arXiv subjects
Publications and source records attributed to Eric Leichtnam.
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
Let X be a compact manifold with boundary, and suppose that the boundary is the total space of a fibration with base Y and fibre Z. Let D be a generalized Dirac operator associated to a Phi-metric g on X. Under the assumption that D is fully elliptic we prove an index formula for D. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain b-metric h(s), with s a positive real number. Next we write the b- (i.e. the APS) index formula for h(s); the Phi-index formula follows by analyzing the limiting behaviour as s goes to zero of the two terms in the formula. The interior term is studied directly whereas the adiabatic limit formula for the eta invariant follows from work of Bismut and Cheeger.
In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function $\hatζ(s)$ (resp. the zeta function $ζ_Y(s)$ of a smooth projective curve $Y$ over a finite field $\mathbb{F}_q$, $q=p^f$)) one could possibly associate a foliated Riemannian laminated space $(S_{\mathbb{Q}}, \mathcal{F}, g, ϕ^t)$ (resp. $(S_Y, \mathcal{F}, g, ϕ^t)$) endowed with an action of a flow $ϕ^t$ whose primitive compact orbits should correspond to the primes of $\mathbb{Q}$ (resp. $Y$). The existence of such a foliated space and flow $ϕ^t$ is still unknown except when $Y$ is an elliptic curve (see Deninger). Being motivated by this latter case, we introduce a class of foliated laminated spaces $(S=\frac{\mathcal{L}\times \R^{+*}}{q^\Z}, \mathcal{F}, g, ϕ^t)$ where $\mathcal{L} $ is locally $D\times \Z_p^m$, $D$ being an open disk of $\mathbb{C}.$ Assuming that the leafwise harmonic forms on $ \mathcal{L}$ are locally constant transversally, we prove a Lefschetz trace formula for the flow $ϕ^t$ acting on the leafwise Hodge cohomology $H^j_τ$ ($0\leq j \leq 2$) of $(S,\mathcal{F})$ that is very similar to the explicit formula for the zeta function of a (general) smooth curve over $\mathbb{F}_q$. We also prove that the eigenvalues of the infinitesimal generator of the action of $ϕ^t$ on $H^1_τ$ have real part equal to ${1/2}.$ Moreover, we suggest in a precise way that the flow $ϕ^t$ should be induced by a renormalization group flow "à la K. Wilson". We show that when $Y$ is an elliptic curve over $\mathbb{F}_q$ this is indeed the case.
Let X be a complex manifold with strongly pseudoconvex boundary M. If u is a defining function for M, then -log u is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form s = i \del \delbar(-log u) is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M. The Poisson structure obtained by inverting s extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M. In addition, when -log u is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Englis for the Berezin-Toeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.
We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative index theorem for the difference of two APS-index classes associated to different boundary conditions; we prove a gluing formula on closed foliated bundles that are the union of two foliated bundles with boundary; we establish a variational formula for APS-index classes of a 1-parameter family of Dirac-type operators on foliated bundles (this formula involves the noncommutative spectral flow of the boundary family). All these formulas take place in the $K$-theory of a suitable cross-product algebra. We then apply these results in order to find sufficient conditions ensuring the equality of the signature index classes of two cut-and-paste equivalent foliated bundles. We give applications to the question of when the Baum-Connes higher signatures of closed foliated bundles are cut-and-paste invariant.
Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; - the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.
Let $Γ$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a $Γ$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $Γ\to \hat{M}\to \hat{M}/Γ$ is a Galois covering of a compact manifold with boundary. Let $(D^+ (θ))_{θ\in T}$ be a $Γ$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r Γ)$. If, in addition, $Γ$ is of polynomial growth, we define higher indeces by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indeces: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assumptions we extend our theorem to any $G$-proper manifold, with $G$ an étale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose $b$-pseudodifferential calculus as well as the superconnection proof of the index theorem on $G$-proper manifolds recently given by Gorokhovsky and Lott.
We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $ϕ$ of cosphere bundles of two Riemannian manifolds X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\expθ - \int_{B^*Y}\hat{A}(T^*Y)\expθ$. Here $B^*$ stands for the unit coball bundle and $θ$ is a certain characteristic class depending on the principal symbol of the Fourier integral operator. In the special case when X=Y we obtain a different proof of the theorem of Epstein and Melrose.
We extend the notion of the symmetric signature $σ(\bar{M},r)$ in L^n(R) for a compact n-dimensional manifold M without boundary, a reference map r from M to BG and a homomorphism of rings with involutions from ZG to R to the case with boundary $\partial M$, where $(\bar{M},\bar{\partial M}) \to (M,\partial M)$ is the G-covering associated to r. We need the assumption that $C_*(\bar{\partial M}) \otimes_{\zz G} R$ isR-chain homotopy equivalent to a R-chain complex D_* with trivial m-th differential for n = 2m resp. n = 2m+1. Let Z be a closed oriented manifold with reference map BG. Let F be a cutting codimension one submanifold in Z and let $\bar{F} \to F$ be the associated $G$-covering. Denote by $α_m(\bar{F})$ the m-th Novikov-Shubin invariant and by $b_m^{(2)}(\bar{F})$ the m-th L^2-Betti number. We use $σ(\bar{M},r)$ to prove the additivity (or cut and paste property) of the higher signatures of Z if we have $α_m(\bar{F}) = \infty^+$ in the case n = 2m and, in the case n = 2m+1, if we have $α_m(\bar{F}) = \infty^+$ and $b_m^{(2)}(\bar{F}) = 0$. We give examples, where these conditions are not satisfied and additivity fails. Our work is motivated by the one of Leichtnam-Lott-Piazza, Lott and Weinberger.
If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants.