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 operators 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 L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call ‘Cheeger spaces’, we show that these Hodge/de Rham cohomology groups satisfy Poincare' Duality.

Hodge theory on Cheeger spaces / P., Albin; E., Leichtnam; R., Mazzeo; Piazza, Paolo. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - (2018), pp. 1-74. [10.1515/crelle-2015-0095]

Hodge theory on Cheeger spaces

PIAZZA, Paolo
2018

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 operators 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 L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call ‘Cheeger spaces’, we show that these Hodge/de Rham cohomology groups satisfy Poincare' Duality.
2018
Hodge theory; Stratified spaces; signature operator
01 Pubblicazione su rivista::01a Articolo in rivista
Hodge theory on Cheeger spaces / P., Albin; E., Leichtnam; R., Mazzeo; Piazza, Paolo. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - (2018), pp. 1-74. [10.1515/crelle-2015-0095]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/535075
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