Elliptic operators in subspaces and the eta invariant

The Analogue of the Dedekind Eta Function for CY Manifolds I

This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an explicit formula for the regularized determinants of the flat metrics with fixed volume on the elliptic curves. The following formula holds in this case; the regularized determinant is the product of the imaginary part of the complex number in the Siegel upper half plane with the De…

Hyperbolicity, irrationality exponents and the eta invariant

We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents.

Boundaries, eta invariant and the determinant bundle

Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit version of this at the level of the determinant bundle. Their result, that the eta invariant of the interior family trivializes the determinant bundle of the boundary family, is extended here to the wider context of pseudodifferential families of cusp type.

Eta forms and the odd pseudodifferential families index

Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $ϕ$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $π_{\cA}:\cA(ϕ)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total e…

Some remarks on equivariant elliptic operators and their invariants

In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and the equivariant index. Many researchers have previously considered these invariants before. What makes this work different is that we are evaluating integer-valued indices corresponding to multiplicities of group representations, and our eta invariant is a number dependent on the entire group at once. Moreover, the techniques of proof and formulas obtained ar…

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford r…

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\widetilde M$ is a $Γ$-regular covering space of $M$. Let $\langle α\rangle$ be the conjugacy class of a non-identity element $α\in Γ$. Suppose $\{Γ_i\}$ is a sequence of finite-index normal subgroups of $Γ$…

The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years. This paper summarizes the work in the papers arXiv:1005.3845 [math.DG] and arXiv:1008.1757 [math.DG].

Continuity of family of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces; e…

Jumps of the eta invariant

We study the eta-invariant, defined by Atiyah-Patodi-Singer a real valued invariant of an oriented odd-dimensional Riemannian manifold equipped with a unitary representation of its fundamental group. When the representation varies analytically, the corresponding eta-invariant may have an integral jump, known also as the spectral flow. The main result of the paper establishes a formula for this spectral jump in terms of the signatures of some homological forms, defined naturally by the path of representations. These signatures may also be computed by means of a spectral sequence of Hermitian fo…

Limits of functions and elliptic operators

We show that a subspace $S$ of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are that $S$ is closed in $L^2(M)$ and that if a sequence of functions $f_n$ in $S$ converges in $L^2(M)$, then so do the partial derivatives of the functions $f_n$.

Spectral Asymmetry, Zeta Functions and the Noncommutative Residue

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local independence with respect to the cutting, the regularity at integer points of eta functions and a geometric expression for the spectral asymmetry of Dirac operators which, in particular, yields a new spectral interpretation of the Einstein-Hilbert action in gravity.

Selected Aspects of the Mathematical Work of Krzysztof P. Wojciechowski

I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of elliptic boundary problems. The theme of the 90s is the eta invariant. The paper with Douglas was fundamental for establishing spectral invariants for manifolds with boundary and for the investigation of the behavior of spectral invariants under analytic surgery. This was so influ…

Kähler度量与toric拉格朗日纤维化

Kähler metrics and toric Lagrangian fibrations

论文扩展了Abreu-Guillemin理论，将不变Kähler度量从toric辛流形推广到具有toric作用的辛流形，揭示了这些流形恰好是具有椭圆奇点的拉格朗日纤维化，并建立了不变Kähler度量与椭圆连接和混合b度量的对应关系。