Index in K-theory for families of fibred cusp operators

Pseudodifferential operators on manifolds with fibred corners

One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ell…

Hodge cohomology of some foliated boundary and foliated cusp metrics

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of…

Families index for manifolds with hyperbolic cusp singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered b…

Cusp singularities of plane envelopes

We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity of envelopes (two transversal semicubic cusps), and we describe its perestroikas under generic small deformations.

轨形消解上的狄拉克算子：一致椭圆理论

Dirac Operators on Orbifold Resolutions: Uniform Elliptic Theory

本文通过将轨形视为锥纤维奇异空间并用渐近锥纤维化进行消解，发展了一致椭圆理论，从而得到轨形消解上狄拉克算子的Fredholm性质、一致有界性及指标分解公式。

带有尖点的黎曼面奇异族的Quillen度量及紧致扰动定理

Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

本文研究了带有尖点的黎曼面奇异族的Quillen度量在尖点退化时的行为，证明了Quillen度量与紧化黎曼面的正常化度量的兼容性，并给出了分析扭量与Takhtajan-Zograf定义的兼容性结果。

Families Index for Pseudodifferential Operators on Manifolds with Boundary

An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of order $-\infty$ such that each $P_b+Q_b$ is invertible. Thus any elliptic family of symbols has a realization as an invertible family of cusp pseudodifferential operators, which is a form of the cobordism invariance of the index. A crucial role is played by the weak contractibil…

不完备尖点边缘空间上的指标理论

Index Theory on Incomplete Cusp Edge Spaces

该文研究边界族可逆的不完备尖点边缘空间上的Dirac型算子，构造相关Laplace型算子的热核，证明算子的本质自伴性与Fredholm性，并借助热核渐近性与推广的Getzler标度论证建立含Hodge-de Rham算子符号差公式的指标公式。

Pseudodifferential operators on manifolds with foliated boundaries

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…

Some index formulae on the moduli space of stable parabolic vector bundles

We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.