The aim of this article is to give for each dimension d≥2 an infinite series of rigid compact complex manifolds which are not infinitesimally rigid and, hence, to give an exhaustive answer to a problem of Morrow and Kodaira stated in the famous book Complex Manifolds.

We consider hedgehog solutions in the .3 C 1/-dimensional Skyrme model, which is an energy-supercritical problem. We introduce a new strategy to prove global well-posedness for arbitrarily large initial data. © 2021 Duke University Press. All rights reserved.

By applying the residue method for period integrals, we prove identities between period integrals on cuspidal automorphic representations and period integrals on residual representations for seven spherical varieties. Combining this with the Langlands–Shahidi theory for residues of Eisenstein series, for each case, we prove some relations between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.

We prove a strong version of a theorem of Siu and Beauville on morphisms to higher genus curves, and we use it to show that if a variety X in characteristic p lifts to characteristic 0, then any morphism X→C to a curve of genus g≥2 can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over F¯p that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty $C^1$-interior.

Let X be a smooth curve over a finitely generated field k, and let l be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod l representations of the geometric fundamental group of X. Using this analysis, we prove several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey–Mazur conjecture for function fields in characteristic 0. For example, we show that if X is a normal, connected variety over C, then the (typically infinite) set of representations of π1(Xan) into GLn(Ql‾), which come from geometry, has no limit points. As a corollary, we deduce that if L is a finite extension of Ql, then the set of representations of π1(Xan) into GLn(L), which arise from geometry, is finite.

We compute the integral $p$-adic \'etale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational $p$-adic \'etale cohomology from Colmez-Dospinescu-Nizio{\l}. The main tools are: the computation of the integral de Rham cohomology from CDN and the integral $p$-adic comparison theorems of Bhatt-Morrow-Scholze and \v{C}esnavi\v{c}ius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used in CDN.

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

We introduce an explicit description of the Φ 3 4 measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale regularization parameter. Once small scale singularities have been renormalized by the standard counterterms, Γ -convergence allows us to extend the variational characterization to the unregularized model.

For a semisimple Lie group G satisfying the equal-rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In our work here we study some of the branching laws for discrete series when restricted to a subgroup H of the same type by combining classical results with recent work of Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra’s condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.