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Hodge numbers of a hypothetical complex structure on $$S^{6}$$. (English) Zbl 1387.32033
Summary: These are the notes for the talk “Hodge numbers of a hypothetical complex structure on $$S^6$$” given by the author at the MAM1 “(Non)-existence of complex structures on $$S^6$$” held in Marburg in March 2017. They are based on [A. Gray, Boll. Unione Mat. Ital., VII. Ser., B 11, No. 2, Suppl., 251–255 (1997; Zbl 0891.53018); L. Ugarte, Geom. Dedicata 81, No. 1–3, 173–179 (2000; Zbl 0996.53046)], where Hodge numbers and the dimensions of the successive pages of the Frölicher spectral sequence for $$S^6$$ endowed with a hypothetical complex structure are investigated. We also add results from [A. P. McHugh, Eur. J. Pure Appl. Math. 10, No. 3, 440–454 (2017; Zbl 1366.53053)], where the Bott-Chern cohomology of hypothetical complex structures on $$S^6$$ is studied. The material is not intended to be original.

MSC:
 32Q99 Complex manifolds 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
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References:
 [1] Angella, D., On the Bott-Chern and aeppli cohomology, (2015), in: Bielefeld Geometry & Topology Days [2] Atiyah, M. F.; Singer, I. M., The index of elliptic operators: III, Ann. Math. (2), 87, 3, 546-604, (1968) · Zbl 0164.24301 [3] Barge, J., Structures différentiables sur LES types d’homotopie rationnelle simplement connexes, Ann. Sci. Éc. Norm. Supér. (4), 9, 4, 469-501, (1976) · Zbl 0348.57016 [4] Bigalke, L.; Rollenske, S., Erratum to: the frölicher spectral sequence can be arbitrarily non-degenerate, Math. Ann., 358, 3-4, 1119-1123, (2014) · Zbl 1285.53062 [5] Brown, J. R., Properties of a hypothetical exotic complex structure on $$\mathbb{C} P^3$$, Math. Bohem., 132, 1, 59-74, (2007) · Zbl 1174.53345 [6] Campana, F.; Demailly, J.-P.; Peternell, Th., The algebraic dimension of compact complex threefolds with vanishing second Betti number, Compos. Math., 112, 1, 77-91, (1998) · Zbl 0910.32032 [7] Cordero, L. A.; Fernández, M.; Gray, A., The frölicher spectral sequence and complex compact nilmanifolds, C. R. Acad. Sci. Paris Sér. I Math., 305, 17, 753-756, (1987) · Zbl 0627.32025 [8] Cordero, L. A.; Fernández, M.; Ugarte, L.; Gray, A., A general description of the terms in the frölicher spectral sequence, Differ. Geom. Appl., 7, 1, 75-84, (1997) · Zbl 0880.53055 [9] Deligne, P.; Griffiths, Ph.; Morgan, J.; Sullivan, D., Real homotopy theory of Kähler manifolds, Invent. Math., 29, 3, 245-274, (1975) · Zbl 0312.55011 [10] Etesi, G., Complex structure on the six dimensional sphere from a spontaneous symmetry breaking, J. Math. Phys., J. Math. Phys., 56, 9, (2015), 1 p · Zbl 1322.81079 [11] Frölicher, A., Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Natl. Acad. Sci. USA, 41, 641-644, (1955) · Zbl 0065.16502 [12] Gray, A., A property of a hypothetical complex structure on the six sphere, Boll. Unione Mat. Ital., B (7), 11, Suppl. 2, 251-255, (1997) · Zbl 0891.53018 [13] Griffiths, Ph.; Harris, J., Principles of algebraic geometry, Pure and Applied Mathematics, (1978), Wiley-Interscience/Wiley New York · Zbl 0408.14001 [14] Hirzebruch, F., Topological methods in algebraic geometry, Classics in Mathematics, (1995), Springer-Verlag Berlin, translated from the German and Appendix One by R.L.E. Schwarzenberger, with a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, reprint of the 1978 edition [15] Hirzebruch, F.; Kodaira, K., On the complex projective spaces, J. Math. Pures Appl. (9), 36, 201-216, (1957) · Zbl 0090.38601 [16] Hodge, W. V.D., The theory and applications of harmonic integrals, (1989), Cambridge Mathematical Library, Cambridge University Press Cambridge, reprint of the 1941 original, with a foreword by Michael Atiyah · Zbl 0693.14002 [17] Huckleberry, A.; Kebekus, S.; Peternell, T., Group actions on $$S^6$$ and complex structures on $$\mathbf{P}_3$$, Duke Math. J., 102, 1, 101-124, (2000) · Zbl 0971.53024 [18] C. Lehn, S. Rollenske, C. Schinko, The complex geometry of a hypothetical complex structure on $$S^6$$, Differ. Geom. Appl. · Zbl 1381.53008 [19] McCleary, J., A User’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics, vol. 58, (2001), Cambridge University Press Cambridge · Zbl 0959.55001 [20] McHugh, Andrew, Narrowing cohomologies on complex $$S^6$$, Eur. J. Pure Appl. Math., 10, 3, 440-454, (2017) · Zbl 1366.53053 [21] Andrew McHugh, Bott-Chern/Aeppli cohomology on compact complex 3-folds. [22] Peternell, Th., A rigidity theorem for $$\mathbf{P}_3(\mathbf{C})$$, Manuscr. Math., 50, 397-428, (1985) [23] Pittie, H., The nondegeneration of the Hodge-de Rham spectral sequence, Bull. Am. Math. Soc., 20, 19-22, (1989) · Zbl 0671.32024 [24] Rollenske, S., The frölicher spectral sequence can be arbitrarily non-degenerate, Math. Ann., 341, 3, 623-628, (2008) · Zbl 1188.53083 [25] Schweitzer, M., Autour de la cohomologie de Bott-Chern, (2007), Prépublication de l’Institut Fourier, no. 703 [26] Sullivan, D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47, 269-331, (1977) · Zbl 0374.57002 [27] Ugarte, L., Hodge numbers of a hypothetical complex structure on the six sphere, Geom. Dedic., 81, 1-3, 173-179, (2000) · Zbl 0996.53046
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