### Supersymmetry in complex geometry: PDF files

#### 4-9 January 2009, IPMU, Japan

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### PDF files of the talks

#### Cecilia Albertsson

Doubled geometry and string theory (PDF file, 42 Mb)

Doubled geometry is a geometric way to describe non-geometric
spaces that are relevant when compactifying string theory to
render it consistent with observations. The formalism of doubled
geometry bears many superficial similarities to that of generalised
geometry, which has prompted several attempts to uncover any
actual relations between the two. To date, however, such relations,
if they exist, remain elusive. I will review the motivation and
construction of doubled geometry, and illustrate its application
in string theory.
** Ryushi Goto**

Holomorphic Poisson structures and deformations of generalized
Kahler structures

Lecture 1 (PDF, 144 Kb) |
Lecture 2 (PDF, 160 Kb)

I would like to talk my recent works about unobstructed
deformations of generalized Kahler structures.
Given small deformations {J_t} of generalized complex structures on a
compact generalized Kahler manifolds with one pure spinor, there
exist corresponding deformations {J_t, psi_t} of generalized
Kahler structures. This is an analogous to the fact that small
deformations of a compact Kahler manifold is still Kahlerian.
A holomorphic Poisson structure on a compact Kahler manifold
provides deformations of
generalized complex structures. Thus applying our theorem, we obtain
deformations of
non-trivial generalized Kahler structures starting from the
ordinary Kahler structure
by using Poisson structure.
Unobstructed deformations of generalized Kahler structures on
Fano surfaces,
Hirzebruch surfaces and several 3-folds will be discussed.
I also obtain a correspondence between generalized Kahler
submanifolds and Poisson submanfolds.

#### Geo Grantcharov

Neutral hypercomplex
structures (PDF file, 112 Kb)

A neutral hypercomplex structure consists of a complex structure and a couple of product structures, mutually anti-commuting. Such structure is naturally associated with a metric of neutral signature, which in dimension four is anti-self-dual. A special case of this structure also is the hypersymplectic structure introduced by N.Hitchin. We notice first that most of the compact complex surfaces with vanishing first Chern class admit such a structure. Then we will discuss the reduction of neutral hypercomplex structures and its relation with generalized geometry.
####
Akito Futaki

Hilbert series and obstructions to asymptotic semistability (PDF file, 148 Kb)

Given a polarized manifold there are obstructions for asymptotic
semistability described as integral invariants. One of them is an
obstruction to the existence for the first Chern class of the
polarization to admit a constant scalar curvature Kahler (cscK)
metric. A natural question is whether or not the other obstructions
are linearly dependent on the obstruction to the existence of a cscK
metric. The purpose of this talk is to show that this is not the case
by exhibiting toric Fano threefolds in which these obstructions span
at least two dimension. To see this we show that on toric Fano
manifolds these obstructions are obtained as derivatives of the
Hilbert series.
#### Luca Martucci

Flux vacua in string theory, generalized
calibrations

and supersymmetry breaking
(PDF file, 1648 Kb)

I will discuss how supersymmetric type II string vacua
with generic fluxes can by completely characterized in
terms of appropriately defined generalized calibrations,
that naturally fit into the context of Generalized Complex
Geometry. These structures turn out to be crucial to prove
that supersymmetric flux compactifications to 4D are
solutions of the full set of 10D supergravity equations of
motion with D-branes and orientifolds as localized
sources. Furthermore, I will show how, within this same
framework, one can construct non-supersymmetric vacua
where supersymmetry is broken in a controlled way,
violating only part of the underlying generalized
calibration structures, but still allowing for a drastic
simplification of the 10D supergravity equations.
#### Liviu Ornea

Holomorphic maps in generalized complex geometry
(PDF file, 384 Kb)

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.

*
In addition to the regular research lectures, a number of review talks
was prepared. We aim to give a clear and elementary introduction to the
physics and mathematics of generalized complex and generalized
Kahler geometry for non-specialists.
*

#### Liviu Ornea

Introduction to Dirac and generalized complex geometry (PDF file, 412 Kb)

I shall provide the necessary background for (real) Dirac
structures and for generalized complex structures, in particular for
generalized Kahler structures.