Given a classical semisimple complex algebraic group $G$ and a symmetric pair $(G,K)$ of Hermitian type, we study the closures of the spherical nilpotent $K$-orbits in the isotropy representation of $K$. We show that all such orbit closures are normal and describe the $K$-module structure of their ring of regular functions.

We consider symmetries of Julia sets of polynomial skew products on ${\mathbb{C}}^2$, which are birationally conjugate to rotational products. Our main result gives the classification of the polynomial skew products whose Julia sets have infinitely many symmetries.

We formulate a multivariable $p$-adic Birch and Swinnerton-Dyer conjecture for $p$-ordinary elliptic curves $A$ over number fields $K$. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case $K=\mathbf{Q}$ and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions, in the following situation: $K$ is imaginary quadratic, $A=E_K$ is the base change to $K$ of an elliptic curve over the rationals, and the rank of $A$ is either 0 or $1$.

The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over $\mathbf{Q}$, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank $1$, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) $p$-adic Gross–Zagier and Waldspurger formulas in families.

We study deformations of nonsingular Poisson varieties and Poisson invertible sheaves, which extend the flat deformation theory of nonsingular varieties and invertible sheaves. In the Appendix, we study deformations of Poisson vector bundles. We identify first-order deformations and obstructions.

The main purpose of this article is to study $C^k$-regularity ($k=0,1,\dots$) for the Cauchy–Riemann equation $$\overline{\partial }u=\varphi$$ on a smoothly bounded, convex domain of infinite type in ${\mathbb{C}}^2$.

In this paper, we obtain boundedness of the maximal operator associated with multilinear Calderón–Zygmund singular integral operators from a product of variable Hardy spaces into variable Lebesgue spaces.

We prove that the equivariant higher-index map is injective for a bounded geometry metric space with a proper and isometric group action which can be equivariantly embedded into a simply connected, complete, and nonpositively curved Riemannian manifold with a proper and isometric group action.

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, $1$-motives.

The second de Rham cohomology groups of nilpotent orbits in noncompact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in noncompact classical simple Lie algebras are computed, and we prove them to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of the first two authors, may be of independent interest.