Hodge theory on Cheeger spaces
-
Pierre Albin
,
Eric Leichtnam
Abstract
We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities,
as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds.
We introduce a class of ideal boundary conditions and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent.
We also prove an isomorphism between the resulting Hodge
and
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1104533
Award Identifier / Grant number: DMS-1105050
Funding statement: P.A. was partly supported by NSF Grant DMS-1104533 and an IHES visiting position and thanks Sapienza Università di Roma, Stanford, and Institut de Mathématiques de Jussieu for their hospitality and support. R.M. acknowledges support by NSF Grant DMS-1105050. P.P. thanks the “Projet Algèbres d’Opérateurs” of Institut de Mathématiques de Jussieu for hospitality during several short visits and a two months long visit in the spring of 2013; financial support was provided by Université Paris 7, Istituto Nazionale di Alta Matematica (INDAM) and CNRS (through the bilateral project “Noncommutative Geometry”) and Ministero dell’Università e della Ricerca Scientifica (through the project “Spazi di Moduli e Teoria di Lie”). E.L. thanks Sapienza Università di Roma for hospitality during several week-long visits; financial support was provided again by INDAM and CNRS through the bilateral project “Noncommutative Geometry”.
Acknowledgements
The authors are happy to thank Jesus Alvarez-Lopez, Markus Banagl, Francesco Bei, Jeff Cheeger, Michel Hilsum, Thomas Krainer, Richard Melrose and Gerardo Mendoza for many useful and interesting discussions.
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The spectral decomposition of shifted convolution sums over number fields
- Hodge theory on Cheeger spaces
- Geometric structures of collapsing Riemannian manifolds II
- Combinatorics and topology of proper toric maps
- On the long time behaviour of the conical Kähler–Ricci flows
- On invariance of plurigenera for foliations on surfaces
- Behavior of canonical divisors under purely inseparable base changes
- Algebraic nature of singular Riemannian foliations in spheres
- On the complex dynamics of birational surface maps defined over number fields
Articles in the same Issue
- Frontmatter
- The spectral decomposition of shifted convolution sums over number fields
- Hodge theory on Cheeger spaces
- Geometric structures of collapsing Riemannian manifolds II
- Combinatorics and topology of proper toric maps
- On the long time behaviour of the conical Kähler–Ricci flows
- On invariance of plurigenera for foliations on surfaces
- Behavior of canonical divisors under purely inseparable base changes
- Algebraic nature of singular Riemannian foliations in spheres
- On the complex dynamics of birational surface maps defined over number fields
