Considérons un espace de Berkovich sur un bon anneau de Banach et la droite projective relative sur celui-ci. (C’est un espace dont les fibres sont des droites projectives sur différents corps valués complets.) Pour tout endomorphisme polarisé de cette droite, nous montrons que la famille des mesures d’équilibre associées aux restrictions de l’endomorphisme aux fibres est continue. Le résultat vaut, par exemple, lorsque l’anneau de Banach est un corps valué complet, un corps hybride, un anneau de valuation discrète complet ou un anneau d’entiers de corps de nombres.

For every , we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures, and thus, the ergodic measure space is not connected, while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti [11, Conjecture 2] in $C^r$-topology for Lorenz attractors.

We compute the $K$-theory of the ${C}^{*}$-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group $G$. Our result uses the Connes-Kasparov isomorphism for $G$, and in fact it is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan’s well-known parametrization of the tempered irreducible representations of $G$ with real infinitesimal character. When the reductive group $G$ has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a $K$-theoretic version of Vogan’s theorem.

This is the first part of a series of papers devoted to the study of linear cocycles over chaotic systems. In the present paper, we establish the existence of such cocycles that $\mathcal {C}^\alpha $-stably exhibit fiberwise bounded orbits ($\alpha>0$). The proof is based on a new mechanism which yields stable elliptic-type behavior in $\mathrm {GL}(d,\mathbb {R})$ or $\mathrm {SL}(d,\mathbb {R})$ cocycles. Moreover, we show that this phenomenon is $\mathcal {C}^0$-dense among $\mathrm {SL}(d,\mathbb {R})$ cocycles over a shift of finite type without dominated splitting.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

For a uniformly locally finite metric space $(X, d)$, we investigate coarse flows on its uniform Roe algebra $\mathrm {C}^*_u(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on X. We first show that any flow $\sigma $ on $\mathrm {C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator h on $\ell _2(X)$ such that $\sigma _t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb {R}$, allowing us to focus on operators h that generate flows on $\mathrm {C}^*_u(X)$.

Assuming Yu’s property A, we prove that a self-adjoint operator h on $\ell _2(X)$ induces a coarse flow on $\mathrm {C}^*_u(X)$ if and only if h can be expressed as $h = a + d$, where $a \in \mathrm {C}^*_u(X)$ and d is a diagonal operator with entries forming a coarse function on X. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators h and k that induce coarse flows on $\mathrm {C}^*_u(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate to each other. In particular, if $h - k$ is bounded, then the flow induced by h is a cocycle perturbation of the flow induced by k.

We extend the notion of the J-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the J-invariant.

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

We establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze’s basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results.

Our key new tool is the six functor formalism with solid almost $\mathcal {O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain ‘stacky’ maps. In the language of this extended formalism, we show that if G is a p-adic Lie group, the structure map of the classifying small v-stack $B\underline {G}$ is p-cohomologically smooth.

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity $1$, threefold singularities, and hypersurface singularities. Given a real value $v>0$, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-$1$ and n-dimensional hypersurface singularities for arbitrary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-$1$ torus action. We exhibit a $3$-dimensional example which shows that this last statement is optimal.

We construct a relative version of the $A_{ {\mathrm {inf}}}$-cohomology theory developed by Bhatt-Morrow-Scholze and relate it to the prismatic theory of Bhatt-Scholze. The construction relies on the fiber product of topoi. As an application we show that there is an étale comparison of the q-crystalline pushforward after inverting $\mu \in A_{ {\mathrm {inf}}}$.

In this work, we conclude our study of fibred $\infty $-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $-bicategory of 2-Cartesian fibrations over S and the $\infty $-bicategory of contravariant functors with values in the $\infty $-bicategory of $\infty $-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

We show that any free action of a connected Lie group of polynomial growth on a finite dimensional locally compact space has finite tube dimension. This is shown to imply that the associated crossed product C*-algebra has finite nuclear dimension. As an application we show that C*-algebras associated with certain aperiodic point sets in connected Lie groups of polynomial growth are classifiable. Examples include cut-and-project sets constructed from irreducible lattices in products of connected nilpotent Lie groups.

We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: suppose a collection of Lagrangian branes satisfy Abouzaid’s criterion [Abo10] for split-generation of a bulk-deformed Fukaya category of cleanly intersecting Lagrangian branes. We show (Theorem 1.1) that for a small blowup parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer [Bay04] and is an example of a more general conjectural description of the behaviour of the Fukaya category under transitions occurring in the minimal model program, namely that minimal model program transitions generate additional summands.

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin$^c$ structure determines a family of Dirac operators $\{D_B\}$ parametrized by a $b_1(Y)$-dimensional torus $\mathbb {T}_Y$. In this paper, we develop techniques to study how the topology of the locus $\mathsf {K}\subset \mathbb {T}_Y$ corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus $Y=T^3$ for which the spectral gap $\lambda _1^*$ on coexact $1$-forms is large, after a small perturbation of the family, the locus $\mathsf {K}$ is a two-sphere.

While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak {s})$ with a large $\lambda _1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the topology of the family of Dirac operators $\{D_B\}$.

We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result solves a conjecture proposed at a conference in Göttingen in 2003. This result also leads to a higher Cheeger-Müller/Bismut-Zhang theorem. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten-type deformation on the flat vector bundle.

We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega ,{ \mathcal J})$, allowing $0$ to be a singular value of the moment map ${ \mathcal J}:M\to {\mathbb R}$. Our formula represents an instance of the Guillemin-Sternberg principle, which states that quantization should commute with reduction. The conceptual novelty of our result is that the involved reduced system only depends on the symplectic data of M. To establish this, we derive a complete singular stationary phase expansion of the Witten integral without appealing to any kind of desingularization. As a consequence, our formula expresses the invariant Riemann-Roch number purely in terms of symplectic invariants of the singular symplectic quotient. In particular, it involves a new explicit symplectic invariant of the singularities.

We prove rigidity and gap theorems for self-dual and even Poincaré-Einstein metrics in dimension four. As a corollary, we give an obstruction to the existence of self-dual Poincaré-Einstein metrics in terms of conformal invariants of the boundary and the topology of the bulk. As a by-product of our proof, we identify a new scalar conformal invariant of three-dimensional Riemannian manifolds.

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety A, we can construct a CM abelian variety B such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, B splits over the Galois closure of the CM field associated with A.