We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for non-trivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we confirm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic $K$-theory, Anderson and Payne for operational $K$-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.

We consider Calabi–Yau 3-folds of Borcea–Voisin type, i.e. Calabi–Yau 3-folds obtained as crepant resolutions of a quotient $(S\times E)/(\alpha_S\times \alpha_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $\alpha_S\in \Aut (S)$ and $\alpha_E\in \Aut (E)$ act on the period of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very classical, the case $n=3$ was recently studied by Rohde, the other cases are less known. First, we construct explicitly a crepant resolution, $X$, of $(S\times E)/(\alpha_S\times \alpha_E)$ and we compute its Hodge numbers; some pairs of Hodge numbers we found are new. Then, we discuss the presence of maximal automorphisms and of a point with maximal unipotent monodromy for the family of $X$. Finally, we describe the map $\mathcal{E}_n: X \rightarrow S/\alpha_S$ whose generic fiber is isomorphic to $E$.

A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.

We prove the existence of limiting laws for symmetric stable-like processes penalized by generalized Feynman-Kac functionals and characterize them by the gauge functions and the ground states of Sch\"rodinger type operators.

This paper is devoted to studying the nonlinear problem with subcritical exponent $(S_\varepsilon) : -\Delta_g u+2u = K|u|^2-\varepsilon u$, in $ S^4_+ $, ${\partial u}/{\partial\nu} =0$, on $\partial S^4_+$, where $g$ is the standard metric of $S^4_+$ and $K$ is a $C^3$ positive Morse function on $\overline{S_+^4}$. We construct some sign-changing solutions which blow up at two different critical points of $K$ in interior. Furthermore, we construct sign-changing solutions of $(S_\varepsilon)$ having two bubbles and blowing up at the same critical point of $K$.

We study many properties concerning weak Kählerianity on compact complex manifolds which admits a holomorphic submersion onto a Kähler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kähler case), Michelsohn (the balanced case), Popovici (the sG case) and others.

Wintgen ideal submanifolds in space forms are those ones attaining the equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Möbius geometry, we show that in the codimension two case, the mean curvature spheres of the Wintgen ideal submanifold correspond to a 1-isotropic holomorphic curve in a complex quadric. Conversely, any 1-isotropic complex curve in this complex quadric describes a 2-parameter family of spheres whose envelope is always a Wintgen ideal submanifold of codimension two at the regular points. Via a complex stereographic projection, we show that our characterization is equivalent to Dajczer and Tojeiro’s previous description of these submanifolds in terms of minimal surfaces in the Euclidean space.

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and aN with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.