In the present paper, we give a characterization of the supersingular divisors [i.e., the zero loci of the Hasse invariants] of nilpotent admissible/ordinary indigenous bundles on hyperbolic curves. By applying the characterization, we also obtain lists of the nilpotent indigenous bundles on certain hyperbolic curves. Moreover, we prove the hyperbolic ordinariness of certain hyperbolic curves.

In this paper, we give a necessarly and sufficient condition for orbits of linear isotropy representations of Riemannian symmetric spaces are biharmonic submanifolds in hyperspheres in Euclidean spaces. In particular, we obtain examples of biharmonic submanifolds in hyperspheres whose co-dimension is greater than one.

We give some sufficient conditions for stochastically complete Riemannian manifolds with parallel Cotton tensor to be either Einstein or of constant sectional curvature, and obtain an optimal pinching theorem. In particular, when $n$ = 4, we give a full classification.

The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in $\mathbf{P}^2$. The first generalization of this theorem to singular (quasi-projective) varieties was given by the first author. In both cases, the relations are generated by the standard monodromy variation operators associated with the special members of a generic pencil of hyperplane sections. In the present paper, we give a new generalization in which the relations are generated by the relative monodromy variation operators introduced by D. Chéniot and the first author. The advantage of using the relative operators is not only to cover a larger class of varieties but also to unify the Zariski-van Kampen type theorems for the fundamental group and for higher homotopy groups. In the special case of non-singular varieties, the main result of this paper was conjectured by D. Chéniot and the first author.

For a fixed integer $n \geq 1$, let $p=2n\ell+1$ be a prime number with an odd prime number $\ell$, and let $F=F_{p,\ell}$ be the real abelian field of conductor $p$ and degree $\ell$. We show that the class number $h_F$ of $F$ is odd when 2 remains prime in the real $\ell$th cyclotomic field $\mathbf{Q}(\zeta_{\ell})^+$ and $\ell$ is sufficiently large.

We show a de Rham theorem for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in [8, 9] produce cocycles of quandles.

Let $G$ be a compact Lie group and $T$ its maximal torus. In this paper, we try to compute $gr_{\gamma}^*(G/T)$ the graded ring associated with the gamma filtration of the complex $K$-theory $K^0(G/T)$. We use the Chow rings of corresponding versal flag varieties.

We obtain two characterizations of an odd-dimensional unit sphere of dimension $>3$ by proving the following two results: (i) If a complete connected $\eta$-Einstein $K$-contact manifold $M$ of dimension $>3$ admits a conformal vector field $V$, then either $M$ is isometric to a unit sphere, or $V$ is an infinitesimal automorphism of $M$. (ii) If $V$ was a projective vector field in (i), then the same conclusions would hold, except in the first case, $M$ would be locally isometric to a unit sphere.

This note is about the Hilbert square $X=S^{[2]}$, where $S$ is a general $K3$ surface of degree 10, and the anti-symplectic birational involution $\iota$ of $X$ constructed by O'Grady. The main result is that the action of $\iota$ on certain pieces of the Chow groups of $X$ is as expected by Bloch's conjecture. Since $X$ is birational to a double EPW sextic $X^\prime$, this has consequences for the Chow ring of the EPW sextic $Y\subset\mathbf{P}^5$ associated to $X^\prime$.