﻿ Moduli Spaces in Mathematics Workshop & Conference-清华丘成桐数学科学中心  ## Activities

1. YMSC Courses Mini Courses Seminars Lectures Conferences

## Moduli Spaces in Mathematics Workshop & Conference

Organizer：Jungkai Chen, Keiji Oguiso, De-Qi Zhang, Caucher Birkar, Ivan Fesenko
Time：April 4 - April 7, 2023 Schedule

# 2023-04-04

08:00-09:00

Shigeru Mukai (Kyoto University, MCM)

Quartics and moduli I

Even restricted to surfaces (of Kod. dim 0), moduli has various viewpoints. I will discuss them centering three famous quartic hypersurfaces.

1. Igusa 3-fold and Enriques surfaces: This quartic was first found as the projective dual of Segre's 10-nodal cubic, the moduli of 6 points on the projective line. It was re-discovered as moduli of p.p.a.s’s by Igusa(1962). I explain its new interpretation (Contemp. Math., 2012) as the moduli of Enriques surfaces of certain root type, together with a similar one for the Coble 4-fold, the double P4 with branch Igusa.

09:30-10:30

Shigeru Mukai (Kyoto University, MCM)

Quartics and moduli II

Even restricted to surfaces (of Kod. dim 0), moduli has various viewpoints. I will discuss them centering three famous quartic hypersurfaces.
2. Coble 6-fold and K3 surfaces: The moduli space of polarized K3 surfaces is of general type except for finitely many genus. As an example of exception I explain the unirationality for genus 13 using the Coble quartic in P7, which is the moduli space N+ of 2-bundles on a plane quartic curve. The generic K3 is described explicitly in the moduli space N-, the Hecke partner of N+ (joint work with A. Kanemitsu).

11:00-12:00

Yongnam Lee (Korea Advanced Instituteof Science and Technology)

Q-Gorenstein deformations and moduli of hypersurfaces I

Lecture I: Q-Gorenstein deformations
The notion of Q-Gorenstein variety and Q-Gorenstein deformation are important for the minimal model theory of algebraic varieties and construction of compact moduli space of varieties. In this lecture, I will give an overview of the theory of Q-Gorenstein deformations with some applications.

Lunch Break

13:30-14:30

Caucher Birkar (YMSC, BIMSA)

Classification and moduli of algebraic varieties I

In these lectures, I will describe the general framework of the classification of algebraic varieties through birational geometric techniques, then introduce the concept of stable minimal models of arbitrary Kodaira dimension and explain some of the main ideas of the proofs of boundedness and existence of compact moduli spaces of such models

15:00-16:00

Caucher Birkar (YMSC, BIMSA)

Classification and moduli of algebraic varieties II

In these lectures, I will describe the general framework of the classification of algebraic varieties through birational geometric techniques, then introduce the concept of stable minimal models of arbitrary Kodaira dimension and explain some of the main ideas of the proofs of boundedness and existence of compact moduli spaces of such models.

Dinner Break

20:00-21:00

János Kollár (Princeton University)

Moduli spaces of algebraic varieties I

The first lecture will be an introduction, explaining the history and some key examples where the moduli problem has unexpectedly bad behavior. Them, the second lecture shows some of the key ideas that lead to a good compactified moduli theory for varieties of general type.

2023-04-05

08:00-09:00

Yongnam Lee (Korea Advanced Instituteof Science and Technology)

Q-Gorenstein deformations and moduli of hypersurfaces II

Lecture II: Compactifying moduli of plane curves
Compactifying the moduli of plane curves is a good example to understand beautiful relations between several different approaches to compactify moduli spaces: GIT, KSBA, Hodge theory, and K-stability. In this lecture, I will give an overview of these approaches to moduli of plane curves and illustrate the simplest non-trivial case: moduli of plane quartic curves.

09:30-10:30

Yongnam Lee (Korea Advanced Instituteof Science and Technology)

Q-Gorenstein deformations and moduli of hypersurfaces III

Lecture III: Smooth specialization of hypersurfaces
Small deformation of a smooth hypersurface of degree d is again a smooth hypersurface of degree d in projective space (or abelian variety) unless it defines a K3 surface. So smooth hypersurfaces in projective space (or abelian variety) form an open set in the moduli space when they are of general type. By using KSBA-model or GIT-model one can try to compactify this moduli space by adding some singular varieties. Before focusing on varieties with singularities in the boundary of this moduli space, it is natural to ask whether we have a structure theorem for smooth specialization of hypersurfaces. In this lecture, I will discuss this question.

11:00-12:00

Shigeru Mukai (Kyoto University, MCM)

Quartics and moduli III

Even restricted to surfaces (of Kod. dim 0), moduli has various viewpoints. I will discuss them centering three famous quartic hypersurfaces.
3. Inose surfaces and type II degenerations: The 2-parameter family of quartic surfaces found by Inose (1978) contains infinitely many 1-parameter subfamilies which are mirror of polarized K3 surfaces in the sense of Dolgachev(1996). Their Picard lattices and nef cones control the type II degenerations. I discuss the Hilbert square of Inose quartics, first their birational automorphisms and next a connection with type II degenerations of K3 4-folds.

Lunch break

13:30-14:30

Shing-Tung Yau (BIMSA, YMSC)

Metric Geometry of Moduli Spaces of Curves

Canonical metrics play an important role in the study of geometry of moduli spaces. Such metrics with nice curvature properties and precise asymptotic lead to stability of bundles over moduli spaces and vanishing theorems. In this talk we discuss various metrics on the moduli spaces of stable curves and algebraic geometric consequences.

16:00-17:00

Caucher Birkar (YMSC, BIMSA)

Classification and moduli of algebraic varieties III

In these lectures, I will describe the general framework of the classification of algebraic varieties through birational geometric techniques, then introduce the concept of stable minimal models of arbitrary Kodaira dimension and explain some of the main ideas of the proofs of boundedness and existence of compact moduli spaces of such models.

Dinner break

20:00-21:00

János Kollár (Princeton University)

Moduli spaces of algebraic varieties II

The first lecture will be an introduction, explaining the history and some key examples where the moduli problem has unexpectedly bad behavior. Them, the second lecture shows some of the key ideas that lead to a good compactified moduli theory for varieties of general type.

2023-04-06

09:30-10:30

Valery Alexeev (University of Georgia)

Compact moduli spaces of K3 surfaces

I will talk about recent results, jointly with Philip Engel, on compactifications of moduli spaces of K3 surfaces via the moduli of KSBA stable pairs (X,D). These compactifications depend on a canonical choice of polarizing divisor D. Some of the results are general, saying that the compactifications corresponding to many natural choices of polarizing divisors are semitoroidal. Others provide explicit descriptions of these compactifications in interesting cases, such as the moduli spaces of K3 surfaces with an involution and elliptic K3 surfaces.

11:00-12:00

Chin-Lung Wang (Taiwan University)

How many ways can a punctured torus being conformally bended to spherical?

In this talk I will present a conjectural formula of the question asked in the title when the puncture has a cone angle (2n + 1)2\pi. The answer turns out to be sensitive to the moduli. An approach using modular forms will be introduced. This is a work in progress jointly with You-Cheng Chou and Po-Sheng Wu.

Lunch Break

16:00-17:00

Sergio Cecotti (BIMSA)

Moduli spaces of Calabi-Yau's as physical systems

The swampland conjectures suggest that the moduli spaces of Calabi-Yau's of complex dimension 4 or less should have peculiar properties, making them to look as "physical" spacetimes which solve the natural equations of motion. We show that this is actually true.

2023-04-07

08:00-09:00

Stephen Shing-Toung Yau (BIMSA)

Moduli space of strongly psedoconvex CR manifolds

The famous result of Fefferman in 70s asserts that a biholomorphic mapping between two bounded smooth strongly pseudoconvex domains induces a CR-equivalence between their boundaries. Therefore the biholomorphically equivalent problem of bounded strongly pseudoconvex domains is the same as the CR equivalent problem of strongly pseudoconvex compact CR manifolds. We introduce a novel tool, the Bergman function, to attack the equivalent problem of strongly pseudoconvex (smooth or singular) domains. As a result, we can construct a complete family of numerical invariants from the Bergman functions for some special domains. In particular the moduli space of these domains is constructed explicitly as the image of this complete family of numerical invariants.

09:30-10:30

Mark Gross (University of Cambridge)

Mirror symmetry and partial compactifications of K3 moduli

I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighborhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued into the moduli space of polarized K3 surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighborhood, certain canonical bases for sections of powers of the polarizing line bundle.

11:00-12:00

David Morrison

Kähler classes, Kähler cones, and moduli

I will discuss 7 situations in which our understanding of moduli is enhanced by a choice of Kähler class or Kähler cone.  Our basic setting is algebraic geometry over the complex numbers or complex manifolds, but in many cases we can replace “Kähler” by “ample” and obtain an algebraic result.

1) Geometric invariant theory:  given a reductive group G acting on a quasi-projective variety X, one wants to form the quotient X/G.  (Often, in order to construct a moduli space of some kinds of varieties.). The birational equivalence class of X/G is well-defined, but specifying the quotient as a scheme requires more data:  in particular, one needs a linearized ample line bundle L on X.  (This ample class, in the complex case, can be regarded as a Kähler class on X.)

2) Symplectic reduction: given a “Hamiltonian” action of a compact (real) Lie group G on a symplectic manifold X (of which Kähler manifold is a special case), there is a so-called moment map, which is an equivariant map mu: X -> (Lie (G))^*.  A symplectic reduction, mu^{-1}(a)/G, is a  quotient which retains a symplectic structure.  By Kirwan-Ness, in the K\”ahler case, this coincides with a geometric invariant theory quotient of X by the complexification G_C of the group G.  Varying “a” gives a variable Kähler class, and in fact the quotient as a complex manifold is independent of the choice of “a” within an appropriate cone — the Kähler cone of the quotient.

3) As a simple case of this, G=U(1) acting on C^4 by (x1, x2, x3, x4) |-> (t x1, t x2, t^{-1} x3, t^{-1} x4)

has moment map mu(x1, x2, x3, x4) = 1/2( |x1|^2 + |x2|^2 - |x3|^2 - |x4|^2.  When a=0, the symplectic reduction is the singular quadric z_1 z_2 + z_3 z_4 = 0; when a>0, we get one of the small resolutions of that singularity, and when a<0 we get the other small resolution.

4) The variation of small resolution from #3 resolves a difficulty in separateness of the moduli of K3 surfaces.  In fact, in the category of complex manifolds, the (coarse) moduli of pairs (complex K3 surface, Kähler cone) is a Hausdorff space.

5) More generally, the notion of birational cobordism introduced by Wlodarcyzk generalizes #3 in a natural way.  If we enhance our description of complex moduli by including a choice of Kähler class, when that Kähler class approaches the boundary of the Kähler cone, then certain “extremal” curve classes have area approaching 0 and are birationally contracted in the family.  Continuing beyond the boundary, produces a different birational model with a different Kähler cone.

6) Kähler classes appear naturally in Calabi’s conjecture (proven by Yau), i.e., in the study of “moduli” of Ricci-flat metrics on a (topologically fixed) Calabi—Yau manifold.  More generally, we can combine these moduli spaces for different birational models of a fixed Calabi—Yau manifold.  The study of these moduli spaces led me to formulate a “cone conjecture” and a “movable cone conjecture” for Calabi—Yau manifolds.

7) A related moduli problem is the “moduli” of superstring theories related to Calabi—Yau manifolds.  The data is enhanced with an additional 2-form known as the “B-field” (with the Kahand the corresponding moduli space, according to the mirror symmetry conjecture, should be built out of moduli of a mirror pair of Calabi—Yau manifolds.

Lunch break

14:30-15:30

Yuji Odaka

K-stability, Kahler-Einstein metrics and Moduli compactifications

In the former half of the talk, we first review our existence conjecture of 'K-moduli'(=moduli of K-polystable varieties, possibly with singularities) and survey some of our recent developments in the case with Kahler-Einstein metrics. In the negative Ricci case, it reexplains the Kollar-Shepherd-Barron-Alexeev moduli compactification. In the positive Ricci ('Fano') case, it essentially gives the Gromov-Hausdorff compactification. In the Calabi-Yau/Ricci-flat case, it only gives a partial quasi-projective compactification. Algebraic re-approaches to these heavily depend on various boundedness works of Birkar.
In the latter half of the talk we discuss full compactification of the moduli in the Calabi-Yau case by allowing Gromov-Hausdorff collapses (e.g. K3 to S^2 or interval). In this case, the obtained canonical topological compactifications are no longer varieties, but interestingly we still observe some 'algebraic' nature a la Kontsevich-Soibelman / SYZ mirror symmetry conjecture. For instance, the obtained boundary of GH compactification often turns out to be the dual intersection complex of toroidal compactifications (or equivalently, one of Satake’s compactification), and the limit density function along the collapse of K3 to interval gives new explicit invariant for type II degenerations, which is again algebraically recoverable. The latter half is based on joint work with Yoshiki Oshima.

Dinner Break

18:00-19:00

Phillip Griffiths (Institute for Advanced Study)

Hodge theory and moduli

Hodge theory provides a major tool for the study of moduli. Con-versely, moduli have furnished a significant stimulus for the developement of Hodge theory. This talk will center around the questions
Q1 Torelli: To what extent does the Hodge structure on the coho- mology H*(X) determine a smooth projective variety X? Better yet: How can one reconstruct X from Hodge theortic data associated to it?
Q2: To what extent does the structure of a completed image of the period mapping reflect the structure of a completed moduli space M? One knows a lot about how Hodge structures degenerate; can this be used to help understand the boundary M\M of moduli spaces?
If time permits I will make a few comments about Hodge theory and fundamental groups of quasi-projective varieties and the moduli of their linear representations.
Q3: How is the Hodge theory of the fundamental group used in the Shafarevich conjecture? That conjecture is that the universial cover of a smooth projective variety should be holomorphically convex.