A reformulation of the Hodge index theorem within the framework of Atiyah’s L2-index theory is provided. More precisely, given a compact Kähler manifold (M, h) of even complex dimension 2m, we prove that σ(M)=∑p,q=02m(-1)ph(2),Γp,q(M)where σ(M) is the signature of M and h(2),Γp,q(M) are the L2-Hodge numbers of M with respect to a Galois covering having Γ as group of deck transformations. Likewise we also prove an L2-version of the Frölicher index theorem, see (3). Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the L2-Hodge numbers.

Von Neumann dimension, Hodge index theorem and geometric applications / Bei, F.. - In: EUROPEAN JOURNAL OF MATHEMATICS. - ISSN 2199-675X. - 5:4(2019), pp. 1212-1233. [10.1007/s40879-018-0269-2]

Von Neumann dimension, Hodge index theorem and geometric applications

Bei F.
2019

Abstract

A reformulation of the Hodge index theorem within the framework of Atiyah’s L2-index theory is provided. More precisely, given a compact Kähler manifold (M, h) of even complex dimension 2m, we prove that σ(M)=∑p,q=02m(-1)ph(2),Γp,q(M)where σ(M) is the signature of M and h(2),Γp,q(M) are the L2-Hodge numbers of M with respect to a Galois covering having Γ as group of deck transformations. Likewise we also prove an L2-version of the Frölicher index theorem, see (3). Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the L2-Hodge numbers.
2019
Euler characteristic; Hodge index theorem; Kähler manifolds; Kähler parabolic manifolds; L; 2; -Hodge numbers; Von Neumann dimension
01 Pubblicazione su rivista::01a Articolo in rivista
Von Neumann dimension, Hodge index theorem and geometric applications / Bei, F.. - In: EUROPEAN JOURNAL OF MATHEMATICS. - ISSN 2199-675X. - 5:4(2019), pp. 1212-1233. [10.1007/s40879-018-0269-2]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1362882
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