We shall give a complete description of the relatively ample cones and the relatively movable cones of symplectic resolutions of the closures of the nilpotent orbits in complex simple Lie algebras. Moreover, we shall prove that all symplectic resolutions of such nilpotent orbit closures are connected by finite numbers of Mukai flops of type $A$, $D$ and $E_6$.

This is a survey lecture on the "theta map" from the moduli space of $SL_r$-bundles on a curve $C$ to the projective space of $r$-th order theta functions on $JC$. Some recent results and a few open problems about that map are discussed.

In this paper, we propose a definition of modular heights of abelian varieties defined over a field of finite type over $\mathbb{Q}$, and prove its bounding property, that is, the finiteness of abelian varieties with bounded modular height.

In this paper we re-visit the moduli problem for regular holonomic $\mathcal{D}$-modules with normal crossing singularities, and give a new definition of semistability, more general than the older notion, construct the moduli scheme, and describe its points. Independently of this, we introduce another natural parabolic notion of stability for such $\mathcal{D}$-modules, and construct the moduli in a special case.

We study certain degenerate abelian schemes $(Q_0, L_0)$ that are GIT-stable in the sense that their SL-orbits are closed in the semistable locus. We prove the vanishing of the cohomology groups $H^q (Q_0, L_{0}^{n}) = 0$ for $q,n \gt 0$ for a naturally defined ample invertible sheaf $L_0$ on $Q_0$. When $n = 1$, this implies that $H^0 (Q_0, L_0)$, the space of global sections, is an irreducible module of the noncommutative Heisenberg group of $(Q_0, L_0)$.

We prove that the moduli space of stable $n$-pointed curves of genus 1 and the projector associated to the alternating representation of the symmetric group on $n$ letters define (for $n \gt 1$) the Chow motive corresponding to cusp forms of weight $n + 1$ for $\mathrm{SL}(2, \mathbb{Z})$. This provides an alternative (in level 1) to the construction of Scholl.

In this paper we present a survey of rigid geometry. Here, special emphasis is put on the so-called "birational approach" to rigid geometry, which adopts classical methods of birational geometry to the theory of rigid spaces. The paper is divided into three parts. Part I is a general introduction to rigid geometry a la J. Tate and M. Raynaud. In Part II we are to overview the birational approach to rigid geometry, which combines the idea of Raynaud and that of O. Zariski, as one of the conceptual starting points of rigid geometry. In Part III we discuss some applications, which reveal the effectiveness of the ideas in rigid geometry that arise from our viewpoint.

In this paper, we show that the family of moduli spaces of $\boldsymbol{\alpha}'$-stable $(\mathbf{t}, \boldsymbol{\lambda})$-parabolic $\phi$-connections of rank 2 over $\mathbf{P}^1$ with 4-regular singular points and the fixed determinant bundle of degree $-1$ is isomorphic to the family of Okamoto–Painlevé pairs introduced by Okamoto [O1] and [STT]. We also discuss about the generalization of our theory to the case where the rank of the connections and genus of the base curve are arbitrary. Defining isomonodromic flows on the family of moduli space of stable parabolic connections via the Riemann-Hilbert correspondences, we will show that a property of the Riemann-Hilbert correspondences implies the Painlevé property of isomonodromic flows.