no. 1
A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas
[Un contre-exemple à la décomposition lisse de Hodge pour des feuilletages généraux et à une classe de formules de trace dynamique]
Deninger, Christopher1 ; Singhof, Wilhelm2
1 WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne)
2 Universität Düsseldorf, Mathematisches Institut, Universitätsstr. 1, 40225 Düsseldorf (Allemagne)
Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 209-219.

Nous construisons un feuilletage à feuilles denses et de codimension deux sur la variété de Heisenberg tel que la décomposition de Hodge feuilletée ne soit pas vérifiée. Nous démontrons aussi qu'un certain type de formules de trace dynamique reliant les orbites périodiques aux traces sur la cohomologie feuilletée, ne sont pas vraies pour des flots arbitraires.

We construct a two dimensional foliation with dense leaves on the Heisenberg nilmanifold for which smooth leafwise Hodge decomposition does not hold. It is also shown that a certain type of dynamical trace formulas relating periodic orbits with traces on leafwise cohomologies does not hold for arbitrary flows.

DOI : 10.5802/aif.1821
Classification : 53C12, 37C27, 58J60
Keywords: Hodge decomposition, foliation, dynamical trace formula, nilmanifold
Mot clés : théorie de Hodge, formule de trace, feuilletage
Deninger, Christopher 1 ; Singhof, Wilhelm 2

1 WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne)
2 Universität Düsseldorf, Mathematisches Institut, Universitätsstr. 1, 40225 Düsseldorf (Allemagne) 
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     title = {A counterexample to smooth leafwise {Hodge} decomposition for general foliations and to a type of dynamical trace formulas},
     journal = {Annales de l'Institut Fourier},
     pages = {209--219},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     doi = {10.5802/aif.1821},
     mrnumber = {1821074},
     zbl = {0997.58017},
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Deninger, Christopher; Singhof, Wilhelm. A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 209-219. doi : 10.5802/aif.1821. http://www.numdam.org/articles/10.5802/aif.1821/

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