Let $X$ be a Calabi-Yau threefold fibred over $\mathbb{P}^{1}$ by non-constant semi-stable K3 surfaces and reaching the Arakelov-Yau bound. In [25], X. Sun, Sh.-L. Tan, and K. Zuo proved that $X$ is modular in a certain sense. In particular, the base curve is a modular curve. In their result they distinguish the rigid and the non-rigid cases. In [17] and [28] rigid examples were constructed. In this paper we construct explicit examples in non-rigid cases. Moreover, we prove for our threefolds that the “interesting” part of their $L$-series is attached to an automorphic form, and hence that they are modular in yet another sense.

In their paper [LY] Livné and Yui discuss several examples of nonrigid Calabi-Yau varieties which admit semi-stable $K3$-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the modularity of the $L$-function of these examples. The purpose of this note is to point out that the examples which were listed in [LY] but which do not lead to semi-stable fibrations are still modular in the sense that their $L$-function is associated to modular forms. We shall treat the case associated to the group $\Gamma _{1}(7)$ in detail, but our technique also works in the other cases given in [LY]. We shall also make some comments concerning the Kummer construction for fibre products of elliptic surfaces in general.

In this paper, we collect basic properties of the Albanese dimension and explain how to generalize the main theorem of [F2]: Algebraic fiber spaces whose general fibers are of maximal Albanese dimension. This paper is a supplement and a generalization of [F2]. We also prove an inequality of irregularities for algebraic fiber spaces in the appendix, which is an exposition of Fujita-Kawamata’s semi-positivity theorem.

We give a new proof of some characteristic-free fundamental theorems in invariant theory first proved in C. De Concini and C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976), 330–354. We treat the action of the general linear group and the symplectic group. Our approach is geometric, and utilizes the fact that the categorical quotients are principal fiber bundles off codimension two or more.

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules. This pairing happens to be a special representative in a general family of pairings involving standard, costandard and tilting modules. In the graded case, under some “Koszul-like” assumptions (which we prove are satisfied for example for the blocks of the category $\mathcal{O}$), we obtain a non-degenerate pairing between certain graded homomorphism and graded extension spaces. This motivates the study of the category of linear complexes of tilting modules for graded quasi-hereditary algebras. We show that this category realizes the module category for the quadratic dual of the Ringel dual of the original algebra. As a corollary we obtain that in many cases Ringel and Koszul dualities commute.

We give a new version of W. L. Edge’s construction of the linear system of plane sextics containing Wiman’s sextic, by means of configuration space of 5 points on projective line. This construction reveals out more of the inner beauty of the hidden geometry of Wiman’s sextic. Furthermore, it allows one to give a friendly proof for the fact that the linear system is actually a pencil, the fact that is important in both Edge’s and our constructions.

The CR geometry is applied to the representation theory of the group $\mathrm{SU}(p, q)$. We prove that the kernel of the CR Yamabe operator on a CR manifold $M$ is a representation of the conformal CR automorphism group of M. So we can construct a representations of $\mathrm{SU}(p, q)$ on the kernel of the CR Yamabe operator on the projective hyperquadric $\Bar{Q}_{p,q}$. This is a complex version of Kobayashi-Orsted's model of the minimal irreducible unitary representation $\varpi _{p,q}$ of $\mathrm{SO}(p, q)$ on $S^{p-1} \times S^{q-1}$.

We show the existence and continuity of the density and obtain its formula for the law of the fraction of time spent on the positive side of a very general class of one-dimensional (generalized or gap) diffusion processes. We also study the asymptotic properties of the density at the extreme values $x = 0$ and $x = 1$, which extend, in particular, some recent results by Kasahara and Yano [8].

Concerning the uniqueness in the Cauchy problem at doubly characteristic points, discrete conditions on lower order terms are known. In this paper, uniqueness is studied when those conditions are not satisfied.

This paper collects 26 conditions for the reverse-order law $(AB)^{\dagger} = B^{\dagger}(A^{\dagger}ABB^{\dagger})^{\dagger}A^{\dagger}$ to hold for the Moore-Penrose inverse of matrix.

Let $M$ be a complex projective Fano manifold whose Picard group is isomorphic to $\mathbb{Z}$ and the tangent bundle $TM$ is semistable. Let $Z \subset M$ be a smooth hypersurface of degree strictly greater than degree($TM$)$(\mathrm{dim}_{\mathbb{C}} Z-1)/(2\mathrm{dim}_{\mathbb{C}} Z-1)$ and satisfying the condition that the inclusion of $Z$ in $M$ gives an isomorphism of Picard groups. We prove that the tangent bundle of $Z$ is stable. A similar result is proved also for smooth complete intersections in $M$. The main ingredient in the proof of it is a vanishing result for the top cohomology of the twisted holomorphic differential forms on $Z$.