Masa-Hiko Saito

Moduli of Stable Parabolic Connections, Riemann–Hilbert Correspondence and Geometry of Painlevé Equation of Type VI, Part I

In this paper, we will give a complete geometric background for the geometry of Painleve VI and Garnier equations. By geometric invariant theory, we will construct a smooth fine moduli space M ¸ n (t,- ,L ) of stable parabolic connections on P 1 with log- arithmic poles at D(t )= t1+ ··· +tn as well as its natural compactification. Moreover the moduli space R(Pn,t)a of Jordan equivalence classes of SL2(C)-representations of the fundamental group π1(P 1 \ D(t), ∗) are defined as the categorical quotient. We define the Riemann-Hilbert correspondence RH : M ¸ n (t,- ,L ) −→ R(Pn,t)a and prove that RH is a bimeromorphic proper surjective analytic map. Painlevea nd Garnier equations can be derived from the isomonodromic flows and Painlevep rop- erty of these equations are easily derived from the properties of RH .W e also prove

MODULI OF UNRAMIFIED IRREGULAR SINGULAR PARABOLIC CONNECTIONS ON A SMOOTH PROJECTIVE CURVE.

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann- Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painleve property as in the regular singular cases proved generally in (8).

Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence

It is well known that the sixth Painleve equation PVI admits a group of Backlund transformations which is isomorphic to the affine Weyl group of type D (1) . Although var- ious aspects of this unexpectedly large symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of charac- terizing the Backlund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the Backlund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations by the Riemann- Hilbert correspondence. This result gives a natural and clear picture of the Backlund

PREPOTENTIALS OF YUKAWA COUPLINGS OF CERTAIN CALABI-YAU 3-FOLDS AND MIRROR SYMMETRY

In this note, we will give a rather naive mathematical approach to verifying the Mirror Symmetry Conjecture for certain Calabi-Yau 3-folds. Though the origin of Mirror symmetry is superstring theory in mathematical physics ([16], [37], [40], [42]), we will not discuss any background material from physics. Instead, we will focus our attention on prepotentials of A-model and B-model Yukawa couplings for certain Calabi-Yau 3-folds. The prepotential of the A-model Yukawa coupling is related to the number of holomorphic maps from a curve of genus g with n-marked points into a Calabi-Yau manifold. In this note, we will only consider the prepotential in the case when (g, n) = (0, 3) which is the simplest non-trivial case. However it is still far from complete mathematical understanding at this moment. The B-model Yukawa couplings of the mirror pair are also 3-points correlation functions.

Generic Torelli Theorem for Hypersurfaces in Compact Irreducible Hermitian Symmetric Spaces

Publisher Summary This chapter discusses Generic Torelli theorem for hypersurfaces in compact irreducible Hermitian Symmetric spaces. It focuses on hypersurfaces of degree ≥ 3 in compact irreducible Hermitian symmetric spaces Y. Such symmetric spaces are classified into six classes. The ingredients of the proof of generic Torelli theorem in the projective case are as follows: (1) an interpretation of the IVHS of smooth hypersurfaces by means of their Jacobian rings, (2) a symmetrizer lemma, (3) the polynomial structure (the defining ideal of the Veronese embedding of PN). The chapter discusses the vanishing theorem, construction of Kahler C-spaces, homogeneous vector bundles and the compact irreducible Hermitian symmetric spaces. It also discusses Kostants decomposition of ΩPY and the properties of weights and roots of G.