### 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.

#### Liviu Ornea Holomorphic maps in generalized complex geometry

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.

#### 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.