arXiv subjects

Masa-Hiko Saito

Publications and source records attributed to Masa-Hiko Saito.

23 records · Page 2Linked to original sources

arXiv2000

Deformation of Okamoto-Painlevé Pairs and Painlevé equations

In this paper, we introduce the notion of generalized rational Okamoto-Painlevé pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlevé equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlevé differential equations from the deformation of Okamoto-Painlevé pairs by using the local cohomology groups. Moreover the reason why the Painlevé equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlevé pairs of type $\tilde{E}_7 (= P_{II})$ and $\tilde{D}_8 (= P_{III}^{\tilde{D}_8})$ are calculated explicitly as examples of our theory.

math.AG↗

arXiv2000

Painlevé equations and deformations of rational surfaces with rational double points

In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.

math.AG↗

arXiv2000

Classification of Okamoto-Painlevé Pairs

In this paper, we introduce the notion of an Okamoto-Painlevé pair (S, Y) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlevé equations introduced by K. Okamoto give examples of Okamoto-Painleve pairs, we find a new example of Okamoto-Painlevé pairs not listed in \cite{Oka}. We will give the complete classification of Okamoto-Painlevé pairs.

math.AG↗

arXiv1997

On Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3 folds

In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the Calabi-Yau 3-fold directly by means of a theta function and Dedekind's eta function. This gives infinitely many Gromov-Witten invariants, and equivalently infinitely many sets of rational curves in the Calabi-Yau 3-fold. Using the toric mirror construction, we also calculate the prepotential of the B-model Yukawa couplings of the mirror partner. Comparing the expansion of the B-model prepotential with that of the A-model prepotential, we check a part of the Mirror Symmetry Conjecture up to a high order.

alg-geom↗

arXiv1996

d-gonality of modular curves and bounding torsions

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this result, we prove an analogue of the strong Uniform Boundedness Conjecture for elliptic curves defined over the function fields of curves. If a base curve is $d$-gonal, we can bound orders of torsions of Mordell-Weil groups in terms of $d$ uniformly.

alg-geom↗