### Supersymmetry in complex geometry: talks and abstracts

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

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#### Cecilia Albertsson

Doubled geometry and string theory

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.
#### Gil Cavalcanti

Reduction of generalized Kahler structures

I will recall the idea of action of a group on a Courant
algebroid over a manifold M. Then I will show how to reduce the
algebroid and the manifold by a given action and how to transport
invariant structures on the algebroid over M to the algebroid over
the reduced manifold. I will pay particular attention to the problem
of reducing generalized complex and generalized Kahler structures.
#### Ryushi Goto

Holomorphic Poisson structures and deformations of generalized
Kahler structures

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

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.
#### Akira Fujiki

Anti-self-dual bihermitian structures on compact complex surfaces of class VII

I would like to talk about the joint work with Pontecorvo on anti-self-dual
bihermitian
structures on compact complex surfaces of class VII.
Except for those coming from hyperhermitian structures
the anti-self-dual bihermitian structures are only possible on surfaces on
class VII,
i.e., the surfaces with first betti number one and with Kodaira dimension
$-\infty$.
Moreover, by Pontecorvo the existence of such a structure forces the
existence of effective and
disconnected anti-canonical divisor on the surface. When the surface is
minimal and with
positive second betti number, this implies that the surface is either a
hyperbolic or parabolic Inoue
surface. For certain parabolic Inoue surfaces there exists such an example
constructed by LeBrun in
'91, but since then no other examples have been discovered. In our work we
have constructed
also some anti-self-dual bihermitian structures on parabolic Inoue surfaces,
which should still to be
compared with the examples of LeBrun.
On the other hand, we obtained a more complete result for the hyperbolic
case:
Namely we can show the existence of anti-self-dual bihermitian structures
on any given hyperbolic Inoue surface depending on real $m$ dimensional
parameter, where $m$ is
the second betti number of the surface. Moreover, we show that any small
deformation of
the surface also carries an anti-self-dual bihermitian structure as long as
it still admits an effective and
disconnected anti-canonical divisor, thus giving the sufficiency of the
necessary condition of
Pontecorvo mentioned above for these surfaces.
####
Akito Futaki

Hilbert series and obstructions to asymptotic semistability

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.
#### Dmitry Kaledin

Generalized complex structures and Fedosov quantization

Abstract: Generalized complex structures are supposed to be related
in some fashion to non-commutative deformation of the ordinary
complex structures, but what this fashion might be, exactly, seems
to be a great mystery. I will discuss some parallels between
deformation theory of generalized complex structure, on one hand,
and deformation quantization, on the other hand. I will only
consider the case of Fano manifolds, where one can use a version of
the well-known Fedosov quantization procedure.
####
Ulf Lindstrom

Non-linear sigma models and generalized complex geometry

Supersymmetric non-linear sigma models serve as an efficient tool
for investigating complex geometry.
For example, two-dimensional models with N=(2,2) supersymmetry have target
spaces that carry generalized
Kahler geometry. In my talk I will give examples of how such relations
have been explored to yield interesting results, including
the existence of a generalized Kahler potential.
#### Toshiki Mabuchi

Stability and extremal Kahler metrics for
projective bundles over curves

I would like to review a recent result of V. Apostolov,
C. Tonnesen-Friedman, P. Gauduchon and D. Calderbank
on the existence of extremal Kahler metrics for projective
bundles over curves. In this talk, the method of approach
is quite different from the original one.
#### Luca Martucci

Flux vacua in string theory, generalized calibrations and supersymmetry
breaking

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.
It is a joint work with Radu Pantilie.
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
is 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.
*

#### Gil Cavalcanti

Introduction to generalized Kaehler geometry

In these lectures I will introduce generalized complex and
generalized Kaehler structures and prove Gualtieri's theorem which
states that generalized Kahler structures are equivalent to a special
type of bi-Hermitian structure which arises naturaly when studying
(2,2) supersymmetric sigma models. Time allowing, I will try to
present some examples of nontrivial generalized Kaehler structures.
#### Liviu Ornea

Introduction to Dirac and generalized complex geometry

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

A Review of Supersymmmetry, Sigma Models and Bihermitian Geometry

#### Alessandro Tomasiello

Introduction to supersymmetry in supergravity

I will review the basics of supergravity, and what
it means for a solution of its equations of motion
to be supersymmetric. This will lead us to systems
of equations that can be interpreted using complex
or generalized complex geometry. I will also review
the relations to the sigma model results.
#### Alessandro Tomasiello

Generalized complex/Kaehler geometry in supergravity

Starting from the results of my previous lecture, I will
review the geometrical consequences of the systems
we derived, and what is known about existence of
solutions.