×

zbMATH — the first resource for mathematics

The geometry of three-forms in six dimensions. (English) Zbl 1036.53042
In this paper the author studies special algebraic properties of alternating 3-forms in six dimensions and introduces a diffeomorphism- invariant functional on the space of differential 3-forms on a closed 6-manifold \(M\). Restricting the functional to a de Rham cohomology class in \(H^3(M,{\mathbb R})\), he finds that a critical point which is generic in a suitable sense defines a complex threefold with trivial canonical bundle. This approach gives a direct method of showing that an open set in \(H^3(M,{\mathbb R})\) is a local moduli space for this structure and introduces in a natural way the special pseudo-Kähler structure on it.

MSC:
53C43 Differential geometric aspects of harmonic maps
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58D27 Moduli problems for differential geometric structures
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid
References:
[1] P. Candelas & X. C. de la Ossa, Moduli space of Calabi-Yau manifolds, Nuclear Phys. B 355 (1991) 455–481.
[2] D. S. Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999) 31–52.
[3] R. Hartshorne, Stable vector bundles and instantons, Comm. Math. Phys. 59 (1978) 1–15.
[4] N. J. Hitchin, The moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999) 77–92.
[5] P. S. Howe, E. Sezgin & P. C. West, The six-dimensional self-dual tensor, Phys. Lett. B 400 (1997) 255–259.
[6] K. Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Vol. 283, 1986.
[7] P. Lu & G. Tian, The complex structures on connected sums of S 3 × S 3 , Manifolds and geometry (Pisa, 1993), Sympos. Math. XXXVI, Cambridge Univ. Press, Cambridge, 1996, 284–293.
[8] S. Merkulov & L. Schwachhöfer, Classification of irreducible holonomies of torsionfree affine connections, Ann. of Math. 150 (1999) 77–149, 1177–1179 (addendum).
[9] A. Nijenhuis & W. B. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. 77 (1963) 424–489.
[10] W. Reichel, Über die Trilinearen Alternierenden Formen in 6 und 7 Veränderlichen, Dissertation, Greifswald, 1907. · JFM 38.0674.03
[11] G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory, (ed. S.-T. Yau), Adv. Ser. Math. Phys. Vol. 1, World Scientific Publishing Co., Singapore, 1987, 629–646.
[12] A. N. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989) 325–346.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.