaloghiros), ii) K-polystable degenerations of quartic 3-folds which are not quartic 3-folds and are not double covers of the smooth quadric 3-fold (joint work with Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, Yuchen Liu). The quest for such examples, both in i) and ii), is done by using mirror symmetry expectations arising in the Fanosearch programme of Coates, Corti, Galkin, Golyshev, Kasprzyk and others.

An I-surface is a minimal complex surface of general type with $K^2=1$ and $p_g=2.$ I will report on recent joint work with Stephen Coughlan, Marco Franciosi, Julie Rana and Soenke Rollenske (in various combinations) giving a partial description of the boundary points of the KSBA compactification of the moduli space of I-surfaces.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will report on joint work in progress with Luca Giovenzana. After giving a very brief introduction to the theory of pseudolattices, I will describe how they show up naturally in the context of Type II degenerations of K3 surfaces, and show how this can be used to recover some classical results on Type II degenerations due to Friedman. If I have sufficient time, I will also discuss how these structures seem to provide a natural setting to study mirror symmetry for Type II degenerations of K3 surfaces.

Over the complex numbers, rationality of conic bundles over $P^2$ is well understood: it is characterized by the Clemens–Griffiths intermediate Jacobian obstruction to rationality. In this talk, we investigate the rationality of these conic bundles over non-closed fields. We study the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg, which extends the classical obstruction over C. We focus on the case when the discriminant curve has degree 4, which is the first case where geometric rationality and existence of a k-rational point do not characterize k-rationality. This talk is based on joint work with S. Frei–S. Sankar–B. Viray–I. Vogt and joint work with M. Ji.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular compactification $\overline{\mathbf{M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of $\mathbf{M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parametrize. This is joint work with Patricio Gallardo, Gregory Pearlstein, and Zheng Zhang.

Compact Hyperkähler manifolds are one of the building blocks of Kähler manifolds with trivial first Chern class, but very few examples are known. One strategy for potentially finding new examples is to look at finite groups of symplectic automorphisms of the known examples and study the fixed loci or quotient. In this talk, we will obtain a classification of birational symplectic involutions of manifolds of OG10 type. We do this from three vantage points: via involutions of the Leech lattice, via involutions of cubic fourfolds, and lattice enumeration via a modified Kneser’s neighbor algorithm. In particular, we exhibit geometric realizations of these involutions in three cases. If time permits, we will mention ongoing work to identify the fixed loci in one of these examples.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

The moduli space $\overline{M}_{0,n}$ of stable n-pointed rational curves admits a combinatorial description as a tropical compactification of a complement of a hyperplane arrangement. This perspective leads, for instance, to an essentially combinatorial proof of Keel's famous presentation of the Chow ring of $\overline{M}_{0,n}$. I will discuss a generalization of the theory of tropical compactifications of complements of hyperplane arrangements to a broader class of varieties, and how this generalization leads to higher-dimensional analogues of Keel's presentation---in particular, to descriptions of the intersection theory of compactifications of moduli spaces of hyperplane arrangements and marked del Pezzo surfaces.