Gothenburg Complex Geometry 2024

Monday August 26

11.15–12.05 Zakarias Sjöström Dyrefelt, Aarhus University: On generalised Monge-Ampère equations

Abstract: Generalised Monge-Ampère (gMA) equations were introduced by Pingali and capture several well-known families of PDE on compact Kähler manifolds, including the J-equation, inverse Hessian equations, and certain deformed Hermitian-Yang-Mills and Z-critical equations. By results of Datar-Pingali and Fang-Ma (extending those of Gao Chen, Song, and others) solvability of gMA equations is equivalent to a Demailly-Păun type positivity condition tested on subvarieties, reminiscent of slope stability. In general this slope condition can be violated by infinitely many subvarieties, of any codimension. However, as a main result we deduce from the divisorial Boucksom-Zariski decomposition that the destabilising subvarieties form a finite set in the semistable case, if we impose enough positivity on a certain (1,1) - class. This leads to first existence results and first examples toward the following conjectural picture: Under suitable assumptions, the “space of gMA equations” admits a locally finite wall-chamber decomposition, in arbitrary dimension.

This is ongoing joint work with Sohaib Khalid (SISSA).

14.00–14.50 Michela Zedda, University of Parma: The coefficients of TYCZ-expansion for Kaehler-Einstein metrics

Abstract: Prescribing the coefficients arising from the asymptotic expansion of the epsilon function of a polarized Kaehler metric, gives rise to a set of partial differential equations which generalize in a natural way the equation for finding Kaehler metrics with constant scalar curvature. The talk introduces the epsilon function and its asymptotic expansion for compact and noncompact manifolds, focusing on the geometric properties related to the vanishing of the third coefficient for Kaehler-Einstein metrics.

15.20–16.10 Jakob Hultgren, Umeå University: Real Monge-Ampère equations and the SYZ conjecture for toric hypersurfaces

Abstract: Abstract: It is well known since a few years that a weak version of the metric SYZ conjecture follows from existence and/or structural results about Monge-Ampère equations. For general maximal degenerations of Calabi-Yau manifolds, this principle is formulated in terms of non-Archimedean geometry, but for certain families of hypersurfaces in toric Fano manifolds, we can be a bit more concrete and the weak SYZ conjecture reduces to solvability of a real Monge-Ampère equation on the boundary of a polytope. I will not present a lot of new results, but I will briefly explain this setting and then talk about the question of regularity of solutions to this equation, and what kind of geometric information is contained in the answer to this question. This is based on joint work with Rolf Andreasson, Thibaut Delcroix, Mattias Jonsson, Enrica Mazzon and Nick McCleerey.

Tuesday August 27

10.00–10.50 Léonard Pille-Schneider, University of Regensburg: The SYZ conjecture for families of hypersurfaces

Abstract: Let X -> D* be a polarized family of Calabi-Yau manifolds, whose complex structure degenerates in the worst possible way. The SYZ conjecture predicts the behavior of the fibers X_t, endowed with their Ricci-flat Kähler metric, as t -> 0, and in particular the program of Kontsevich and Soibelman relates it to the non-archimedean analytification of X, viewed as a variety over the non-archimedean field of complex Laurent series. I will explain the ideas of this program and some progress in the case of hypersurfaces.

11.15–12.05 Catriona Maclean, Université Grenoble Alpes: Infinite divisors appearing in approximable algebras in the complex case

Abstract: Approximable algebras are graded algebras whose rate of growth of graded pieces resembles that of section algebras of big divisors or their full dimensional sub algebras. They are not necessarily associated to big divisors, but can be associated to certain infinite divisors. In the work that will be presented in this talk, we wrap up our study of such algebras by showing that in the complex case, full dimensional sub algebras of infinite divisors are approximable if and only if the associated series of cohomology classes is convergent.

14.00–14.50 Ruadhaí Dervan, University of Glasgow: Arcs and the Mabuchi functional

Abstract: Donaldson introduced arcs as a generalisation of test configurations, as used in K-stability. I will discuss some connections between the Mabuchi functional and arcs, with applications. This is joint work with Rémi Reboulet.

15.20–16.10 Quang-Tuan Dang, ICTP: Analytic Nakai-Moishezon's criterion on compact complex varieties

Abstract: The classical Nakai-Moishezon-Kleiman ampleness criterion characterizes ample line bundles on a projective variety as those which have positive intersection against all subvarieties.

In a groundbreaking paper, Demailly and Păun proved a vast generalization of this result, which holds for all real closed (1,1) classes on a compact Kähler manifold. Generalizing the result of Demailly-Păun, we establish Nakai-Moishezon type criterion on a singular complex compact variety under some geometric conditions.

Wednesday August 28

09.00–09.50 Siarhei Finski, École Polytechnique: Hermitian Yang-Mills functionals on direct images

Abstract: For a polarized family of complex projective manifolds, we study the Hermitian Yang-Mills functionals on the sequence of vector bundles over the base of the family associated with direct image sheaves of the tensor powers of polarization. We make a connection between the asymptotic minimization of these functionals and the minimization of the so-called Wess-Zumino-Witten functional defined on the space of all relatively Kähler (1, 1)-forms on the fibration. We establish the sharp lower bounds on the latter functional and give some applications towards an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.

10.00–10.50 Xu Wang, Norwegian University of Science and Technology: Ross-Witt Nyström correspondence and Ohsawa-Takegoshi extension

Abstract: (Joint work with Yan He and Johannes Testorf) We obtain a general Ohsawa-Takegoshi extension theorem by using the Ross-Witt Nyström correspondence for deformation to the normal cones. In particular, our result gives a sharp lower bound of the Bergman kernel in terms of Witt Nyström's canonical growth condition for ample line bundles.

11.15–12.05 Masafumi Hattori, Kyoto University: Positivity of CM line bundles on K-moduli of Calabi-Yau fibrations over curves

Abstract: CM line bundle is the line bundle canonically defined on the family of polarized varieties, which was named by Paul-Tian and related to K-stability and the generalized Weil-Petersson metric. Reflecting the works of Fujiki-Schumacher, Kollár-Shepherd-Barron and Alexeev, Odaka predicted that there would exist a quasi-projective moduli parametrizing K-polystable objects and it would admit the ample CM line bundle. In recent years, this conjecture was solved by in many cases including log Fano pairs (cf. Xu-Zhuang) and cscK manifolds (Dervan-Naumann). In this talk, we would like to explain the notion of uniform adiabatic K-stability and the result that Hashizume and I constructed the moduli space of uniform adiabatic K-stable Calabi-Yau fibrations over curves and positivity of the CM line bundle on all proper subspaces. If time permits, I would like to explain the ongoing work with Kenta Hashizume for entire quasi-projectivity of our moduli space.

Thursday August 29

10.00–10.50 Vlassis Mastrantonis, University of Maryland: A convex-complex approach to Mahler’s Conjectures

Abstract: By applying techniques from convex geometry, we establish shard bounds on the Bergman kernels of tube domains.

In 2012, Nazarov discovered that the Mahler volume of a convex body K is bounded from below by the Bergman kernel of the tube domain over K. He suggested that finding an optimal lower bound on such Bergman kernels could lead to the resolution of the celebrated Mahler’s conjecture from the 1930s. In 2014, Błocki conjectured that this optimal bound is achieved by a cube. We confirm Błocki’s conjecture in dimension n=2, using techniques inspired by shadow systems in convex geometry with a twist to handle the technically challenging nature of the Bergman kernel.

This work is part of our broader program to combine convex and complex techniques to tackle the Mahler Conjectures. We discuss how Nazarov’s and Błocki’s methods do not directly address the original Mahler conjectures, but rather lead to a “L1-Mahler conjecture”. We then explain how this can be modified to give a new convex-complex approach to the original Mahler conjecture.

Partly based on joint work with B. Berndtsson and Y. Rubinstein.

11.15–12.05 Lucie Devey, University of Edinburgh: Stability of toric vector bundles

Abstract: Stability has been introduced in order to classify vector bundles. What about the toric case? Every toric object has a combinatorial description, a toric variety has its fan; a toric divisor has its moment polytope; a toric vector bundle has its parliament of polytopes. The stability of a toric vector bundle has to be encoded combinatorially... In this talk, we explore toric geometry and explain a visualization and an algorithm to check stability of toric bundles.

14.00–14.50 Carl Tipler, Université de Bretagne Occidentale: Foldable fans and cscK surfaces

Abstract: We study the moduli space of constant scalar curvature Kähler surfaces around the toric ones. To this aim, we introduce the class of foldable surfaces : smooth toric surfaces whose lattice automorphism group contain a non trivial cyclic subgroup. We classify such surfaces and show that they all admit a constant scalar curvature Kähler metric (cscK metric). We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modeled on a finite quotient of a toric affine variety with terminal singularities.

15.20–16.10 Prakhar Gupta, University of Maryland: Complete Geodesic Metrics on Finite Energy Space of Big Cohomology Classes

Abstract: Complete geodesic metrics on finite energy spaces have played an important role in the study of canonical metrics on compact Kähler manifolds. In this talk, I'll describe how to construct the complete geodesic metrics on the finite energy spaces for big cohomology classes. I'll also describe certain negative curvature properties of these metric spaces, and use them to construct a complete geodesic metric on the space of finite energy geodesic rays.

Friday August 30

10.00–10.50 Tristan Collins, University of Toronto: Complete Calabi-Yau metrics, optimal transport and free boundaries

Abstract: I will describe progress towards the construction of complete Calabi-Yau metrics on the complement of ample, simple normal crossings anti-canonical divisors. This talk will discuss joint works with Y. Li, and F. Tong and S.-T. Yau.

11.15–12.05 Carlo Scarpa, Université du Québec à Montréal: Z-critical equations on projective surfaces

Abstract: The Z-critical equations are geometric PDEs for a Hermitian metric on a holomorphic vector bundle, proposed by Dervan, McCarthy, and Sektnan as a possible differential-geometric counterpart of Bridgeland stability conditions. A notable special case is the deformed Hermitian Yang-Mills equation. In this talk, I will present some results in joint work with Julien Keller, arXiv:2405.03312[math.DG]. Specifically, I will show that any bundle over a projective surface that admits a Z-positive, Z-critical Hermitian metric must satisfy some algebraic stability conditions. Time permitting, we will discuss a possible converse to this result, along the lines of the Kobayashi-Hitchin correspondence between Hermitian-Einstein metrics and Mumford-stable bundles.