*Humboldt-Universität zu Berlin *

*Mathem.-Naturwissenschaftliche Fakultät*

*Institut für Mathematik*

Sommersemester 2020

Das Forschungsseminar findet mittwochs in der Zeit von 16:00 - 17:00 Uhr in Zoom statt.

Registration required here

Seminar: Algebraic Geometry an der FU

29.04.2020 | Arnaud Beauville (Université de Nice) | |

Title: Vector bundles on Fano threefolds and K3 surfaces. |
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Abstract: Let \(X\) be a Fano threefold, and let \(S\) be a smooth
anticanonical K3 surface in \(X\) . Any moduli space \(M\) of simple vector
bundles on \(S\) carries a holomorphic symplectic structure. Following an
idea of Tyurin, I will show that in some cases, those vector bundles
which come from \(X\) form a Lagrangian subvariety of \(M\) . Most of the
talk will be devoted to concrete examples of this situation.
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06.05.2020 | Daniel Erman (University of Wisconsin - Madison) | |

Title: Limits of polynomials rings. |
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Abstract: I will discuss two ways to think about a limit of a
polynomial ring in \(n\) variables as \(n\) goes to infinity, and how this can
be used to study complexity problems in algebraic geometry. This is
joint work with Steven V Sam and Andrew Snowden.
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13.05.2020 | Riccardo Salvati Manni (Università degli studi di Roma "La Sapienza") | |

Title: The Kodaira dimension of the moduli space of abelian varieties of
dimension 6. |
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Abstract: The moduli space \(\mathcal{A}_n\) of principally polarized complex abelian varieties
of dimension \(n\) was believed for a long time to be unirational. This was
first disproved by Freitag for \(n\) congruent to 1 mod 8 and n at least 17.
Then Tai, Freitag and Mumford proved that the moduli space is of general
type for all \(n\) at least 7. On the other hand, this
moduli space is unirational for \(n\) at most 5. Recently M. Dittmann and
N. R. Scheithauer and I proved that when \(n=6\) the Kodaira dimension is
non-negative. I will give a general overview on the topic.
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20.05.2020 | Patrick Brosnan (University of Maryland) | |

Title: Fixed points in toroidal compactifications and essential dimension of covers. |
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Abstract: We (Najmuddin Fakhruddin and I) prove a fixed point theorem for the action of certain local monodromy groups on etale covers and use it to deduce lower bounds on essential dimension of covers. In particular, we can give geometric proofs of many (but definitely not all) of the results in a recent preprint of Farb, Kisin and Wolfson. Moreover, we can extend their incompressibility results for Hodge type Shimura varieties to certain Shimura varieties for E7 (those with 0-dimensional rational boundary components).
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27.05.2020 | Adrien Sauvaget (Université de Cergy Pontoise) | |

Title: Moduli spaces of large pluricanonical divisors. |
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Abstract: We consider the moduli spaces of curves endowed with a \(k\)-canonical divisor. The purpose of the talk is to explain that the geometry of moduli spaces of metric surfaces can be studied by considering the large \(k\)-behavior of these moduli spaces. The first application of this strategy is a proof of the finiteness of the volumes of moduli spaces of flat surfaces with conical singularities. The second one is a conjecture on the volumes of moduli spaces of hyperbolic cone surfaces.
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03.06.2020 | András Lőrincz (HU Berlin) | |

Title: Equivariant D-modules. |
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Abstract: In this talk, I will discuss some results concerning equivariant
D-modules. Under suitable finiteness conditions, the category of such
objects on a stratification is equivalent to the category of
finite-dimensional representations of a quiver. We describe such
quivers explicitly for some irreducible representations of complex
reductive groups and toric varieties. In these cases we use the
results to find the explicit D-module structure of local cohomology
modules supported in orbit closures. Other applications include
Lyubeznik numbers and intersection cohomology groups of orbit
closures.
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10.06.2020 | 17:00 - 18:00 Mihnea Popa (Northwestern University) | |

Title: Minimal exponents of singularities. |
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Abstract: The minimal exponent of a function is the negative of the largest root of its reduced Bernstein-Sato polynomial. It refines the notion of log canonical threshold, and it is related (sometimes conjecturally) to other interesting objects, for instance the Igusa zeta function. I will describe some results towards understanding minimal exponents, based on viewing them in the context of D-modules and Hodge theory on one hand, and birational geometry on the other. This is joint work with Mircea Mustata.
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17.06.2019 | 16:00 - 17:00 Yuri Tschinkel (New York University) | |

Title: Equivariant birational geometry. |
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Abstract: I will discuss new invariants in equivariant birational
geometry (joint work with Kontsevich and Pestun).
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17:15 - 18:15 Mircea Mustaţă (University of Michigan) | ||

Title: Minimal exponents of hypersurfaces and a conjecture of Teissier. |
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Abstract: The minimal exponent of a hypersurface is a refinement of the log canonical threshold, a fundamental
invariant of singularities in birational geometry. After a brief introduction to these invariants, I will discuss work on
a conjecture of Teissier, relating the invariant of a hypersurface with that of a hyperplane section. This is joint work
(partially in progress) with Eva Elduque and Bradley Dirks.
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24.06.2020 | Egor Yasinsky (Universität Basel) | |

Title: Cremona groups and their subgroups. (Slides) |
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Abstract: The Cremona group is a group of birational automorphisms of a
projective space. Its study goes back to classical works of Cremona,
Noehter, Castelnuovo and Bertini, and it has been a subject of very
intensive research during the last 15 years. This study employs
different techniques - from modern birational geometry (e.g. Mori
theory, Sarkisov program, and Caucher Birkar's works) to geometric
group theory. In this talk, I will try to overview three plots on
Cremona groups: classification of finite subgroups, topological
properties, infinite subgroups and quotients of Cremona groups.
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01.07.2020 15:00 Uhr | Rahul Pandharipande (ETH Zürich) | |

Title: The moduli spaces of differentials on curves. |
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Abstract: The moduli space of pairs \( (C,f) \) where \( C \) is a curve and \( f \) is a rational
function leads to the well-developed theory of Hurwitz spaces. The study of the moduli of \( (C,\omega) \) where \( C \) is a curve and \( \omega\) is a
meromorphic differential is a younger subject. I will discuss recent developments in the study of the moduli spaces
of holomorphic/meromorphic differentials on curves. Many of the basic questions about cycle classes and integrals have now been solved
(through the work of many people) -- but there are also several interesting open directions.
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08.07.2020 | Elisabetta Colombo (Università degli studi di Milano Statale) | |

Title: The dimension of the Voisin sets in the moduli space of abelian
varieties. |
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Abstract: I will discuss reseach in collaboration with Naranjo and
Pirola on the subsets \(V_k(A)\) of an abelian variety \(A\), consisting of
those points \(x\) such that the
zero-cycle \(x-O\) is \(k\)-nilpotent with respect to the Pontryaghin product.
These sets were initially introduced and studied by Claire Voisin. We
find a sharp upper bound of the dimension in moduli of the locus of
those abelian varieties \(A\) for which \(V_2(A)\) is positive dimensional.
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15.07.2020 17:00 Uhr | Alexandru Suciu (Northeastern University Boston) | |

Title: Poincaré duality and cohomology jump loci. |
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Abstract: The cohomology jump loci of a space \(X\) are of two basic
types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for \(X\). We will explore in this talk the geometry of these varieties and the delicate interplay between them in the context of compact, orientable 3-manifolds. In the process, we will arrive at a fairly precise geometric description of the resonance varieties \(R^i_k(A)\) of a 3-dimensional Poincaré duality algebra \(A\).
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