# Joint Seminar on Complex Algebraic Geometry and Complex Analysis (Bochum - Essen - Köln - Wuppertal)

Photo by Max Greve and Timo Bobert

# The Seminar

The "Joint Seminar on Complex Algebraic Geometry and Complex Analysis" is a research seminar, organised by the algebraic and complex geometry research groups in Bochum, Essen, Cologne, and Wuppertal. The seminar meets roughly twice per semester for a full day in Bochum, Essen, Cologne, or Wuppertal. There are three talks per meeting, both by invited guests and by speakers from the organising universities. We aim to leave ample room for discussions.

The talks are open for everyone. Contact one of the organisers if you are interested in attending the meeting. We have some funds that might help to support travel for some junior participants.

Meetings of the seminar are partially supported by the DFG via

• GRK 2553 "Symmetries and Classifying Spaces - analytic, arithmetic, and derived" (Essen)
• SFB/TR 191 "Symplectic structures in geometry, algebra and dynamics" (Bochum-Cologne)

# Next Meeting

### DATE AND PLACE

The next meeting will take place on Friday, May 7th, 2021 in a virtual format, hosted by Wuppertal.

### TALKS

• Siarhei Finski (Grenoble): On Monge-Ampère volumes of direct images

Abstract:
Recently, Demailly introduced an elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor of a vector bundle. This system is designed so that existence of a solution to it implies existence of a dual Nakano positive metric on the vector bundle. This has motivated the study of Monge-Ampère volumes for vector bundles, defined analogously to the volumes of line bundles. In this talk we will discuss the asymptotic properties of Monge-Ampère volumes of direct images associated with high tensor powers of an ample line bundle, and explain how in the special case of high symmetric powers of ample vector bundles, this provides a criteria of projective flatness. The theory of Toeplitz operators and Szegő-type asymptotic theorems plays a fundamental role in our approach.

• Laszlo Lempert (Purdue): Asymptotics of Hermitian--Yang--Mills metrics (after Kuang--Ru Wu)

Abstract:
Consider a compact complex manifold X and a holomorphic line bundle L over it, endowed with a positively curved hermitian metric. An old theme, going back to Hilbert, is to investigate asymptotic properties of the space of holomorphic sections of the line bundles $L^k$ as $k$ tends to infinity. Hilbert computed the dimension of the space of sections, but in the past 50 years more refined geometrical questions have been asked and answered. Following and generalizing some of these developments, Wu considers so called Hermitian--Yang--Mills metrics associated with the spaces of sections, and determines their limit as k tends to infinity. The talk will introduce the relevant notions, discuss motivation, formulate Wu's theorem and, if all goes well, indicate the approach to the proof.

• Katharina Neusser (Masaryk University): Cone structures and parabolic geometries

Abstract:
(based on a joint work with J.-M. Hwang)
A cone structure on a complex manifold M is a closed submanifold C of the projectivized tangent bundle of M which is submersive over M. Such structures arise naturally in differential and algebraic geometry, and when they do, they are typically equipped with a conic connection that specifies a distinguished family of curves on M in direction of C. In differential geometry, as we will see, cone structures with conic connections arise from so-called holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure with the conic connection given by the null-geodesics. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. In this talk we will discuss various examples of cone structures and will introduce two important invariants for conic connections. As an application of the study of these invariants, we obtain a local-differential-geometric version of the global algebraic-geometric recognition theorem of Mok and Hong–Hwang, which recognizes certain rational homogeneous spaces from their VMRT-structures. In our local version, the role of rational curves is completely replaced by appropriate torsion conditions on the conic connection.

### SCHEDULE

10:15 - 11:15 Finski

11:30 - 12:30 Neusser

afterwards Lunch break

14:00 - 15:00 Lempert

### VENUE

Talks will take place using ZOOM. Access data will be distributed via the usual channels. In case you don't have the information and want to participate, please write an eMail to one of the organisers in order to gain access.

# Organisers

Valdemar Tsanov
Ruhr-Universität Bochum
Valdemar [dot] Tsanov [at] ruhr-uni-bochum [dot] de

Daniel Greb
Universität Duisburg-Essen
daniel [dot] greb [at] uni-due [dot] de

George Marinescu
Universität zu Köln
gmarines [at] math [dot] uni-koeln [dot] de

Jean Ruppenthal
Bergische Universität Wuppertal
jean [dot] ruppenthal [at] math [dot] uni-wuppertal [dot]de

# Past Meetings

26.4.2013, Bochum: Tim Kirschner, Christian Miebach, Henrik Seppänen

28.6.2013, Wuppertal: Rafael Andrist, Immanuel Stampfli, Matei Toma

22.11.2013, Bochum: Sébastien Boucksom, Egmont Porten, Sönke Rollenske

24.1.2014, Wuppertal: Clemens Jörder, Helge Ruddat, Jörg Winkelmann

16.5.2014, Bochum: Alberto Abbondandolo, Judith Brinkschulte, Julius Ross

20.6.2014, Wuppertal: Julie Déserti, Slawomir Dinew, Slawomir Rams

24.10.2014, Bochum: Henri Guenancia, Shulim Kaliman, Adriano Tomassini

15.12.2014, Wuppertal: Richard Lärkäng, Christian Miebach, Marco Spinaci

7.5.2015, Essen: Brent Doran, Erwan Rousseau, Emanuel Scheidegger

15.6.2015, Bochum: Roya Beheshti, Daniel Greb, Hossein Raufi

27.11.2015, Wuppertal: Francesco Bei, Kai Cieliebak, Tyson Ritter

1.2.2016, Essen: Ursula Ludwig, George Marinescu, Wenhao Ou

25.4.2016, Wuppertal: Gian Maria Dall’Ara, Christian Lehn, Luca Tasin

26.6.2016, Cologne: Chin-Yu Hsiao, Stefan Nemirovski, Nikhil Savale

25.11.2016, Bochum: Ana-Maria Brecan, Antonio di Scala, Dan Popovici

27.1.2017, Essen: Tim Kirschner, Viêt Anh Nguyên, Stefan Schreieder

21.4.2017, Wuppertal: Vicente Cortés, Philipp Naumann, Markus Reineke

7.7.2017, Cologne: Roger Bielawski, Dan Coman, Martin Schwald

3.11.2017, Bochum: Junyan Cao, Sönke Rollenske, Robert Szöke

1.2.2018, Essen: Gilberto Bini, Ruadhaí Dervan, Hendrik Herrmann

27.4.2018, Wuppertal: Hugues Auvray, Bernd Stratmann, Michael Lennox Wong

6.7.2018, Cologne: Turgay Bayraktar, Jarek Buczynski, Andrew Dancer

23.11.2018, Bochum: Ben Anthes, Kevin Fritsch, Ariyan Javanpeykar

18.1.2019, Essen: André Belotto, Andreas Höring, Valdemar Tsanov

12.4.2019, Wuppertal: Ya Deng, Christian Miebach, Duc Viet Vu

5.7.2019, Cologne: Tien-Cuong Dinh, Xiaonan Ma, Nessim Sibony

21.11.2019, Bochum: Patrick Graf, Ulrike Rieß, Jonas Stelzig

16.1.2020, Essen: Anda Degeratu, Takeo Ohsawa, Mihai Paun

14.5.2020, Wuppertal (via ZOOM): Lynn Heller, Simone Marchesi, Jan Swoboda

16.7.2020, Cologne (via ZOOM): Shin-ichi Matsumura, Sönke Rollenske, Lars Martin Sektnan

17.11.2020, Bochum (via ZOOM): Rafael Andrist, Nicholas Buchdahl, Eleonora di Nezza

29.1.2021, Essen (via ZOOM): Robert Berman, Simone Diverio, Ilaria Mondello