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

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

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

(4)
BatyrevBorisov mirror
construction, Chapter 1 of [CR14] sec. 1,2 &3 ,
as well as
sec. 4.1 of [CK00]

(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)
HoriVafa mirror construction I, Chapter 2 of [CR14] sec. 4 &5

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

(8) GrossSiebert
mirror construction, based on [Au] as
well as section 8.4 of [Clay]

(9)
Introduction to SYZ
Literature:
 Tuesdays, 10:15  11:45 in room 241 (Geomatikum)
Sessions this term:
Date

Speaker

Topic

Room

28.03.2017

Everyone

Prediscussion

Geo 431

04.04.2017

Murad Alim

(1)

Geo 241

11.04.2017

Immanuel van Santen

(2)

Geo 241

25.04.2017

Immanuel van Santen

(2)

Geo 241

02.05.2017

Martin Vogrin

(3)

Geo 241

09.05.2017

Victoria
Hoskins

Group actions on quiver moduli spaces

Geo 241

16.05.2017

Maxim
Smirnov

On quantum cohomology of isotropic
Grassmanians

Geo 241

18.05.2017!

Mathieu
Florence

Lifting theorems in Galois cohomology

Geo H2

23.05.2017

Arpan Saha

(5)

Geo 241

30.05.2017

Tim Gabele

(6)

Geo 241

13.06.2017

Vladimir
Lazic

The existence of morphisms from a
CalabiYau variety

Geo 241

20.06.2017

Raffaele Caputo

(7)

Geo 241

27.06.2017

Michel
van Garrel 
(8)

Geo 241

04.07.2017

Florian Beck

(9)

Geo 241

11.07.2017

Hossein
Movasati

NoetherLefschetz and Hodge loci

Geo 241

Talk titles and abstracts:
09.05.2017
Speaker: Victoria Hoskins
Title: Group actions on quiver moduli spaces
Abstract:
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 krational
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.
13.06.2017
Speaker: Vladimir Lazic
Title: The existence of morphisms from a CalabiYau variety
Abstract:
Given an algebraic variety, finding nontrivial 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.
11.07.2017
Speaker: Hossein Movasati
Title: NoetherLefschetz and Hodge loci
Abstract:
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 NoetherLefschetz loci. The main
tools are the infinitesimal variation of Hodge
structures and the notion of modular foliations.
 
