Recent zbMATH articles in MSC 58Bhttps://zbmath.org/atom/cc/58B2022-07-25T18:03:43.254055ZWerkzeugNoncommutative differential \(K\)-theoryhttps://zbmath.org/1487.190102022-07-25T18:03:43.254055Z"Park, Byungdo"https://zbmath.org/authors/?q=ai:park.byungdo"Parzygnat, Arthur J."https://zbmath.org/authors/?q=ai:parzygnat.arthur-j"Redden, Corbett"https://zbmath.org/authors/?q=ai:redden.corbett"Stoffel, Augusto"https://zbmath.org/authors/?q=ai:stoffel.augustoThis paper introduces the notion of noncommutative differential \(K\)-theory associated to a (not necessarily commutative) \(k\)-algebra \(A\) and a differentially graded algebra \(\Omega_*(A)\) satisfying \(\Omega_0(A)=A\), called a DGA on top of \(A\). In the special case where \(A=C^{\infty}(X,\mathbb{C})\) and \(\Omega_*(A)=\Omega^*(X;\mathbb{C})\), with \(X\) a smooth manifold, it is shown that this noncommutative differential \(K\)-theory reduces to the usual differential \(K\)-theory of the smooth manifold \(X\).
The paper begins with a review of differential \(K\)-theory and Karoubi's noncommutative chern character, which takes values in the abelianization of a DGA on top of \(A\). Taking the DGA on top of \(A\) to be the exterior algebra \(\Omega_*^u(A)\) of (the noncommutative generalization of) Kahler differentials provides a universal example: given any DGA \(\Omega_*(A)\) that is functorial in \(A\), there is a canonical map \(\Omega_*^u(A)\to \Omega_*(A)\) that arises as the \(A\) component of the initial morphism in the category of DGA's on top of algebras.
The paper proceeds by defining the Karoubi-Chern-Simons transgression form \(KCS(D_t)\), associated to a polynomial path of connections on a finitely generated projective module over \(A\). The Karoubi-Chern-Simons form takes values in the abelianization of \(\Omega_*(A)\) and it is shown that \(dKCS(D_t)=\mathrm{ch}(D_1)-\mathrm{ch}(D_0)\), where \(\mathrm{ch}\) denotes the Karoubi Chern character, as expected. Further properties of \(KCS\) are also explored.
Given a DGA \(\Omega_*(A)\) over \(A\), the noncommutative differential \(K\)-theory \(\widehat{K}_0(A)\) is then defined to be the Grothendieck group completion of the monoid whose undelying set is given by taking equivalence classes of triples \((M,D,\omega)\), where \(M\) is a finitely generated projective module over \(A\), \(D\) is a connection on \(A\) and \(\omega\) is an odd form in the abelianization of \(\Omega^*(A)\). Two triples \((M_0,D_0,\omega_0)\) and \((M_1,D_1,\omega_1)\) are equivalent if there is a module \(N\) with connection \(D\) and an isomorphism \(\phi:M_0\oplus N\to M_1\oplus N\) such that \(KCS(D_0\oplus D,\phi^*(D_1\oplus D))=\omega_1-\omega_0\), modulo exact forms. The monoid operation is given by taking the direct sum of modules and connections and the sum of differential forms. The map that sends a triple \((M,D,\omega)\) to the module \(M\) isnduces a morphism \(I:\widehat{K}_0(A)\to K_0(A)\) and the Karoubi Chern character refines to a homomorphism \(R:\widehat{K}_0(A)\to \Omega_*(A)_{\mathrm{ab}}\).
The paper concludes by showing that differential noncommutative \(K\)-theory fits into a hexagon diagram similar to the hexagon diagram for commutative differential \(K\)-theory.
Reviewer: Daniel Joseph Grady (New York)A symplectic form on the space of embedded symplectic surfaces and its reduction by reparametrizationshttps://zbmath.org/1487.321532022-07-25T18:03:43.254055Z"Kessler, Liat"https://zbmath.org/authors/?q=ai:kessler.liatSummary: Let \((M, \omega)\) be a symplectic manifold, and \((\Sigma, \sigma)\) a closed connected symplectic 2-manifold. We construct a weakly symplectic form \({\omega^D}\) on \(\operatorname{C}^{\infty}(\Sigma,M)\) which is a special case of Donaldson's form. We show that the restriction of~\({\omega^D}\) to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding \((\Sigma, \sigma) \hookrightarrow (M, \omega)\) descends to a weakly symplectic form on the quotient by \(\operatorname{Sympl}(\Sigma, \sigma)\), and that the symplectic space obtained is a symplectic quotient of the subspace of symplectic embeddings with respect\- to the \(\operatorname{Sympl}(\Sigma, \sigma)\)-action. We also compare \({\omega^D}\) to another 2-form. We conclude with a result on the restriction of \({\omega^D}\) to moduli spaces of holomorphic~curves.Lipschitz isomorphisms of compact quantum metric spaceshttps://zbmath.org/1487.460772022-07-25T18:03:43.254055Z"Long, Botao"https://zbmath.org/authors/?q=ai:long.botao"Wu, Wei"https://zbmath.org/authors/?q=ai:wu.wei.3Suppose that \(A\) is a real partially ordered vector space which contains an element \(1\) satisfying the following two properties: (i) for every \(a\) in \(A\), there exists a real number \(r\) such that \(a \leq r1\); and (ii) if \(a \leq r1\) for all positive real numbers \(r\), then \(a \leq 0\). Then we say that \(A\) is an order unit space. The formula \(\Vert a \Vert := \operatorname{inf}\{r \in [0, \infty) : -r1 \leq a \leq r1\}\) makes \(A\) into a real normed linear space; if \(A\) is complete in this norm, we say that \(A\) is a complete order unit space. The prototypical example of a complete order unit space is the real vector space of all self-adjoint elements in a unital \(C^*\)-algebra.
A Lipschitz seminorm on an order unit space (not necessarily complete) is a function \(L: A \longrightarrow [0, \infty]\) such that (i) the set \(\mathcal{A} := \{a \in A: L(a) < \infty\}\) is dense in \(A\), and (ii) \(L(a) = 0\) if and only if a is a real multiple of \(1\). If \(L\) is lower semicontinuous, the formula \(\Vert a \Vert_1 := \Vert a \Vert + L(a)\) makes \(\mathcal{A}\) into a real Banach space which is called a compact quantum metric space.
In the paper under review, the authors study morphisms and isomorphisms of such objects. They begin by stating and proving a variety of results about compact quantum metric spaces in general, and then consider the class of asymptotically \(k\)-nonexpansive maps from a compact quantum metric space \(\mathcal{A}\) to itself; these are maps \(\phi\) for which there exists a sequence \(\{k_n\}\) of positive real numbers going to infinity such that \(L(\phi^n(a)) \leq k_nL(a)\) for all \(a\) in \(\mathcal{A}\) and all natural numbers \(n\). In the final section of the paper, the authors consider asymptotically \(k\)-expansive maps between compact quantum metric spaces.
Reviewer: Efton Park (Fort Worth)Quasifolds, diffeology and noncommutative geometryhttps://zbmath.org/1487.580022022-07-25T18:03:43.254055Z"Iglesias-Zemmour, Patrick"https://zbmath.org/authors/?q=ai:iglesias-zemmour.patrick"Prato, Elisa"https://zbmath.org/authors/?q=ai:prato.elisaAuthors' abstract: After embedding the objects quasifolds into the category \(\{Diffeologyg\}\), we associate a \(C^{\star}\)-algebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry -- beginning with the today classical example of the irrational torus -- which associates a Morita class of \(C^{\star}\)-algebras with a diffeomorphic class of quasifolds.
Reviewer: Yong Wang (Changchun)On analytic Todd classes of singular varietieshttps://zbmath.org/1487.580042022-07-25T18:03:43.254055Z"Bei, Francesco"https://zbmath.org/authors/?q=ai:bei.francesco"Piazza, Paolo"https://zbmath.org/authors/?q=ai:piazza.paoloSummary: Let \((X,h)\) be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of \((X,h)\). In the 1st part, assuming either \(\dim (\mathrm{sing}(X))=0\) or \(\dim(X)=2\), we show that the rolled-up operator of the minimal \(L^2-\overline{\partial}\) complex, denoted here \(\overline{\eth}_{\mathrm{rel}}\), induces a class in \(K_0 (X)\equiv KK_0(C(X),\mathbb{C})\). A similar result, assuming \(\dim (\mathrm{sing}(X))=0\), is proved also for \(\overline{\eth}_{\mathrm{abs}}\), the rolled-up operator of the maximal \(L^2-\overline{\partial}\) complex. We then show that when \(\dim(\mathrm{sing}(X))=0\) we have \([\overline{\eth}_{\mathrm{rel}}]=\pi_*[\overline{\eth}_M]\) with \(\pi :M\rightarrow X\) an arbitrary resolution and with \([\overline{\eth}_M]\in K_0 (M)\) the analytic K-homology class induced by \(\overline{\partial}+\overline{\partial}^t\) on \(M\). In the 2nd part of the paper we focus on complex projective varieties \((V,h)\) endowed with the Fubini-Study metric. First, assuming \(\dim (V)\leq 2\), we compare the Baum-Fulton-MacPherson K-homology class of \(V\) with the class defined analytically through the rolled-up operator of any \(L^2-\overline{\partial}\) complex. We show that there is no \(L^2-\overline{\partial}\) complex on \((\mathrm{reg}(V),h)\) whose rolled-up operator induces a K-homology class that equals the Baum-Fulton-MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on \(V\) the push-forward of \([\overline{\eth}_{\mathrm{rel}}]\) in the K-homology of the classifying space of the fundamental group of \(V\) is a birational invariant.