International Journal of Mathematics

This paper concerns hyperkähler fourfolds $X$ having a non-symplectic involution $\iota$. The Bloch–Beilinson conjectures predict the way $\iota$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/\iota$.

We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank $9$. We give a fundamental system of solutions to this system in terms of this hypergeometric series. We give circuit matrices along generators of the fundamental group of the complement of its singular locus with respect to our fundamental system.

During the development of the theory of slice regular functions over the real algebra of quaternions $ℍ$ in the last decade, some natural questions arose about slice regular functions on the open unit ball $𝔹$ in $ℍ$. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of $𝔹$ fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps $𝔹\to 𝔹$ that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps $f$ of the complex unit disk with $f(0)=0$. Landau had computed, in terms of $a:=|f^{\prime }(0)|$, a radius $\rho$ such that $f$ is injective at least in the disk $\mathrm{\Delta }(0,\rho )$ and such that the inclusion $f(\mathrm{\Delta }(0,\rho ))\supseteq \mathrm{\Delta }(0,{\rho }^2)$ holds. The analogous result proven here for slice regular functions $𝔹\to 𝔹$ allows a new approach to the study of Bloch–Landau-type properties of slice regular functions $𝔹\to ℍ$.

In this paper, we study the convergence in the capacity of sequence of plurisubharmonic functions. As an application, we prove stability results for solutions of the complex Monge–Ampère equations.

We consider holomorphic foliations of dimension $k>1$ and codimension $\ge 1$ in the projective space ${\mathbb{P}}^n$, with a compact connected component of the Kupka set. We prove that if the transversal type is linear with positive integer eigenvalues, then the foliation consists of the fibers of a rational fibration $\mathrm{\Phi }:{\mathbb{P}}^n--\to {\mathbb{P}}^{n-k}$. As a corollary, if $dim({ℱ})\ge \text{cod}({ℱ})+2$ and has a transversal type diagonal with different eigenvalues, then the Kupka component $K$ is a complete intersection and the leaves of the foliation define a rational fibration. The same conclusion holds if the Kupka set has a radial transversal type. Finally, as an application, we find a normal form for non-integrable codimension-one distributions on ${\mathbb{P}}^n$.

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $𝔟(p,q)$ is minimal where $q\le 6$ or $p\le 100$.

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces ${\mathbb{P}}^n({𝒯})$ constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra $C({\mathbb{P}}^1({𝒯}))$ realized as a concrete groupoid C*-algebra, and find its $K$-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra $C({\mathbb{P}}^1({𝒯}))$, we identify those quantum principal $U(1)$-bundles introduced by Hajac and collaborators among the projections classified.