# Article

Full entry | PDF   (0.3 MB)
Keywords:
complex structure; projective space; Frölicher spectral sequence; Hodge numbers
Summary:
We consider almost-complex structures on $\mathbb{C}\text{P}^3$ whose total Chern classes differ from that of the standard (integrable) almost-complex structure. E. Thomas established the existence of many such structures. We show that if there exists an “exotic” integrable almost-complex structures, then the resulting complex manifold would have specific Hodge numbers which do not vanish. We also give a necessary condition for the nondegeneration of the Frölicher spectral sequence at the second level.
References:
[1] M. F. Atiyah, I. M. Singer: The index of elliptic operators: III. Ann. of Math. 87 (1968), 546–604. MR 0236952
[2] F. Hirzebruch: Topological Methods in Algebraic Geometry. Springer, Berlin, 1966. MR 1335917 | Zbl 0138.42001
[3] F. Hirzebruch, K. Kodaira: On the complex projective spaces. J. Math. Pures Appl. 36 (1957), 201–216. MR 0092195
[4] A. Gray: A property of a hypothetical complex structure on the six sphere. Bol. Un. Mat. Ital. 11 Suppl. fasc. 2 (1997), 251–255. MR 1456264 | Zbl 0891.53018
[5] P. Griffiths, J. Harris: Principles of Algebraic Geometry. Wiley, New York, 1978. MR 0507725
[6] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry: I, II. Wiley, New York, 1969.
[7] T. Peternell: A rigidity theorem for ${\mathbb{C}}\text{P}^3$. Manuscripta Math. 50 (1985), 397–428. MR 0784150
[8] Y. T. Siu: Nondeformability of the complex projective space. J. Reine Angew. Math. 399 (1989), 208–219. MR 1004139 | Zbl 0671.32018
[9] Y. T. Siu: Global nondeformability of the complex projective space Proceedings of the 1989 Taniguchi International Symposium on “Prospect in Complex Geometry” in Katata, Japan, Lecture Notes Math. vol. 1468, Springer, Berlin, 1991, pp. 254–280. MR 1123546
[10] E. Thomas: Complex structures on real vector bundles. Amer. J. Math. 89 (1967), 887–908. MR 0220310 | Zbl 0174.54802
[11] L. Ugarte: Hodge numbers of a hypothetical complex structure on the six sphere. Geom. Dedicata 81 (2000), 173–179. MR 1772200 | Zbl 0996.53046
[12] S. T. Yau: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74 (1977), 1798–1799. MR 0451180 | Zbl 0355.32028

Partner of