16-20 June 2025 

Higher Dimensional Geometry and Foliations

Organisers: Paolo Cascini and Calum Spicer

Registration is not required, everyone is welcome to attend.

Location: Room 340 – Huxley Building – Department of Mathematics – Imperial College London

Directions: Google Map

Speakers:

Hamid Abban

Title: K-stability of Fano 3-folds, rational points, and Condition (A)

Abstract: K-stability is an algebraic condition that detects existence of Kähler-Einstein metrics on Fano manifolds. Although several methods are in hand to check K-stability of a given Fano variety, some cases resist all attempts. In this talk I report on recent joint work with Ivan Cheltsov, Takashi Kishimoto, and Frederic Mangolte, where we examine pointlessness over subfields of the complex numbers, or failure to satisfy the so-called Condition (A), for smooth Fano 3-folds to make conclusions about their K-stability.

André Belotto

Title: Resolution of foliations via principalization

Abstract: I will discuss resolution of singularities for foliations and present our approach via a weighted principalization theorem for ideals on smooth orbifolds equipped with a foliation. As an application, I will describe the resolution of certain foliations in arbitrary dimensions, including Darboux totally integrable foliations. This is joint work with D. Abramovich, M. Temkin, and J. Wlodarczyk.

Fabio Bernasconi

Title: A strong counterexample to the log canonical Beauville–Bogomolov decomposition

Abstract: In recent years, the work of Druel, Guenancia, Greb, Kebekus, Höring, and Peternell has shown that an analogue of the BB decomposition holds for $K$-trivial varieties with klt singularities over the complex numbers. Motivated by the MMP and the study of open varieties, it is natural to ask whether the BB decomposition can be extended to the log canonical setting. In this talk, I will give a negative answer by constructing, for each integer $d > 3$, a $K$-trivial log canonical variety of dimension $d$ that does not admit a BB-type decomposition. More precisely, its Albanese morphism is not birationally isotrivial (so the abelian part cannot be split after an étale cover), and its tangent sheaf is not polystable. This is joint work with S. Filipazzi, Zs. Patakfalvi, N. Tsakanikas, and N. Müller.

Junyan Cao

Title: The positivity of the relative canonical bundle in the Kähler setting and applications

Abstract: Let p: X → Y be a fibration between two compact Kähler manifolds, and assume that the canonical bundle of the generic fiber is pseudo-effective. A conjecture states that the relative canonical bundle K_X/Y is also pseudo-effective. In the first part, we will discuss progress toward this conjecture. In the second part, we will present an application to the algebraicity of holomorphic foliations with positive minimal slope. The proof relies on the positivity of the relative canonical bundle and a recent breakthrough due to Wenhao Ou on the algebraic criterion in the Kähler setting. This work is joint with Mihai Paun.

Giulio Codogni

Title: Positivity of Hodge bundles and applications – Part I.

Abstract: Given a family f: X \to T, the positivity properties of the direct images of relative pluricanonical sheaves -known as Hodge bundles- play a fundamental role in moduli theory and birational geometry, and are intimately related to the stability properties of the family. In these talks, we will present several effective positivity results for Hodge bundles and discuss how they can be combined with higher-dimensional generalizations of the Xiao–Cornalba–Harris slope inequality. These techniques lead to new lower bounds for the volume of algebraically integrable foliations. Additional applications include results on the volume and the structure of the ample cone of KSBA moduli spaces, as well as uniform bounds on the automorphism groups of KSBA fibrations. The talk is based on joint works with F. Viviani and Zs. Patakfalvi.

Simon Donaldson

Title: Exceptional holonomy and complex geometry.

Abstract: The talk will be a survey of various connections between complex algebraic geometry and the study of Riemannian manifolds of exceptional holonomy. Most  of the discussion will involve 7-dimensional manifolds with the exceptional holonomy group G_2.  We will review the twisted connected sum construction of Kovalev and Corti et al., based on Fano and semi-Fano threefolds,  and a more recent construction of Joyce and Karigiannis,  based on Calabi-Yau threefolds with real structure. We will discuss some known singularity models and “folklore conjectures” about the interactions of these. In the last part of the talk we will outline the role of multivalued harmonic 1-forms in this area, the construction of  model solutions using complex geometry methods (twistor theory) and recent work of Dashen Yan.

Christopher Hacon

Title: Recent progress in the Kähler minimal model program

Anne Sophie Kaloghiros

Title: K-polystable degenerations of prime Fano threefolds of genus 12.

Abstract: In the past couple of years, many components of K-moduli spaces parameterising K-polystable Fano threefolds with a smoothing in one of the 105 deformation families classified by Mori, Mukai and Iskovskikh have been explicitly described. In many cases, we now know which smooth Fano threefolds in the family are K-polystable and have some geometric information on K-(poly/ semi)stable degenerations.The case of prime Fano threefolds of genus 12 remains mysterious – we don’t even know which smooth prime Fano 3-folds are K-polystable (Donaldson’s conjecture). The associated component of the K-moduli space is 6-dimensional. In this talk, I will show that general one-nodal prime Fano threefolds of genus 12 are K-polystable. Prokhorov showed that there are 4 families of one-nodal prime Fano threefolds of genus 12 , and I will show that these 4 families correspond to 4 boundary components of the associated K-moduli.This is joint work with Elena Denisova.

Stefan Kebekus

Title: Irregularities and Albanese constructions within Campana’s theory of C-pairs

Abstract: We address the problem of constructing Albanese maps within Campana’s theory of C-pairs (or “geometric orbifolds”). A first application extends the fundamental theorem of Bloch–Ochiai to the context of C-pairs. The proof builds on work of Kawamata, Ueno, and Noguchi, recasting parabolic Nevanlinna theory as a “Nevanlinna theory for C-pairs” and might be of independent interest.

Jihao Liu

Title: Adjoint Foliated Structures in Birational Geometry

Abstract: The birational geometry of foliations differs significantly from that of projective varieties. Many classical results, such as Bertini-type theorems, finite generation of the canonical ring, abundance, effective birationality, and boundedness properties, fail to hold for foliations in general. To address these challenges, we study adjoint foliated structures, a notion first introduced by Pereira and Svaldi (2019) in their study of birational invariants for foliations on projective surfaces. In this talk, I will explain why many classical theorems, though invalid for arbitrary foliations, are expected to remain true for adjoint foliated structures. In particular, I will discuss my recent joint works with Cascini, Han, Meng, Spicer, Svaldi, and Xie on the finite generation of the canonical ring and boundedness results for algebraically integrable adjoint foliated structures. Finally, I will outline potential future directions in the study of this new structure in birational geometry.

Michael McQuillan

Title: Modular variation of leaves

Abstract: On the complement U of the singularities of a foliation in curves, F, its holonomy groupoid, in the analytic topology, affords a map U->[U/F] exhibiting U as a fibration in Riemann surfaces, whose modular variation has sense. Indeed if U is a surface, then the foliations of general type are exactly
those with hyperbolic leaves and modular variation. Better still the modular variation even takes place on a set of positive Lebesgue measure in U.

Maria Pe Pereira

Title: A theory to study metric degenerations: Moderately Discontinuous Algebraic Topology

Abstract: In the same way algebraic topology gives a language to talk about properties of topological spaces up to homeomorphism (more precisely, up to homotopy), we give a theory to talk about metric degenerations where the metric and dynamical information matters.
In the works [1] and [2] we developed a theory that applies to analytic/algebraic germs and that gives analytic invariants. First examples are the computation for plane curve singularities, for which the invariant gives the Puiseux exponents, or for normal surface singularities, from which one can read information about a concrete resolution of singularities that codifies the inner metric up to bilipschitz homeomorphism.
In the setting of [1] and [2] we impose a subanalytic hypothesis (which is roughly speaking as asking the spaces and mappings to be triangulable). In a work in progress we enlarge the framework of the theory to continuous families of metric spaces. I will report on that. In particular this would allow the theory to express properties of very general degenerations of metric spaces, such as degenerations of riemannian metrics.
[1] with J. Fernandez de Bobadilla, S. Heinze, E. Sampaio, Moderately Discontinuous Homology,
Communications on Pure and Applied Mathematics, volume 75.
https://doi.org/10.1002/cpa.22013 Also available in arXiv:1910.12552v3
[2] with J. Fernandez de Bobadilla and S. Heinze. Moderately Discontinuous Homotopy. International Mathematics Research Notices, Volume 2022,
https://doi.org/10.1093/imrn/rnab225 Available in ArXiv:2007.01538.

Jorge Pereira

Title: Numerically flat foliations 

Abstract: I will report on joint work with Druel, Pym, and Touzet on the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles

Quentin Posva

Title: Some recent developments on the singularities of 1-foliations in positive characteristic

Abstract: 1-foliations are the naive analogue of complex foliations in positive characteristic. Despite the similarity in the definitions, 1-foliations behave quite differently from complex foliations: they are intimately linked with inseparable morphisms, and lead to many surprising, pathological examples. In this talk, I will report on some recent results of mine on the singularities of 1-foliations. I will explain how, building on the results of Giraud and Cossart on resolution of singularities, one can resolve 1-foliations in dimensions up to three. I will also present a way to understand the MMP singularities of infinitesimal quotients arising from 1-foliations, and show pathological examples of terminal singularities constructed in this way.

Helena Reis

Title: On singularity analysis and global dynamics of uniformizable vector fields

Abstract: Holomorphic/meromorphic vector fields all of whose solutions are univalued as functions of the time T \in C, (aka semicomplete vector fields) are somehow rare but they tend to be associated with remarkable situations not only in Mathematics but also in Physics. In a sense, they are akin – albeit not identical – to vector fields possessing the “Painleve property” which, over the years, have been more popular especially among physicists. However, in the past 20 years, the interest in semicomplete vector fields has steadily grown and, more recently, significant attention has been paid to the problem of finding examples of semicomplete vector fields exhibiting complicated dynamical behavior. I will discuss this question and report on some recent joint work with J. Rebelo and L. Rosales-Ortiz.

Calum Spicer

Title: Moduli of Foliations

Abstract: We will survey some approaches to constructing moduli of foliations and then turn to a description of a KSB approach to constructing (compact) moduli spaces of foliations.  Our main result is the existence of a proper moduli space for surface foliations with prescribed invariants.  Time permitting we will also explain some applications of these ideas.  This is part of ongoing work in progress which is joint work with M. McQuillan, R. Svaldi and S. Velazquez.

Behrouz Taji

Title: Negativity in the direct image of relative anti-canonical sheaf in families of Fano varieties

Abstract: We know that positivity or negativity properties of the canonical line bundle encode a significant amount of geometric data about the underlying projective variety. It is therefore unsurprising to expect that the same should be true for the relative canonical divisor in families of projective varieties. For projective families of varieties whose canonical divisor is ample (canonically polarized) or numerically trivial (Calabi-Yau), important positivity properties of the pushforward of the relative (pluri)canonical was discovered by Fujita, Kawamata, Kollár and Viehweg. For families of Fano varieties however less is known. In this talk I will discuss how one can complement some of these classical results in the Fano case, with a view towards alternative constructions of the moduli of K-stable varieties. This is based on ongoing joint work with Sándor Kovács.

Luca Tasin

Title: Positivity of Hodge bundles and applications – Part II.

Abstract: Given a family f: X \to T, the positivity properties of the direct images of relative pluricanonical sheaves -known as Hodge bundles- play a fundamental role in moduli theory and birational geometry, and are intimately related to the stability properties of the family. In these talks, we will present several effective positivity results for Hodge bundles and discuss how they can be combined with higher-dimensional generalizations of the Xiao–Cornalba–Harris slope inequality. These techniques lead to new lower bounds for the volume of algebraically integrable foliations. Additional applications include results on the volume and the structure of the ample cone of KSBA moduli spaces, as well as uniform bounds on the automorphism groups of KSBA fibrations. The talk is based on joint works with F. Viviani and Zs. Patakfalvi.

Frédéric Touzet

Title: Foliations with small singular set in positive characteristic

Abstract: In this talk, we will address several elementary specificities of foliations in positive characteristic, trying to highlight the similarities and differences with the characteristic-0 case. This will notably include Bott’s vanishing theorem for regular foliations, regular foliations on rational surfaces, Camacho-Sad index, and foliations with “few” singularities on invariant subvarieties. This is a work in progress with T. Fassarella, W. Mendson, and J. P. dos Santos.

Sebastian Velazquez

Title: Extensions of foliations

Abstract: Let $\mathcal{F}$ be a foliation on a smooth variety $X$ and let $X \hookrightarrow Y$ be a regular embedding. A natural question is whether $\mathcal{F}$ extends to a foliation on $Y$, and if so, in how many ways. In this talk, we will focus on the case where $\codim(\mathcal{F}) = \codim(X,Y) = 1$, paying special attention to the cases where $X$ is an ample divisor or the central fibre of a deformation. We will present sufficient conditions that guarantee the existence of (unique) extensions, and we will also show that in some cases, the existence of extensions imposes strong constraints on the ambient geometry, such as the existence of tubular neighbourhoods. This is joint work with P. Perrella.

Ziquan Zhuang

Title: Boundedness in general type MMP

Abstract: I will explain how local volumes (an invariant related to K-stability) can be used to show that in any general type MMP, the minimal log discrepancy of singularities takes only finitely many values, and the fibers of all flipping contractions and flips fall into finitely many deformation types. An application is the effective termination of fivefold general type MMP. This is based on joint work with Jingjun Han, Jihao Liu and Lu Qi.