Complex Geometry Seminar

This is the homepage of a complex geometry seminar organized by Junyan Cao, Ya Deng and Mingchen Xia. We aim to provide a worldwide platform of online discussion for working mathematicians in complex geometry, algebraic geometry and related fields in mathematics and physics.

For videos of our talks, see our YouTube channel.

This page is currently maintained by Ya Deng (email: deng at ihes dot fr) and Mingchen Xia (email: xiam at chalmers dot se).

If you want to recieve announcements of this seminar, please send an email to Mingchen.

Next talk

• Title: Coherent sheaves, superconnection, and R.R.G.

Speaker: Shu Shen (Sorbonne university)

Time: Mar 9, 10:30--11:30 (CET)

Abstract: In this talk, I will explain a construction of Chern character for coherent sheaves on a closed complex manifold with values in Bott-Chern cohomology. I will also show a corresponding Riemann-Roch-Grothendieck formula, which holds for general holomorphic maps between closed non-Kahler manifolds. This is a joint work with J.-M. Bismut and Z. Wei.

Zoom link: https://chalmers.zoom.us/j/64730780014

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Past talks

• Title: Plurisubharmonic geodesics in non-Archimedean geometry

Speaker: Rémi Reboulet (Institut Fourier)

Time: Feb 9, 9:30--10:30 (CET)

Abstract: We study the space of finite-energy plurisubharmonic metrics on the Berkovich analytification of an ample line bundle on a variety over a non-Archimedean field. We discuss the construction of plurisubharmonic geodesics in this space, in parallel with classical results in the complex setting.

• Title: Convergence of Bergman measure on a family of Riemann surfaces

Speaker: Sanal Shivaprasad (University of Michigan)

Time: Jan 12, 15:30--17:00 (CET)

Abstract: We study how the Bergman measure behaves in a degenerating family of Riemann surfaces. We show that the measure converges, in a suitable sense, to a measure on a non-Archimedean space (in the sense of Berkovich). No knowledge of non-Archimedean geometry will be assumed.

• Title: Invariance of plurigenera and Chow-type lemma

Speaker: Sheng Rao (Wuhan University)

Time: Dec 1, 9:30--10:30 (CET)

Abstract: I will report my recent joint work with I-Hsun Tsai, which answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only canonical singularities and uncountable ones therein of general type and also two Chow-type lemmata on the structure of family of projective varieties.

• Title: On projective manifolds with pseudo-effective tangent bundle

Speaker: Shin-ichi Matsumura (Tohoku University)

Time: Nov 24, 9:30--10:30 (CET)

Abstract: In this talk, I would like to discuss projective manifolds whose tangent bundle is pseudo-effective or admits a positively curved singular metric. I will explain a structure theorem for such manifolds and the classification in the two-dimensional case, comparing our theory with classical results for nef tangent bundle or non-negative bisectional curvature. Related open problems will be discussed if time permits. This is joint work with Genki Hosono (Tohoku University) and Masataka Iwai (Osaka City University).

YouTube Link: https://www.youtube.com/watch?v=ryehJlYDVxc

• Title: On CM points away from the Torelli locus

Speaker: Ke Chen (Nanjing University)

Time: Nov 17, 9:30--11:00 (CET)

Abstract: The Coleman-Oort conjecture predicts that up to isomorphism there are at most finitely many CM Jacobians for smooth projective curves of genus g, when g is large enough. We show that this is true when suitable conditions are put on the CM type, and discuss its connection with the Ekedahl-Serre problem. This is a joint work with X. Lu and K. Zuo.

• Title: Reflexive sheaves, Hermitian-Yang-Mills connections, and tangent cones

Speaker: Xuemiao Chen (University of Maryland)

Time: 16:00--17:00 (CET) November 2, 2020

Abstract: The Donaldson-Uhlenbeck-Yau theorem confirms the existence of a Hermitian-Yang-Mills connection on a given slope stable holomorphic vector over a compact Kahler manifold. Later, Bando and Siu generalized this to stable reflexive sheaves by using natural singular Hermitian-Yang-Mills connections. In this talk, I will talk about how to characterize the analytic tangent cones of a singular Hermitian-Yang-Mills connection by using certain new local complex geometric data extracted from the corresponding reflexive sheaf. (Based on joint works with Song Sun.)

• Title: On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

Speaker: Takayuki Koike (Osaka city university)

Time: 9:30--10:30 (CET) October 27, 2020

Abstract: We investigate the complex analytic structure of the complement of a non-singular hypersurface with unitary flat normal bundle when the corresponding line bundle admits a Hermitian metric with semipositive curvature.

• Title: The Coleman-Oort conjecture on the finiteness of CM curves

Speaker: Xin Lv (East China Normal University)

Time: 9:30--10:30 (CET) October 20, 2020

Abstract: An algebraic curve is called CM if its Jacobian is an abelian variety with complex multiplication. The Coleman-Oort conjecture predicts that the number of CM algebraic curves of a fixed sufficiently high genus, up to isomorphism, is finite. In this talk, I will give a brief introduction to this conjecture, and report its recent progress.

YouTube link: https://www.youtube.com/watch?v=Ro7UcgOQsoQ

• Title: Special subvarieties of non arithmetic ball quotients

Speaker: Gregorio Baldi (IHES)

Time: 9:30--11:00 (CET) October 6, 2020

Abstract: We study complex hyperbolic lattices and prove that, if the associated ball quotient contains infinitely many maximal totally geodesic subvarieties, then the lattice is arithmetic. The idea is to realise such quasi-projective varieties inside a period domain for polarised integral Hodge structures and interpret totally geodesic subvarieties as unlikely intersections. Our theorem is indeed a special case of Klingler’s generalised Zilber-Pink conjecture. This is joint work with Emmanuel Ullmo.

• Title: Adiabatic limits of ASD connections on K3 surfaces

Speaker: Yuguang Zhang (Leibniz University Hannover)

Time: 9:30--10:30 (CET) September 29, 2020

Abstract: In this talk, we will study the limit behaviour of anti-self-dual SU(n)-connections on elliptic fibred K3 surfaces, and show that limits are the Fourier-Mukai transforms of holomorphic curves equipped with flat U(1)-connections.

YouTube link: https://www.youtube.com/watch?v=tUN29hfiRr0

• Title: Stability thresholds and canonical metrics

Speaker: Kewei Zhang (Beijing University)

Time:10:00--11:00 (CET) September 21, 2020.

Abstract: In this talk I will discuss several aspects of Fujita--Odaka's delta invariant in the study of Kahler--Einstein problem. Then I will focus on the quantization approach. In particular, I will show that the quantized delta invariant is the coercivity threshold of the quantized Ding energy, which characterizes the existence of balanced metrics. Our approach also works other canonical metrics in Kahler geometry, e.g. Kahler Ricci solitons and coupled soliton metrics. This talk is based on my recent joint work with Rubinstein and Tian.

YouTube link: https://www.youtube.com/watch?v=uAbc907ztcw&t=1449s

Notes: https://drive.google.com/file/d/18SKYYHF2qEgeuf5aKNjrlpW6-v4EnwIG/view?usp=sharing

• Title: Currents with minimal singularities in nef (1,1) classes

Speaker: Valentino Tosatti (Northwestern University)

Time: 16:00--17:00 (CET) July 27, 2020.

Abstract: Every pseudoeffective (1,1) class on a compact Kahler manifold contains a closed positive current with minimal singularities, as was observed some time ago by Demailly-Peternell-Schneider. These currents are in general singular, and understanding precisely how singular they are is an important question with connections to many other topics. I will discuss the case when the (1,1) class is nef, which arises naturally when taking the limit of Kahler metrics satisfying geometric PDEs (e.g. Ricci-flat, Ricci flow, etc), discuss some natural questions and present some recent results on K3 surfaces, joint with Simion Filip.

YouTube link: https://www.youtube.com/watch?v=KlKKAIF-F2U

Notes: https://drive.google.com/file/d/1XbxUzSJgD1E3lp-QMp6qiCfqnzKQWiDD/view?usp=sharing

• Title: Canonical metrics on holomorphic submersions

Speaker: Ruadhaí Dervan (University of Cambridge)

Time: 10:00--11:00 (CET) July 20, 2020

Abstract: Constant scalar curvature Kähler (cscK) metrics can be viewed as canonical metrics on Kähler manifolds, in the sense that they are unique, up to biholomorphisms, when they exist. On a holomorphic submersion, when the fibres have large biholomorphism group, a choice of fibrewise cscK metric is not unique. I will describe a new notion of a canonical relatively Kähler metric on a holomorphic submersion, which is a fibrwise cscK metric satisfying an additional PDE called the optimal symplectic connection equation. One can view Hermite-Einstein metrics as a special case. I will then describe some foundational aspects of optimal symplectic connections, for example their uniqueness and the link with an algebro-geometric notion of stability. This is joint work with Lars Sektnan.

YouTube link: https://www.youtube.com/watch?v=biZbe96OK6Q

• Title: Minimizing normalized volume

Speaker: Chenyang Xu (MIT)

Time: 15:00--16:00 (CET) July 6, 2020

Abstract: For klt singularities, the normalized volume function was defined by Chi Li about five years ago. Investigating the geometry of its minimizer provides a local model for K-stability type problems of Fano varieties. One guiding question is the Stable Degeneration Conjecture, by now a large part of which has been settled but one central piece is still open.

In this talk, I will present the progress of the Stable Degeneration Conjecture, which builds on the machinery of higher dimensional geometry. If time permits, I will also discuss some applications to the global questions of K-stability of Fano varieties.

YouTube link: https://www.youtube.com/watch?v=gEy3G-7Kmec&t=3s

Notes: https://drive.google.com/file/d/1WRsNsx5gkGEqP57Fz4rQ2TRm4iQlTtQ5/view?usp=sharing

• Title: Recent progress on the Yau-Tian-Donaldson conjecture

Speaker: Sébastien Boucksom (École Polytechnique)

Time: 10:30--11:30 (CET) June 29, 2020

Abstract: One version of the Yau-Tian-Donaldson (YTD) conjecture states the equivalence between the existence of a constant scalar curvature Kähler (cscK) metric on a polarized complex projective manifold and an algebro-geometric condition known as K-stability. On the analytical side of the story, tremendous progress was recently accomplished by Chen and Cheng, reducing the existence of a cscK metric to the study of the K-energy functional along geodesic rays in a space of finite energy metrics. On the algebraic side, K-stability admits a formulation involving non-Archimedean pluripotential theory, and very recent new insights of Chi Li, based in part on joint work of mine with Berman and Jonsson, gets very close to a complete proof of the conjecture. The purpose of this talk is to review these new developments.

YouTube link: https://www.youtube.com/watch?v=jNyOtwpXDZY&list=PL_CYAV-d9z897oN_OJ9_sDX74KBdyi8WH&index=8&t=1602s

• Title: Finite entropy vs Finite energy (Joint work with Eleonora Di Nezza and Vincent Guedj)

Speaker: Hoang-Chinh Lu(Université Paris-Saclay)

Time: 10:00--11:00 (CET) June 15, 2020

Abstract: Probability measures with either finite Monge-Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry.

We make a systematic study of quasi-plurisubharmonic potentials on a $n$-dimensional compact Kähler manifold whose Monge-Ampère measures have finite entropy.

We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.

YouTube link: https://www.youtube.com/watch?v=R331FqZOrZE

• Title: Variation of Kähler-Einstein metrics

Speaker: Henri Guenancia (Institut de Mathématiques de Toulouse Université Paul Sabatier)

Time: 10:00--11:00 (CET) June 8, 2020

Abstract: Let f:X\to Y be a projective fibration between Kähler manifolds with general fiber F. I will explain the following results:

1/ If K_F is big, then the relative singular Kähler-Einstein metric induces a positive current on X.

2/ If K_F is numerically trivial, the relative Ricci-flat metric needs not be semipositive.

This is joint work with Junyan Cao and Mihai Paun.

Notes: https://drive.google.com/file/d/1PQYm31UgFNd68-PESRvSx-WDeMiQ7JHx/view?usp=sharing

YouTube link: https://www.youtube.com/watch?v=X80gRpt1kYU&t=24s

• Title: Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles

Speaker: Jean-Pierre Demailly (Université Grenoble Alpes)

Time: 10:15--11:15 (CET) June 2, 2020

Abstract: Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed s o that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths-and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled.

• Title: Albanese maps and fundamental groups of varieties with many rational points over function fields

Speaker: Ariyan Javanpeykar(Johannes Gutenberg-Universität)

Time: 10:00--11:00 (CET) May 26, 2020

Abstract:

In this talk we will discuss topological properties of varieties with many rational points over a function field, and present joint work-in-progress with Erwan Rousseau. More precisely, we define a smooth projective variety X over the complex numbers to be geometrically-special if there is a dense set of closed points S in X such that, for every x in S, there is a pointed curve (C,c) and a sequence of morphisms (C,c)->(X,x) which covers C x X, i.e., the union of their graphs is Zariski-dense in C x X. Roughly speaking, a variety is geometrically-special if it satisfies density of "pointed" rational points over some function field. Inspired by conjectures of Campana on special varieties and Lang on hyperbolic varieties, we prove that every linear quotient of the fundamental group pi_1(X) of such a variety is virtually abelian.

Link to notes for an earlier talk: click here.

• Title: Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Speaker: Wenhao Ou (Chinese Academy of Science)

Time: 9:30--11:00 (CET) May 18, 2020

YouTube Link: https://www.youtube.com/watch?v=EmEvEP2RFVM&t=4085s

Abstract: After a theorem of Andreatta and Wisniewski, if the tangent bundle of a projective manifold $X$ contains an ample subsheaf, then $X$ is isomorphic to the projective space. We show that, if the tangent bundle contains a strictly nef subsheaf, then X is a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group of $X$ is virtually abelian, then $X$ is isomorphic to a projective space. This is joint with Jie Liu and Xiaokui Yang.

• Title: Limits of Hodge structures Part II (after Steenbrink)

Speaker: Feng Hao (KU Leuven)

Time: 9:30--11:00 (CET) May 4, 2020

YouTube Link: Missing

Abstract: For a family projective varieties degenerating to a singular fiber over a disc, a limit of pure Hodge structures of general fibers exists as a mixed Hodge when general fibers approach to the singular fiber. The existence of the limit is first given by Schmid in his celebrated paper “Variation of Hodge Structure: The Singularities of the Period Mapping”. There are many applications of the existence of limit mixed Hodge structures in the study of singular fibers of degenerations, compactification of moduli spaces, milnor fibers associated to isolated singular points, cycle theory, etc. In this learning seminar, I will intoduce the algebraic construction of limit mixed Hodge structures given by Steenbrink. The weight filtration and Hodge filtration are defined over a double complex, which resolves the cohomology of nearby fiber. Also, I will cover some basic properties of the limit mixed Hodge structure, and the integral structure via log structures.

References: 1. Steenbrink, Joseph. "Limits of Hodge structures." Inventiones mathematicae 31.3 (1976): 229-257.

2. Steenbrink, Joseph. "Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures." Mathematische Annalen 301.1 (1995): 105-118

• Title: Limits of Hodge structures Part I (after Steenbrink)

Speaker: Feng Hao (KU Leuven)

Time: 9:30--11:00 (CET) Apr 27, 2020

Notes: https://drive.google.com/open?id=1z6SbjXGs5pT8BUok0mQgcPWHvhkHUqqb

YouTube link: https://www.youtube.com/watch?v=f1tq1wePyXs

Abstract: For a family projective varieties degenerating to a singular fiber over a disc, a limit of pure Hodge structures of general fibers exists as a mixed Hodge when general fibers approach to the singular fiber. The existence of the limit is first given by Schmid in his celebrated paper “Variation of Hodge Structure: The Singularities of the Period Mapping”. There are many applications of the existence of limit mixed Hodge structures in the study of singular fibers of degenerations, compactification of moduli spaces, milnor fibers associated to isolated singular points, cycle theory, etc. In this learning seminar, I will intoduce the algebraic construction of limit mixed Hodge structures given by Steenbrink. The weight filtration and Hodge filtration are defined over a double complex, which resolves the cohomology of nearby fiber. Also, I will cover some basic properties of the limit mixed Hodge structure, and the integral structure via log structures.

References: 1. Steenbrink, Joseph. "Limits of Hodge structures." Inventiones mathematicae 31.3 (1976): 229-257.

2. Steenbrink, Joseph. "Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures." Mathematische Annalen 301.1 (1995): 105-118

• Title: O-minimality and its applications Part II. (after Pila--Zannier, Bakker--Brunebarbe--Klingler--Tsimerman)

Speaker: Jiaming Chen (Humboldt Universität)

Time: 10:00--11:00 (CET) Apr 20, 2020

Notes: https://drive.google.com/open?id=1XFM9QL2yV4-lVjqAn638Q17UNk5SsNuh

YouTube link: https://www.youtube.com/watch?v=g443weFNRsI&t=1972s

Abstract: O-minimal structures, originally developed by model-theorists, provide an excellent framework for developing tame topology which was prophesied by Grothendieck in his “Esquisse d’un Programme” as a way to amend the inadequacy of the foundations of general topology.

Recent applications of o-minimality has revealed its powerful capabilities in understanding some transcendental phenomena appeared in arithmetic and complex algebraic geometry. For example,

(1) it plays a crucial role, via the celebrated Pila-Wilkie counting theorem, in the Pila-Zannier’s strategy to attack the Andr ́e- Oort (more general Zilber-Pink) conjecture.

(2) it can be used to prove some global algebraic results without renouncing the local flexibility of analytic varieties, for instance, the o-minimal Chow theorem of Peterzil-Starchenko and the very recent applications in classical Hodge theory (a new proof of a fundamental theorem of Cattani-Deligne- Kaplan on the algebraicity of Hodge loci by Bakker- Klingler- Tsimerman and a resolution of the Griffiths conjecture on the quasiprojectivity of period images by Bakker-Brunebarbe- Tsimerman).

In the first talk, I will give a brief introduction to o-minimal struc- tures and outline the proof of Manin-Mumford conjecture (originally proved by Raynaud using p-adic method) by Pila-Zannier using o-minimality (after Pila-Zannier).

In the second talk, I will discuss the idea of the proof of the above- mentioned Griffiths conjecture (after Bakker, Brunebarbe, Klingler and Tsimerman).

• Title: O-minimality and its applications Part I. (after Pila--Zannier, Bakker--Brunebarbe--Klingler--Tsimerman)

Speaker: Jiaming Chen (Humboldt Universität)

Time: 10:00--11:00 (CET) Apr 13, 2020

Notes: https://drive.google.com/open?id=1zy632VQMLK28oLMcY6sZ7m4kjLAZvV31

YouTube Link: https://www.youtube.com/watch?v=bIuFcp6ppw8&t=19s

Abstract:

O-minimal structures, originally developed by model-theorists, provide an excellent framework for developing tame topology which was prophesied by Grothendieck in his “Esquisse d’un Programme” as a way to amend the inadequacy of the foundations of general topology.

Recent applications of o-minimality has revealed its powerful capabilities in understanding some transcendental phenomena appeared in arithmetic and complex algebraic geometry. For example,

(1) it plays a crucial role, via the celebrated Pila-Wilkie counting theorem, in the Pila-Zannier’s strategy to attack the Andr ́e- Oort (more general Zilber-Pink) conjecture.

(2) it can be used to prove some global algebraic results without renouncing the local flexibility of analytic varieties, for instance, the o-minimal Chow theorem of Peterzil-Starchenko and the very recent applications in classical Hodge theory (a new proof of a fundamental theorem of Cattani-Deligne- Kaplan on the algebraicity of Hodge loci by Bakker- Klingler- Tsimerman and a resolution of the Griffiths conjecture on the quasiprojectivity of period images by Bakker-Brunebarbe- Tsimerman).

In the first talk, I will give a brief introduction to o-minimal struc- tures and outline the proof of Manin-Mumford conjecture (originally proved by Raynaud using p-adic method) by Pila-Zannier using o- minimality (after Pila-Zannier).

In the second talk, I will discuss the idea of the proof of the above- mentioned Griffiths conjecture (after Bakker, Brunebarbe, Klingler and Tsimerman).

• Title: A complex analytic proof of the Bourgain--Milman theorem (after F. Nazarov)

Speaker: Jian Xiao (YMSC, Tsinghua)

Time: 10:00--11:00 (CET) Apr 6, 2020

Notes: https://drive.google.com/open?id=1qBzWyAkm9uSrEmWulHV5BBdTfOhJDipe

Youtube Link: https://www.youtube.com/watch?v=lsqblFA5S0w

• Title: Asymptotic expansion of non-Archimedean L-functionals and applications

Speaker: Mingchen Xia (CTH)

Time: 10:00--11:00 (CET) Mar 30, 2020

Notes: https://drive.google.com/open?id=1Wi5uK7j0MDo5IAtdPU2N-9HotKSSmLnV

YouTube Link: https://www.youtube.com/watch?v=vXBFPOGu0YE

Abstract:

This is a joint work with Tamás Darvas (arXiv: 2003.04818). Given a compact Kähler manifold $X$ with an ample line bundle $L$, we consider the non-Archimedean analogue of Donaldson’s $\mathcal{L}$-functionals on the space of maximal finite energy geodesic rays associated to the $c_1(L)$ in the sense of Berman—Boucksom—Jonsson. We prove that the leading order term in the asymptotic expansion of these functionals is given by the non-Archimedean Monge—Ampère energy. I will explain various applications of this result in pluripotential theory and in Kähler geometry.

Reference(s): The closures of test configurations and algebraic singularity types https://arxiv.org/abs/2003.04818