Article

Full entry | PDF   (0.2 MB)
Keywords:
$\partial\overline{\partial}$-lemma; Hodge-de Rham spectral sequence; $E_1$-degeneration; bi-generalized Hermitian manifold
Summary:
For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.
References:
[C] Cavalcanti G.: New aspects of the $dd^c$-lemma. Oxford Univ. DPhil. thesis, arXiv:math/0501406v1[math.DG].
[Ca07] Cavalcanti G.: Introduction to generalized complex geometry. impa, 26-Col´oquio Brasileiro de Matem´atica, 2007. MR 2375780 | Zbl 1144.53090
[CHT] Chen T.W., Ho C.I., Teh J.H.: Aeppli and Bott-Chern cohomology for bigeneralized Hermitian manifolds and $d'd”$-lemma. J. Geom. Phys. 93 (2015), 40–51. DOI 10.1016/j.geomphys.2015.03.006 | MR 3340172
[DGMS] Deligne P., Griffiths P., Morgan J., Sullivan D.: Real homotopy theory of Kähler manifolds. Invent. Math.29 (1975), no. 3, 245–274. MR 0382702 | Zbl 0355.55016
[G1] Gualtieri M.: Generalized complex geometry. Ann. of Math. 174 (2011), 75–123. DOI 10.4007/annals.2011.174.1.3 | MR 2811595 | Zbl 1235.32020
[M] McCleary J.: A User's Guide to Spectral Sequences. 2nd edition, Cambridge studies in advanced mathematics, 58, Cambridge University Press, Cambridge, 2001. MR 1793722 | Zbl 0959.55001

Partner of