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Paolo Piazza

Publications and source records attributed to Paolo Piazza.

At least 19 recordsLinked to original sources
arXiv2026

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps. Part 2

In this Part 2 of our article we give a detailed discussion of the compatibility between the analytic Gysin maps we have defined in Part 1 and the topological Gysin maps defined by the second author. A significant role is played by a bordism-like description of K-homology due to Jakob which is closely related to the geometric K-homology theory of Baum and Douglas. We give a self-contained proof of the equivalence of the former with the analytic K-homology theory of Kasparov. As an intermediate step towards proving our main result we use Thom's transversality theorem to describe Gysin maps compatibly with Jakob's definition of K-homology.

math.AT↗
arXiv2025

Higher Lefschetz formulas on {\Gamma}-proper manifolds

Let $\Gamma$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $\Gamma$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $\tau$ in $HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$. Our formula takes place on the fixed point manifold $M^\gamma$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_\gamma$-invariant form on $M^\gamma$ that is naturally associated to $\tau\in HP^\bullet (\mathbb{C}\Gamma,\langle \gamma \rangle)$

math.DG↗
arXiv2025

The G-signature Theorem on Witt spaces

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.

math.DG↗
arXiv2025

Classification of spin$^c$ manifolds with generalized positive scalar curvature

Suppose $M$ is a closed $n$-dimensional spin$^c$ manifold with spin$^c$ structure $\sigma$ and associated spin$^c$ line bundle $L$. If one fixes a Riemannian metric $g$ on $M$ and a connection $\nabla_L$ on $L$, the generalized scalar curvature $R^{\text{gen}}$ of $(M,L)$ is $R_g - 2|\Omega_L|_{\text{op}}$, where $|\Omega_L|_{\text{op}}$ is the pointwise operator norm of the curvature $2$-form $\Omega_L$ of $\nabla_L$, acting on spinors. In a previous paper, we showed that positivity of $R^{\text{gen}}$ is obstructed by the non-vanishing of the index of the spin$^c$ Dirac operator on $(M,g,L,\nabla_L)$, and that in some cases, the vanishing of this index guarantees the existence of a pair $(g,\nabla_L)$ with positive generalized scalar curvature. Building on this and on surgery techniques inspired by those that have been developed in the theory of positive scalar curvature on spin manifolds, we show that if $\dim M = n \ge 5$, if the fundamental group $\pi$ of $M$ is in a large class including surface groups and finite groups with periodic cohomology, and if $M$ is totally non-spin (meaning that the universal cover is not spin), then $(M,L)$ admits positive generalized scalar curvature if and only if the generalized $\alpha$-invariant of $(M,L)$ vanishes in the $K$-homology group $K_n(B\pi)$. We also develop an analogue of Stolz's sequence for computing the group of concordance classes of positive generalized scalar curvature metrics, and connect this to the analytic surgery sequence of Roe and Higson. Finally, we give a number of applications to moduli spaces of positive generalized scalar curvature metrics.

math.DG↗
arXiv2025

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps

We introduce smooth atlas stratified spaces. We show that this class is closed under cartesian products; consequently, it is possible to define fiber bundles of smooth atlas stratified spaces. We describe the resolution of such a space to a manifold with fibered corners and use this result in order to prove that the class of smooth atlas stratified spaces coincides with that of Thom-Mather stratified spaces. We then consider Witt pseudomanifolds (such as singular complex algebraic varieties) where it is well-known that a bordism invariant signature is available and equal to the Fredholm index of a realization of the signature operator. To each oriented fiber bundle of stratified spaces, with Witt fibers, we assign a class in bivariant KK-theory (with 2 inverted). Kasparov multiplication by this element defines a Gysin map in analytic K-homology and one of our main results is that this Gysin map preserves the analytic signature class of Witt spaces. We prove in fact a more general result: functoriality for fiber bundles in the sense that if one fiber bundle is the composition of two others then the KK-class of the former is the Kasparov product of the classes of the latter. We also discuss this result for other Dirac-type operators satisfying an analytic Witt condition, for example the spin-Dirac operator on a fibration of psc-Witt spin pseudomanifolds. We next define the analytic Gysin map associated to an oriented normally non-singular inclusion of Witt spaces and prove that it also preserves the signature class. Finally, we relate the analytic signature class of a Witt space with the topological Siegel-Sullivan orientation. Specifically we show that if one applies the inverse of the second Adams operation to the Sullivan orientation and complexifies then one obtains our analytic signature class under the natural identification between analytic and topological K-homology.

math.DG↗
arXiv2023

Stability of $L^2-$invariants on stratified spaces

Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_\Gamma$ be a Galois $\Gamma$-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^2$-Betti numbers and the Novikov-Shubin invariants are well defined. We then establish their invariance under a smoothly stratified, strongly stratum preserving homotopy equivalence, thus extending results of Dodziuk, Gromov and Shubin to these pseudo-manifolds.

math.DG↗
arXiv2023

Delocalized eta invariants of the signature operator on G-proper manifolds

Let $G$ be a connected, linear real reductive group and let $X$ be a cocompact $G$-proper manifold without boundary. We define delocalized eta invariants associated to a $L^2$-invertible perturbed Dirac operator $D_X+A$ with $A$ a suitable smoothing perturbation. We also investigate the case in which $D_X$ is not invertible but $0$ is isolated in the $L^2$-spectrum of $D_X$. We prove index formulas relating these delocalized eta invariants to Atiyah-Patodi-Singer delocalized indices on $G$-proper manifolds with boundary. In order to achieve this program we give a detailed account of both the large and small time behaviour of the heat-kernel of perturbed Dirac operators, as a map from the positive real line to the algebra of Lafforgue integral operators. We apply these results to the definition of rho-numbers associated to $G$-homotopy equivalences between closed $G$-proper manifolds and to the study of their bordism properties. We also define delocalized signatures of manifolds with boundary satisfying an invertibility assumption on the differential form Laplacian of the boundary in middle degree and prove an Atiyah-Patodi-Singer formula for these delocalized signatures.

math.DG↗
arXiv2021

Higher orbital integrals, rho numbers and index theory

Let $G$ be a connected, linear real reductive group. We give sufficient conditions ensuring the well-definedness of the delocalized eta invariant $\eta_g (D_X)$ associated to a Dirac operator $D_X$ on a cocompact $G$-proper manifold $X$ and to the orbital integral $\tau_g$ defined by a semisimple element $g\in G$. Along the way, we give a detailed account of the large time behaviour of the heat kernel and of its short time bahaviour near the fixed point set of $g$. We prove that such a delocalized eta invariant enters as the boundary correction term in an index theorem computing the pairing between the index class and the 0-degree cyclic cocycle defined by $\tau_g$ on a $G$-proper manifold with boundary. More importantly, we also prove a higher version of such a theorem, for the pairing of the index class and the higher cyclic cocycles defined by the higher orbital integral $\Phi^P_g$ associated to a cuspidal parabolic subgroup $P<G$ with Langlands decomposition $P=MAN$ and a semisimple element $g\in M$. We employ these results in order to define (higher) rho numbers associated to $G$-invariant positive scalar curvature metrics.

math.DG↗
arXiv2021

Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_Σ$ with singular stratum $βM$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple compact Lie group $G$ acting transitively on $L$ by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when $M_Σ$ and $βM$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

math.DG↗
arXiv2020

A note on higher Todd genera of complex manifolds

Let $M$ be a compact complex manifold. In this paper we give a simple proof of the bimeromorphic invariance of the higher Todd genera of $M$, a result first proved implicitly by Brasselet-Schürmann-Yokura using algebraic methods.

math.DG↗
arXiv2020

Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

math.KT↗
arXiv2020

Signatures of Witt spaces with boundary

Let M be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce the de Rham signature and the Hodge signature of M, and prove their equality. Next, building also on recent work of Albin and Gell-Redman, we extend the Atiyah-Patodi-Singer index theory established in our previous work under the hypothesis that M has stratification depth 1 to the general case, establishing in particular a signature formula on Witt spaces with boundary. In a parallel way we also pass to the case of a Galois covering M' of M with Galois group Gamma. Employing von Neumann algebras we introduce the de Rham Gamma-signature and the Hodge Gamma-signature and prove their equality, thus extending to Witt spaces a result proved by Lueck and Schick in the smooth case. Finally, extending work of Vaillant in the smooth case, we establish a formula for the Hodge Gamma-signature. As a consequence we deduce the fundamental result that equates the Cheeger-Gromov rho-invariant of the boundary of M' with the difference of the signatures of M and M'. We end the paper with two geometric applications of our results.

math.DG↗
arXiv2019

On positive scalar curvature bordism

Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups of a cartesian product GxZ are infinite in dimension 4n if n>0 G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group GxZ. We get the same result in dimension 4n+2 (n>0) if G is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, "On positive scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds" to arbitrary even dimensions and arbitrary groups with torsion.

math.GT↗
arXiv2019

On analytic Todd classes of singular varieties

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 fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial}$ complex, denoted here $\overlineð_{\mathrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\mathrm{dim}(\mathrm{sing}(X))=0$, is proved also for $\overlineð_{\mathrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex. We then show that when $\mathrm{dim}(\mathrm{sing}(X))=0$ we have $[\overlineð_{\mathrm{rel}}]=π_*[\overlineð_M]$ with $π:M\rightarrow X$ an arbitrary resolution and with $[\overlineð_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial}+\overline{\partial}^t$ on $M$. In the second 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 $[\overlineð_{\mathrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

math.DG↗
arXiv2019

Positive scalar curvature on simply connected spin pseudomanifolds

Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\alpha$-class $\alpha_w (M_\Sigma)\in KO_n$. In order to establish a sufficient condition we need to assume additional structure: we assume that the link of $M_\Sigma$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_\Sigma$ and $\beta M$ are spin, we reinterpret our obstruction in terms of two $\alpha$-classes associated to the resolution of $M_\Sigma$, $M$, and to the singular locus $\beta M$. Finally, when $M_\Sigma$, $\beta M$, $L$, and $G$ are simply connected and $\dim M$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this article, showing that the vanishing of these two $\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

math.DG↗
arXiv2019

Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature

Let $\Gamma$ be a f.g. discrete group and let $\tilde M$ be a Galois $\Gamma$-covering of a smooth closed manifold $M$. Let $S_*^\Gamma(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^\Gamma(\tilde M)\to K_*(M)\to K_*(C_r^*\Gamma)\to$. We prove that for an arbitrary discrete group $\Gamma$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[*-1]}(\mathcal{A}\Gamma)\to H^{del}_{[*-1]}(\mathcal{A}\Gamma)\to H^{e}_{[*]}(\mathcal{A}\Gamma)\to$, with $\mathcal{A}\Gamma$ a dense homomorphically closed subalgebra of $C^*_r\Gamma$. Here, $ H_{*}^{del}(\mathcal{A}\Gamma)$ is the delocalized homology and $H_{*}^{e}(\mathcal{A}\Gamma)$ is the homology localized at the identity element. Then, under additional assumptions on $\Gamma$, we prove the existence of a pairing between $HC^*_{del}(\mathbb{C}\Gamma)$, the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and $H^{del}_{*-1}(\mathcal{A}\Gamma)$. This, in particular, gives a pairing between $S^\Gamma_*(\tilde M)$ and $HC^{*-1}_{del}(\mathbb{C}\Gamma)$. We also prove the existence of a pairing between $S^\Gamma_*(\tilde M)$ and the relative cohomology $H^{[*-1]}(M\to B\Gamma)$. Both these parings are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $\rho(\tilde D)\in S_*^\Gamma(\tilde M)$ of an invertible $\Gamma$-equivariant Dirac type operator on $\tilde M$. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of $M$. Then, we establish new results on the moduli space of metrics of positive scalar curvature when $M$ is spin.

math.KT↗
arXiv2018

Eta and rho invariants on manifolds with edges

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As an application, we discuss stability results for the two rho-invariants we have defined.

math.DG↗