Fachbereich Mathematik 
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Complex Geometry Seminar, University of Hamburg,  Summer Term 2017

The topic of the research seminar on complex geometry for the summer term 2017 is Mirror Symmetry Constructions. It is open to anyone interested in research questions on complex and algebraic geometry and their connections to mathematical physics. It consists of several talks given by the participants of the seminar following some suggested literature on mirror symmetry constructions as well as a number of research talks given by external invited speakers.

Mirror symmetry has its origins in the physics of string theories. The equivalence of physical theories obtained using different deformation families of Calabi-Yau threefolds predicted isomorphisms between variation problems formulated in terms of flat connec- tions in complex geometry on one side and in symplectic geometry on the other side. This seminar will address various constructions of mirror families of geometries and their rela- tions. The guiding literature for the talks will be a set of lecture notes of Clader and Ruan [CR14], supplemented by chapters of a book of Cox and Katz [CK00], Ref. [GS16] of Gross and Siebert, Gross’ lecture notes [Gro] as well as other resources.

Topics of the talks
  1. (1)  Toric varieties and fans, Chapter 0 of [CR14] sec. 1 &2 , as well as sec. 3.1 of [CK00]

  2. (2)  Divisors and orbifolds, Chapter 0 of [CR14] sec. 3 &4

  3. (3)  Polytopes, Chapter 0 of [CR14] sec. 5 &6 , as well as sec. 3.2 of [CK00]

  4. (4)  Batyrev-Borisov mirror construction, Chapter 1 of [CR14] sec. 1,2 &3 , as well as

    sec. 4.1 of [CK00]

  5. (5)  The gauged linear sigma model, Chapter 2 of [CR14] sec. 1,2 &3 , as well as

    sec. 3.3.3 and 3.3.4 of [CK00]

  6. (6)  Hori-Vafa mirror construction I, Chapter 2 of [CR14] sec. 4 &5

  7. (7)  Hori-Vafa mirror construction II, Chapter 2 of [CR14] sec. 6 &7

  8. (8)  Gross-Siebert mirror construction, based on [Au] as well as section 8.4 of [Clay]

  9. (9)  Introduction to SYZ


[Au] Denis Auroux, Special Lagrangian Fibrations, Mirror Symmetry and Calabi-Yau Double Covers
[CK00] D. A. Cox and S. Katz.
Mirror symmetry and algebraic geometry. 2000.
[Clay] Dirichlet Branes and Mirror Symmetry
[CR14] Emily Clader and Yongbin Ruan. Mirror Symmetry Constructions. 2014.
[Gro] Mark Gross.
Tropical Geometry and Mirror Symmetry

The seminar takes place on:

  • Tuesdays, 10:15 - 11:45 in room 241 (Geomatikum)

Sessions this term:

Geo 431
Murad Alim
Geo 241
Immanuel van Santen
Geo 241
Immanuel van Santen
Geo 241
Martin Vogrin
Geo 241
Victoria Hoskins
Group actions on quiver moduli spaces Geo 241
Maxim Smirnov
On quantum cohomology of isotropic Grassmanians
Geo 241
Mathieu Florence
Lifting theorems in Galois cohomology
Geo H2
Arpan Saha
Geo 241
Tim Gabele
Geo 241
Vladimir Lazic
The existence of morphisms from a Calabi-Yau variety
Geo 241
Raffaele Caputo
Geo 241
Michel van Garrel (8)
Geo 241
Florian Beck
Geo 241
Hossein Movasati
Noether-Lefschetz and Hodge loci
Geo 241

Talk titles and abstracts:

Speaker: Victoria Hoskins
Title: Group actions on quiver moduli spaces

We first introduce King's moduli spaces of quiver representations over a
field k. We study two types of actions on such moduli spaces and we
decompose their fixed loci using group cohomology and give modular
interpretations of the components. The first type of action arises by
considering finite groups of quiver automorphisms. The second type of
action is to consider the absolute Galois group of k, for a perfect field
k, acting on the points of this quiver moduli space valued in an algebraic
closure of k; the fixed locus is the set of k-rational points, and we
obtain a decomposition of this fixed locus indexed by the Brauer group of
k. Finally, working over the field of complex numbers, we will describe
the symplectic and holomorphic geometry of these fixed loci in
hyperkaehler quiver varieties in the language of branes. This is joint
work with Florent Schaffhauser.

Speaker: Vladimir Lazic
Title: The existence of morphisms from a Calabi-Yau variety

Given an algebraic variety, finding non-trivial morphisms to other varieties is one of the fundamental problems in algebraic geometry. For varieties with trivial canonical class, this can be formulated as a standard conjecture that a multiple of every nef line bundle is basepoint free, and it is intimately related to the abundance conjecture. In this talk I will report on a recent progress on this problem in a joint work with Thomas Peternell.

Speaker: Hossein Movasati
Title: Noether-Lefschetz and Hodge loci

In this talk I will talk about identifying components of the Hodge loci which  live in the
parameter spaces of hypersurfaces. For surfaces this is known as Noether-Lefschetz loci.  The main
tools are the infinitesimal variation  of Hodge structures and the notion of modular foliations.

  Seitenanfang  Impress 2017-07-07, Murad Alim