RIMS Workshop on Non-abelian Hodge theory and Geometry of Twistor structures

Abstracts

FUJIKI, Akira (Osaka Univ.)

Anti-self-dual bihermitian structures on Inoue surfaces via twistor method

July 15 (Wednesday) 14:30--15:30

Except for those coming from hyperhermitian structures the only examples of anti-self-dual hermitian structures are the well-known examples on Hopf surfaces and the ones constructed by LeBrun in 1991 on certain blown-up Hopf surfaces and certain parabolic Inoue surfaces by using his Hyperbolic Ansatz. I would like to talk about the construction of anti-self-dual bihermitian structures on hyperbolic Inoue surfaces and its small deformations by using a twistor method which is the variation of that of Donaldson and Friedman. This is a joint work with M. Pontecorvo.

MORIYA, Syunji (Kyoto Univ.)

Twistor structures on real pro-algebraic homotopy types of compact Kahler manifolds

July 16 (Thusday) 14:30--15:30

Recently, Mixed twister structures on pro-algebraic homotopy types of compact Kahler manifolds were defined and constructed by Pridham. In this talk, I will explain the correspondence between pro-algebraic homotopy types and dg-categories of flat bundles, and what algebraic topological invariants determine pro-algebraic homotopy types. Then I will explain the above Pridham's result.

SABBAH, Claude (École Polytechnique)

Introduction to twistor D-modules

July 15 (Wednesday) 11:30--12:30

I will explain the notion of (pure polarized) twistor D-module as a convenient generalization of that of a harmonic bundle. I will insist on the integrability condition, and its preservation by various operations.

Twistor D-Modules and Fourier-Laplace transform

July 16 (Thusday) 11:30--12:30

The Fourier-Laplace transform is a way to produce examples of wild twistor D-modules starting from regular ones. I will indicate some of its properties. The integrable case is of special interest, and I will comment on the 'new supersymmetric index' of Cecotti-Vafa and limit properties.

SIMPSON, Carlos (Université de Nice)

Lectures on Nonabelian Hodge Theory

I-- The Hodge filtration on nonabelian cohomology

July 13 (Monday) 10:00--11:00

We will introduce the Betti, de Rham and Hitchin moduli spaces of representations of $\pi _1(X)$, and the deformations between them. Harmonic bundles give prefered sections, and the natural action of ${\mathbb C}^{\ast}$ induces stratifications of the de Rham and Hitchin moduli spaces and we get compactifications. In case of a family of varieties, the isomonodromy equations satisfy a Griffiths transversality property. These basic structures persist for higher nonabelian cohomology, and can lead to the notion of Hodge structure on schematic homotopy types.

II--Local study and the role of variations of Hodge structure

July 14 (Tuesday) 10:00--11:00

The local structure of the family of moduli spaces of $\lambda$-connections around a prefered section includes the twistor property which says that we get a hyperk\"ahler structure on the moduli space. This leads to a mixed Hodge structure on the local ring at a variation of Hodge structure. The ${\mathbb C}^{\ast}$-limit construction can be interpreted in terms of construction of Griffiths-transverse filtrations on flat bundles, the leaves of this construction are Lagrangian subvarieties, and it leads to an {\em oper stratification}. We discuss the interaction between this structure and the isomonodromy equations; and also the classification of rigid representations in rank $2$. These are related to recent work classifying algebraic solutions to the Painlev\'e equations.

III--The noncompact case: weight-two phenomena

July 15 (Wednesday) 10:00--11:00

Harmonic theory on an open variety, now fully developped by Takuro Mochizuki's work, includes notions of parabolic structures on $\lambda$-connections for all $\lambda$. The transformation law for the parabolic weights and the residues of the $\lambda$-connections is related to the local study of the twistor space, but rather than corresponding to a weight $1$ Hodge structure it corresponds to a weight $2$ Hodge structure. We indicate the problems and questions arising in trying to make precise this observation in general.

IV--The noncompact case: classification questions

July 16 (Thusday) 10:00--11:00

The case of flat bundles on ${\mathbb P}^1 - \{ x_1,\ldots , x_k\}$ leads to a number of different classification questions. If we fix unitary conjugacy classes of the monodromy transformations, then the stratification corresponding to the Hodge filtration contains a generic or open stratum corresponding to variations of Hodge structures which may be different from unitary bundles; the Hodge type changes upon wall-crossing. On the other hand, there seem to be interesting relationships between different moduli spaces, conjectured by Boalch-Etingof-Oblomkov-Rains. We will point out how this conjecture can be extended to a conjectural isomorphism between different de Rham and Hitchin moduli spaces, and might be tested by looking at the oper stratification.

TSUCHIMOTO, Yoshifumi (Kochi Univ.)

Endomorphisms of the Weyl algebra An and associated projective modules of rank 1

July 14 (Tuesday) 16:00--17:00

In this talk we will concentrate on a case where characteristic of the base field $k$ is non-zero. (Though the case where $k=\C$ may be treated by our result using a technique of ultra filter.) Then much of the properties of a Weyl algebra $A_n$ are explained in terms of connections and curvatures on a vector bundle on an affine space $X=\A^{2n}$. In particular, an algebra endomorphism $\varphi$ of $A_n$ gives rise to a symplectic endomorphism $f$ of $X$ with a gauge transfomation $g$. Now we study converse problem of finding $\varphi$ from an arbitrary symplectic endomorphism $f$ of $X=\A^{2n}$. It is shown that given such $f$, we may construct a projective left $A_n$-module $W_f$ (``the sheaf of local gauge transformations'') such that its triviality is equivalent to the existence of the ``lift'' $\varphi$. Some properties of such a module will be discussed.