[Remarques sur le complexe de Schweitzer]
Nous prouvons que le complexe de Schweitzer est elliptique et que ses cohomologies définissent des foncteurs cohomologiques. Comme applications, nous obtenons la dimensionnalité finie, une version de la dualité de Serre, des restrictions du comportement de la cohomologie dans les petites déformations, et une formule d’index qui s’avère être équivalente aux relations de Hirzebruch–Riemann–Roch.
We prove that the Schweitzer complex is elliptic and its cohomologies define cohomological functors. As applications, we obtain finite dimensionality, a version of Serre duality, restrictions of the behaviour of cohomology in small deformations, and an index formula which turns out to be equivalent to the Hirzebruch–Riemann–Roch relations.
Révisé le :
Accepté le :
Première publication :
Keywords: Complex manifolds, Cohomology, Index theory, Deformations.
Mot clés : Variétés complexes, cohomologie, théorie de l’indice, déformations.
@unpublished{AIF_0__0_0_A99_0,
author = {Stelzig, Jonas},
title = {Some remarks on the {Schweitzer} complex},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
year = {2024},
doi = {10.5802/aif.3645},
language = {en},
note = {Online first},
}
Stelzig, Jonas. Some remarks on the Schweitzer complex. Annales de l'Institut Fourier, Online first, 13 p.
[1] Some exact sequences in cohomology theory for Kähler manifolds, Pac. J. Math., Volume 12 (1962), pp. 791-799 | DOI | MR | Zbl
[2] A Lefschetz fixed point formula for elliptic complexes. I, Ann. Math., Volume 86 (1967), pp. 374-407 | DOI | MR | Zbl
[3] The index of elliptic operators on compact manifolds, Bull. Am. Math. Soc., Volume 69 (1963), pp. 422-433 | DOI | MR | Zbl
[4] The index of elliptic operators. I, Ann. Math., Volume 87 (1968), pp. 484-530 | DOI | MR | Zbl
[5] Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math., Volume 114 (1965), pp. 71-112 | DOI | MR | Zbl
[6] Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics, J. Geom. Anal., Volume 26 (2016) no. 1, pp. 252-286 | DOI | MR | Zbl
[7] Complex Analytic and Differential Geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
[8] Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Natl. Acad. Sci. USA, Volume 41 (1955), pp. 641-644 | DOI | MR | Zbl
[9] Holomorphic string algebroids, Trans. Am. Math. Soc., Volume 373 (2020) no. 10, pp. 7347-7382 | DOI | MR | Zbl
[10] A faithful braid group action on the stable category of tricomplexes, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), 019, 32 pages | DOI | MR | Zbl
[11] Complex manifolds and deformation of complex structures, Classics in Mathematics, Springer, 2005, x+465 pages | DOI | MR | Zbl
[12] Bigraded notions of formality and Aeppli–Bott–Chern–Massey products (2022) (to appear in Comm. Anal. Geom., https://arxiv.org/abs/2202.08617)
[13] Seminar on the Atiyah–Singer index theorem. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay, Annals of Mathematics Studies, 57, Princeton University Press, 1965, x+366 pages | DOI | MR | Zbl
[14] Autour de la cohomologie de Bott-Chern (2007) (https://arxiv.org/abs/0709.3528)
[15] On the structure of double complexes, J. Lond. Math. Soc., Volume 104 (2021) no. 2, pp. 956-988 | DOI | MR | Zbl
[16] On linear combinations of cohomological invariants of compact complex manifolds, Adv. Math., Volume 407 (2022), 108560, 52 pages | DOI | MR | Zbl
Cité par Sources :
