Mixed Hodge Modules in Birational Geometry

SFB/TRR 45 Summer School, July 9--13 2018, Mainz, Germany

This summer school is intended for advanced master students, PhD students, and younger researchers in algebraic geometry.

The purpose is to familiarize the participants with techniques from Hodge Theory, in particular mixed Hodge modules, and their applications to problems in birational geometry.

Speakers

Mihnea Popa (Northwestern) (mini-course)

Mircea Mustata (Michigan) (mini-course)

Florian Ivorra (Rennes) (mini-course)

Jean Babtiste Teissier (Leuven) (intro nearby cycles)

Nero Budur (Leuven)

Christian Schnell (Stony Brook)

Lei Wu (Utah) (problem sessions)

Organizer

Manuel Blickle (Mainz) (blicklem@uni-mainz.de)

Registration and financial support

Full financial support is available for members of the SFB/TRR 45. There is limited financial support available for local expenses (hotel costs) only. Please indicate on the application form if you require financial support:

Registration is closed.

With questions of non-mathemematical nature please contact our secretarial support Ms Gonska (gonska@uni-mainz.de)

List of participants

Venue

The conference takes place at the Institute of Nuclear Physcis at the Johannes Gutenberg-Universität Mainz (see Campus Map).

Preparation workshop in Freiburg

There is a series of introductory talks about D-modules and Hodge modules shortly before the wrokshop at FRIAS in Freiburg, Germany. Click here for more Information.

Schedule

Monday Tuesday Wednesday Thursday Friday
9:30-10:30 Mustata 1 Ivorra 2 Mustata 3 Popa 2 Schnell 2
10:30-11:00 coffee/
registration
coffee coffee coffee coffee
11:00--12:00 Ivorra 1 Mustata 2 Popa 1 Schnell 1 Popa 4
 12:00-13:00 Lunch Lunch Mustata 4 Lunch Lunch
13:00--14:00 Teyssier Ivorra 3 Bus for Boat Trip
leaves at 13:30
Popa 3 Budur
14:00-15:00 coffee coffee coffee
15:00-17:00 Wu Wu Wu
DINNER

Abstracts

Florian Ivorra: Introduction to mixed Hodge Modules

Mihnea Popa: Hodge modules, birational geometry, and families of varieties

Two lectures will be devoted to various aspects of Hodge ideals, including vanishing theorems and applications. The other two will give an overview of how the theory of
mixed Hodge modules can be used towards various questions regarding what types of parameter spaces can support maximally varying families of varieties.

Mircea Mustata: Hodge modules, birational geometry, and singularities

1. Introduction to D-modules
The goal of this lecture is to overview some basic concepts and results in D-module theory:
the classical Riemann-Hilbert correspondence, filtrations on coherent D-modules and the characteristic variety,
Bernstein's inequality and holonomic D-modules, Bernstein-Sato polynomials, and the V-filtration.
2. Hodge ideals of integral divisors
After a brief review of multiplier ideals, I will give the definition of Hodge ideals for integral divisors,
present the description in terms of a log resolution, and discuss some general properties.
3. Hodge ideals of Q-divisors
I will describe some filtered D-modules that lead to a definition of Hodge ideals for effective divisors
with rational coefficients and discuss some special features in this setting.
4. Local properties of Hodge ideals
This talk will focus on a description of Hodge ideals using the V-filtration and on applications to
triviality criteria

Jean Babtiste Teissier: Introduction to Nearby Cycles for D-modules

We will explain the construction of nearby cycles for D-modules relevant to the theory of Hodge modules, as well as its main properties.

Nero Budur: Absolute sets and the Decomposition Theorem

The celebrated Monodromy Theorem states that the eigenvalues
of the monodromy of a polynomial are roots of unity. In this talk we
give on overview of recent results on local systems giving a
generalization of the Monodromy Theorem. We end up with a conjecture
of André-Oort type for special loci of local systems. The conjecture
is true in rank one. If true in general, it would provide a simple conceptual
proof for all semi-simple perverse sheaves of the Decomposition Theorem, assuming only the geometric case (Beilinson-Bernstein-Deligne-Gabber) or the MHM case (Saito). Joint work with Botong Wang.

Christian Schnell: Extension theorems for holomorphic forms on singular spaces

My two lectures will be about the following problem: Suppose we have an algebraic (or holomorphic) differential form, defined on the smooth locus of an algebraic variety (or analytic space). Under what conditions does it extend to an algebraic (or holomorphic) differential form on a resolution of singularities? In 2011, Greb, Kebekus, Kovacs, and Peternell proved that such an extension always exists on algebraic varieties with klt singularities. I will explain how to generalize their result to a much larger class of singular spaces, with the help of Hodge modules and the decomposition theorem. This is joint work with Kebekus.

Lei Wu: Afternoon Problem Sessions

Hotel

Regardless of whether they receive funding from us or not, participants have to book accommodation themselves. Here are some suggestions:

Travel information:

Closest airport is located at Frankfurt/Main. From there, there are frequent trains to Mainz central station (Hauptbahnhof). One possibility would be to take the S8 from Frankfurt airport to Mainz.

The following lines serve the university from the main station (get off at the stop "Friedrich-von-Pfeiffer-Weg"):

  • tram 51 (towards Lerchenberg)
  • tram 53 (towards Lerchenberg)
  • bus 54 (towards Klein-Winternheim)
  • bus 55 (towards Finthen)
  • bus 56 (towards Finthen/Wackernheim)
  • tram 59 (towards Hochschule Mainz)
  • bus 75 (towards Schwabenheim/Ingelheim)
  • bus 650 (towards Sprendlingen)

From the stop, walk over the pedestrian bridge to the university campus and follow the street to the right. After about 100 meters there is a left curve. After passing the Mensa (campus cafeteria) you see the Mathematics building right in front of you. (The walk is about 10 minutes.)

All lectures will take place at the Institute of Mathematics of the University of Mainz, Staudinger Weg 9, in room 05-426.

Free afternoon on wednesday

On the free afternoon on wednesday you may participate in a trip along the rhine river.

We will organize Bus and Boat transfer from the University to the city of St. Goar up the Rhine valley.
From there you have two options:

  1. Visit the Castle Rheinfels, have dinner at a local restaurant and return to Mainz by yourself with the train
  2. Take a challenging a hike along the Rheinbugenweg from St Goar to Oberwesel then have dinner in Oberwesel and ruturn to Mainz by train.

In any case: after the ca. 2h boat ride up the rhine river, passing the castle and the famous rock of loreley, you are on your own (but information will be provided).

Thursday night dinner

There will be a dinner in a restaurant organized on Thursday (probably at Heiliggeist). We will meet at 18:30 inside the restaurant (so that we do not block the street waiting altogether outside). You can sign up for dinner with your dietary requirements at the summer school until Wednesday at noon.

List of Participants (as of June 20, 2018):

Rodolfo Aguilar Aguilar Institut Fourier
Patricio Almirón Cuadros Universidad Complutense de Madrid
Emelie Arvidsson Ecole Polytechnique Fédérale de Lausanne
Sjoerd Beentjes University of Edinburgh
Fabio Bernasconi Imperial College
Guillem Blanco Polytechnic University of Catalonia
Amaël Broustet Université de Lille
Nero Budur KU Leuven
Alberto Castaño Domínguez Universidade de Santiago de Compostela
Jiaming Chen Université Paris Diderot / IMJ-PRG
Qianyu Chen Stony Brook University
Eric Chen Princeton University
Kiryong Chung Kyungpook National University, Korea
Luis Francisco Curquejo Otero Technische Universität Chemnitz
Ferran Dachs Cadefau Martin-Luther-Universität Halle-Wittenberg
Haohua Deng Washington University in St. Louis
Bradley Drew Universität Freiburg
Yajnaseni Dutta Northwestern Uni
Yanbo Fang Université Paris Diderot
Dino Festi Universität Mainz
Franco Giovenzana Universität Chemnitz
Luca Giovenzana Universität Chemnitz
Aleksei Golota National Research University Higher School of Economics
Brian Hepler Northeastern University
Eriola Hoxhaj Universität Kaiserslautern
Florian Ivorra Université de Rennes
Shuddhodan Kadattur Vasudevan Freie Universität Berlin
Louis-Clément Lefèvre Université Grenoble Alpes
Leonardo Lerer Université Paris-Sud
Yongqiang Liu KU Leuven
Luigi Lombardi Stony Brook University
Victor Lozovanu Leibniz Universität Hannover
Mirko Mauri Imperial College London
Giacomo Mezzedimi University of Pisa
Takumi Murayama University of Michigan
Mircea Mustata University of Michigan
Philipp Naumann Université Grenoble Alpes
Manh Toan Nguyen Universität Osnabrück
The Hoang Nguyen Ecole Polytechnique Fédérale de Lausanne
Thi Quynh Trang Nguyen University of Edinburgh
Sebastián Olano Northwestern University
Anna Piwatz Universität Duisburg-Essen
Mihnea Popa Northwestern University
Quentin Posva Ecole Polytechnique Fédérale de Lausanne
Fei Ren FU Berlin
Fatemeh Rezaee University of Edinburgh
Andrés Rojas Universitat de Barcelona
Marcel Rubió KU Leuven
David Sabonis TU München & Uni Kopenhagen
Takahiro Saito University of Tsukuba
Adrien Sauvaget UPMC
Christian Schnell Stony Brook University
Christian Sevenheck Universität Chemnitz
Jaime Silva Instituto Superior Técnico, Universidade de Lisboa
Andrew Staal Hebrew University of Jerusalem
Axel Stäbler Universität Mainz
Tom Sutherland Universität Mainz
Jean Baptiste Teissier KU Leuven
Benjamin Tighe University of Georgia
Sofia Tirabassi Universiteget I Bergen
Quang Tue Tran KU Leuven
Nikolaos Tsakanikas Universität des Saarlandes
Davide Cesare Veniani Universität Mainz
Federico Venturelli Università di Padova
Alexandra Viktorova Stony Brook University
Duc Viet Vu Universität zu Köln
Haopeng Wang KU Leuven
Juanyong Wang Ecole polytechnique
Chuanhao Wei University of Utah
Jakub Witaszek Imperial College London
Lei Wu University of Utah
Ruijie Yang Stony Brook University
Fei Ye CUNY-QCC
Maciej Zdanowicz EPFL
Mingyi Zhang Northwestern University
Yilong Zhang Ohio State University
Xiaolei Zhao University of California, Santa Barbara
Runhong Zong Universität Mainz

Miscellaneous: