Band 2014, Heft 695

We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from cycle complex mod p r to Moser's complex of logarithmic de Rham–Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex. In the end, we prove that a weak form of the Gersten conjecture implies that Sato's dualizing complex is quasi-isomorphic to Bloch's complex.

In this work we present new tools for studying the variations of the Willmore functional of immersed surfaces into ℝ m . This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in ℝ m , away possibly from isolated branched points, which minimizes the Willmore energy among all possible Lipschitz immersions in ℝ m having an L 2 -bounded second fundamental form and realizing this conformal class. This branched immersion is either a smooth conformal Willmore branched immersion or an isothermic branched immersion. We show that branched points do not exist whenever the minimal energy in the conformal class is less than 8π and that these immersions extend to smooth conformal Willmore embeddings or global isothermic embeddings of the surface in that case. Finally, as a by-product of our analysis, we establish that inside a compact subspace of the moduli space the following holds: the weak limit of Palais Smale Willmore sequences are conformal Willmore, the weak limits of Palais Smale sequences of conformal Willmore are either conformal Willmore or global isothermic and finally we observe also that weakly converging Palais Smale sequences of global isothermic immersions are global isothermic. The analysis developed along the paper – in particular these last results – opens the door to the possibility of constructing new critical saddle points of the Willmore functional without or with constraints using min-max methods.

In this paper we make a detailed comparison between the Paley–Wiener theorems of J. Arthur and P. Delorme for a real reductive Lie group G . We prove that these theorems are equivalent from an a priori point of view. We also give an alternative formulation of the theorems in terms of the Hecke algebra of bi- K -finite distributions supported on K , a maximal compact subgroup of G . Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules.

We give a method to estimate Seshadri constants on toric varieties at any point. By using the estimations and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number 1 in detail.

We study properties of the central sequence algebra Aω∩A'${A_\omega \cap A^{\prime }}$ of a C * -algebra A , and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C * -algebra, whose trace simplex is finite-dimensional, tensorially absorbs the Jiang–Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C * -algebra of having the strict comparison property to a weaker property that we call local weak comparison . Namely, we prove that a unital separable simple nuclear C * -algebra, whose trace simplex has finite-dimensional closed extreme boundary, tensorially absorbs the Jiang–Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II 1 -factor representation of the C * -algebra is a McDuff factor.

Erratum to http://dx.doi.org/10.1515/crelle-2012-0118.

Given a compact manifold with boundary X 0 endowed with a foliation ℱ0${\mathcal {F}_0}$ transverse to the boundary, and which admits a holonomy invariant transverse measure Λ, we define three types of signature for the pair (foliation, boundary foliation): the analytic signature, denoted by σΛ,an(X0,∂X0)${\sigma _{\Lambda ,\operatorname{an}}(X_0,\partial X_0)}$, is the leafwise L 2 -Λ-index of the signature operator on the extended manifold X obtained by attaching cylindrical ends to the boundary; the Hodge signature σΛ,Hodge(X0,∂X0)${\sigma _{\Lambda ,\operatorname{Hodge}}(X_0,\partial X_0)}$ is defined using the natural representation of the Borel groupoid ℛ${\mathcal {R}}$ of X on the field of square integrable harmonic forms on the leaves; and the de Rham signature, σΛ,dR(X0,∂X0)${\sigma _{\Lambda ,\operatorname{dR}}(X_0,\partial X_0)}$, defined using the natural representation of the Borel groupoid ℛ0${\mathcal {R}_0}$ of X 0 on the field of the L 2 -relative de Rham spaces of the leaves. We prove that these three signatures coincide σΛ,an(X0,∂X0)=σΛ,Hodge(X0,∂X0)=σΛ,dR(X0,∂X0).$ \sigma _{\Lambda , \operatorname{an}}(X_0,\partial X_0)= \sigma _{\Lambda ,\operatorname{Hodge}}(X_0,\partial X_0)=\sigma _{\Lambda ,\operatorname{dR}}(X_0,\partial X_0). $ As a consequence of the index formula we proved in [Bull. Sci. Math. (2010), DOI 10.1016/ j.bulsci.2010.10.003], we finally obtain the Atiyah–Patodi–Singer signature formula for measured foliations: σΛ,dR(X0,∂X0)=〈L(Tℱ0),CΛ〉+1/2[ηΛ(Dℱ∂)].$ \sigma _{\Lambda ,\operatorname{dR}}(X_0,\partial X_0)=\langle L(T\mathcal {F}_0),C_{\Lambda }\rangle +1/2[\eta _{\Lambda }(D^{\mathcal {F}_{\partial }})]. $