Acta Mathematica Articles (Project Euclid)
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The latest articles from Acta Mathematica on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 17 Aug 2017 13:00 EDTThu, 17 Aug 2017 13:00 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Global bifurcation of steady gravity water waves with critical layers
https://projecteuclid.org/euclid.acta/1502989201
<strong>Adrian Constantin</strong>, <strong>Walter Strauss</strong>, <strong>Eugen Vărvărucă</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 195--262.</p><p><strong>Abstract:</strong><br/>
We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.
</p>projecteuclid.org/euclid.acta/1502989201_20170817130019Thu, 17 Aug 2017 13:00 EDTMaximum independent sets on random regular graphs
https://projecteuclid.org/euclid.acta/1502989202
<strong>Jian Ding</strong>, <strong>Allan Sly</strong>, <strong>Nike Sun</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 263--340.</p><p><strong>Abstract:</strong><br/>
We determine the asymptotics of the independence number of the random d -regular graph for all ${d\geq d_0}$ . It is highly concentrated, with constant-order fluctuations around ${n\alpha_*-c_*\log n}$ for explicit constants ${\alpha_*(d)}$ and ${c_*(d)}$ . Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.
</p>projecteuclid.org/euclid.acta/1502989202_20170817130019Thu, 17 Aug 2017 13:00 EDTLocal Hodge theory of Soergel bimodules
https://projecteuclid.org/euclid.acta/1502989203
<strong>Geordie Williamson</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 341--404.</p><p><strong>Abstract:</strong><br/>
We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.
</p>projecteuclid.org/euclid.acta/1502989203_20170817130019Thu, 17 Aug 2017 13:00 EDTTits geometry and positive curvaturehttps://projecteuclid.org/euclid.acta/1505401886<strong>Fuquan Fang</strong>, <strong>Karsten Grove</strong>, <strong>Gudlaugur Thorbergsson</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 1--53.</p>projecteuclid.org/euclid.acta/1505401886_20170914111134Thu, 14 Sep 2017 11:11 EDTA hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-foldhttps://projecteuclid.org/euclid.acta/1505401887<strong>Radu Laza</strong>, <strong>Giulia Saccà</strong>, <strong>Claire Voisin</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 55--135.</p>projecteuclid.org/euclid.acta/1505401887_20170914111134Thu, 14 Sep 2017 11:11 EDTEquivariant Dirac operators and differentiable geometric invariant theoryhttps://projecteuclid.org/euclid.acta/1505401888<strong>Paul-Emile Paradan</strong>, <strong>Michèle Vergne</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 137--199.</p>projecteuclid.org/euclid.acta/1505401888_20170914111134Thu, 14 Sep 2017 11:11 EDTHitchin characters and geodesic laminationshttps://projecteuclid.org/euclid.acta/1517426684<strong>Francis Bonahon</strong>, <strong>Guillaume Dreyer</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 201--295.</p><p><strong>Abstract:</strong><br/>
For a closed surface $S$, the Hitchin component $\mathrm{Hit}_n (S)$ is a preferred component of the character variety consisting of group homomorphisms from the fundamental group $\pi_1(S)$ to the Lie group $\mathrm{PSL}_n (\mathbb{R})$. We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination $\lambda$ on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space $\mathcal{T}(S)$ by shearing coordinates associated with $\lambda$, corresponding to the case $n=2$. However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.
</p>projecteuclid.org/euclid.acta/1517426684_20180131142450Wed, 31 Jan 2018 14:24 ESTEnumeration of points, lines, planes, etc.https://projecteuclid.org/euclid.acta/1517426685<strong>June Huh</strong>, <strong>Botong Wang</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 297--317.</p><p><strong>Abstract:</strong><br/>
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(r-p)$-dimensional subspaces as there are $p$-dimensional subspaces, for every $p$ at most $\frac{1}{2} r$. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.
</p>projecteuclid.org/euclid.acta/1517426685_20180131142450Wed, 31 Jan 2018 14:24 ESTThe tempered spectrum of a real spherical spacehttps://projecteuclid.org/euclid.acta/1517426686<strong>Friedrich Knop</strong>, <strong>Bernhard Krötz</strong>, <strong>Henrik Schlichtkrull</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 319--383.</p><p><strong>Abstract:</strong><br/>
Let $G/H$ be a unimodular real spherical space which is either absolutely spherical, i.e. the real form of a complex spherical space, or of wave-front type. It is shown that every tempered representation for $G/H$ embeds into a twisted discrete series for a boundary degeneration of $G/H$. If $G/H$ is of wave-front type it follows that the tempered representation is parabolically induced by a twisted discrete series representation for a real spherical space formed by a Levi subgroup.
</p>projecteuclid.org/euclid.acta/1517426686_20180131142450Wed, 31 Jan 2018 14:24 ESTAsymptotic behavior of flows by powers of the Gaussian curvaturehttps://projecteuclid.org/euclid.acta/1517430210<strong>Simon Brendle</strong>, <strong>Kyeongsu Choi</strong>, <strong>Panagiota Daskalopoulos</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 1--16.</p><p><strong>Abstract:</strong><br/>
We consider a $1$-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $-K^{\alpha} ν$, where ν denotes the outward-pointing unit normal vector and $\alpha \geqslant 1 / (n+2)$. For $\alpha \gt 1 / (n+2)$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha = 1 / (n+2)$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.
</p>projecteuclid.org/euclid.acta/1517430210_20180131152337Wed, 31 Jan 2018 15:23 ESTCorrection to “On the density of geometrically finite Kleinian groups”https://projecteuclid.org/euclid.acta/1517430211<strong>Jeffrey F. Brock</strong>, <strong>Kenneth W. Bromberg</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 17--19.</p><p><strong>Abstract:</strong><br/>
This erratum corrects an error arising in the proof of Proposition 6.4 in the article “On the density of geometrically finite Kleinian groups” [Brock, J. F. & Bromberg, K. W., Acta Math. , 192 (2004), 33–93].
</p>projecteuclid.org/euclid.acta/1517430211_20180131152337Wed, 31 Jan 2018 15:23 ESTBernstein- and Markov-type inequalities for rational functionshttps://projecteuclid.org/euclid.acta/1517430212<strong>Sergei Kalmykov</strong>, <strong>Béla Nagy</strong>, <strong>Vilmos Totik</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 21--63.</p><p><strong>Abstract:</strong><br/>
Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.
</p>projecteuclid.org/euclid.acta/1517430212_20180131152337Wed, 31 Jan 2018 15:23 ESTSingular Ricci flows Ihttps://projecteuclid.org/euclid.acta/1517430213<strong>Bruce Kleiner</strong>, <strong>John Lott</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 65--134.</p><p><strong>Abstract:</strong><br/> We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the $3$-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows. </p>projecteuclid.org/euclid.acta/1517430213_20180131152337Wed, 31 Jan 2018 15:23 ESTQuantum indices and refined enumeration of real plane curveshttps://projecteuclid.org/euclid.acta/1517430214<strong>Grigory Mikhalkin</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 135--180.</p><p><strong>Abstract:</strong><br/>
We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve, and thus has a discrete spectrum of values. We use the quantum index to refine enumeration of real rational curves in a way consistent with the Block–Göttsche invariants from tropical enumerative geometry.
</p>projecteuclid.org/euclid.acta/1517430214_20180131152337Wed, 31 Jan 2018 15:23 ESTGauduchon metrics with prescribed volume formhttps://projecteuclid.org/euclid.acta/1517430215<strong>Gábor Székelyhidi</strong>, <strong>Valentino Tosatti</strong>, <strong>Ben Weinkove</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 181--211.</p><p><strong>Abstract:</strong><br/>
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
</p>projecteuclid.org/euclid.acta/1517430215_20180131152337Wed, 31 Jan 2018 15:23 ESTGlobal solutions of the gravity-capillary water-wave system in three dimensionshttps://projecteuclid.org/euclid.acta/1525299701<strong>Yu Deng</strong>, <strong>Alexandru D. Ionescu</strong>, <strong>Benoît Pausader</strong>, <strong>Fabio Pusateri</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 2, 213--402.</p><p><strong>Abstract:</strong><br/>
In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models.
To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than ${\lvert t \rvert}^{- 5/6}$) and the presence of a large, codimension-$1$, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system.
The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension $1$. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.
</p>projecteuclid.org/euclid.acta/1525299701_20180502182144Wed, 02 May 2018 18:21 EDTThe global non-linear stability of the Kerr–de Sitter family of black holeshttps://projecteuclid.org/euclid.acta/1560966510<strong>Peter Hintz</strong>, <strong>András Vasy</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 1, 1--206.</p><p><strong>Abstract:</strong><br/>
We establish the full global non-linear stability of the Kerr–de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: we develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations. In particular, the iteration scheme used to solve Einstein’s equations automatically finds the parameters of the Kerr–de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.
</p>projecteuclid.org/euclid.acta/1560966510_20190619134844Wed, 19 Jun 2019 13:48 EDTSimple homotopy equivalence of nearby Lagrangianshttps://projecteuclid.org/euclid.acta/1560966691<strong>Mohammed Abouzaid</strong>, <strong>Thomas Kragh</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 2, 207--237.</p><p><strong>Abstract:</strong><br/>
Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.
</p>projecteuclid.org/euclid.acta/1560966691_20190619135141Wed, 19 Jun 2019 13:51 EDTAlgebraic actions of discrete groups: the $p$-adic methodhttps://projecteuclid.org/euclid.acta/1560966692<strong>Serge Cantat</strong>, <strong>Junyi Xie</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 2, 239--295.</p><p><strong>Abstract:</strong><br/>
We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on $p$-adic analysis, the second makes use of isoperimetric inequalities and Lang–Weil estimates. For instance, we show that, if $\mathsf{SL}_n(\mathbf{Z})$ acts faithfully on a complex quasi-projective variety $X$ by birational transformations, then $\mathrm{dim}(X) \geqslant n-1$ and $X$ is rational if $\mathrm{dim}(X) = n-1$.
</p>projecteuclid.org/euclid.acta/1560966692_20190619135141Wed, 19 Jun 2019 13:51 EDTSemiclassical measures on hyperbolic surfaces have full supporthttps://projecteuclid.org/euclid.acta/1560966693<strong>Semyon Dyatlov</strong>, <strong>Long Jin</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 2, 297--339.</p><p><strong>Abstract:</strong><br/>
We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in “Spectral gaps, additive energy, and a fractal uncertainty principle” [Dyatlov, S. & Zahl, J. Geom. Funct. Anal., 26 (2016), 1011–1094] and proved for porous sets in “Spectral gaps without the pressure condition” [Bourgain, J. & Dyatlov, S. Ann. of Math., 187 (2018), 825–867].
</p>projecteuclid.org/euclid.acta/1560966693_20190619135141Wed, 19 Jun 2019 13:51 EDTStable rationality of quadric surface bundles over surfaceshttps://projecteuclid.org/euclid.acta/1560966694<strong>Brendan Hassett</strong>, <strong>Alena Pirutka</strong>, <strong>Yuri Tschinkel</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 2, 341--365.</p><p><strong>Abstract:</strong><br/>
We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.
</p>projecteuclid.org/euclid.acta/1560966694_20190619135141Wed, 19 Jun 2019 13:51 EDTConvergence and divergence of formal CR mappingshttps://projecteuclid.org/euclid.acta/1560966695<strong>Bernhard Lamel</strong>, <strong>Nordine Mir</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 220, Number 2, 367--406.</p><p><strong>Abstract:</strong><br/>
Let $M \subset \mathbb{C}^N$ be a generic real-analytic submanifold of finite type, $M' \subset \mathbb{C}^{N'}$ be a real-analytic set, and $p \in M$, where we assume that $N, N' \geqslant 2$. Let $H: (\mathbb{C}^N, p) \to \mathbb{C}^{N'}$ be a formal holomorphic mapping sending $M$ into $M'$, and let $\mathcal{E}_{M'}$ denote the set of points in $M'$ through which there passes a complex-analytic subvariety of positive dimension contained in $M'$. We show that, if $H$ does not send $M$ into $\mathcal{E}_{M'}$, then $H$ must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when $M'$ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.
</p>projecteuclid.org/euclid.acta/1560966695_20190619135141Wed, 19 Jun 2019 13:51 EDTCharacter bounds for finite groups of Lie typehttps://projecteuclid.org/euclid.acta/1560966800<strong>Roman Bezrukavnikov</strong>, <strong>Martin W. Liebeck</strong>, <strong>Aner Shalev</strong>, <strong>Pham Huu Tiep</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 1, 1--57.</p><p><strong>Abstract:</strong><br/>
We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $\lvert \chi(g) \rvert \leqslant c \chi (1)^{\alpha g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular, we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.
</p>projecteuclid.org/euclid.acta/1560966800_20190619135324Wed, 19 Jun 2019 13:53 EDTOn the structure of band edges of $2$-dimensional periodic elliptic operatorshttps://projecteuclid.org/euclid.acta/1560966801<strong>Nikolay Filonov</strong>, <strong>Ilya Kachkovskiy</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 1, 59--80.</p><p><strong>Abstract:</strong><br/>
For a wide class of $2$-dimensional periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.
</p>projecteuclid.org/euclid.acta/1560966801_20190619135324Wed, 19 Jun 2019 13:53 EDTRestriction estimates using polynomial partitioning IIhttps://projecteuclid.org/euclid.acta/1560966802<strong>Larry Guth</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 1, 81--142.</p><p><strong>Abstract:</strong><br/>
We improve the estimates in the restriction problem in dimension $n \geqslant 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and $n$.
</p>projecteuclid.org/euclid.acta/1560966802_20190619135324Wed, 19 Jun 2019 13:53 EDTExamples of finite free complexes of small rank and small homologyhttps://projecteuclid.org/euclid.acta/1560966803<strong>Srikanth B. Iyengar</strong>, <strong>Mark E. Walker</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 1, 143--158.</p><p><strong>Abstract:</strong><br/>
This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.
</p>projecteuclid.org/euclid.acta/1560966803_20190619135324Wed, 19 Jun 2019 13:53 EDTIsoperimetric characterization of upper curvature boundshttps://projecteuclid.org/euclid.acta/1560966804<strong>Alexander Lytchak</strong>, <strong>Stefan Wenger</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 1, 159--202.</p><p><strong>Abstract:</strong><br/>
We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.
</p>projecteuclid.org/euclid.acta/1560966804_20190619135324Wed, 19 Jun 2019 13:53 EDTOn topological cyclic homologyhttps://projecteuclid.org/euclid.acta/1560967045<strong>Thomas Nikolaus</strong>, <strong>Peter Scholze</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 221, Number 2, 203--409.</p><p><strong>Abstract:</strong><br/>
Topological cyclic homology is a refinement of Connes–Tsygan’s cyclic homology which was introduced by Bökstedt–Hsiang–Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas–Goodwillie–McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum.
The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p : X \to X^{t C_p}$ for all primes $p$. Here, $X^{t C_p} = \mathrm{cofib}(\mathrm{Nm} : X^{h C_p} \to X^{h C_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology.
In order to construct the maps $\varphi_p : X \to X^{t C_p}$ in the example of topological Hochschild homology, we introduce and study Tate-diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular, we prove a version of the Segal conjecture for the Tate-diagonals and relate these Frobenius homomorphisms to power operations.
</p>projecteuclid.org/euclid.acta/1560967045_20190619135728Wed, 19 Jun 2019 13:57 EDT