Hitchin, Nigel The geometry of three-forms in six dimensions. (English) Zbl 1036.53042 J. Differ. Geom. 55, No. 3, 547-576 (2000). 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. Reviewer: Neculai Papaghiuc (Iaşi) Cited in 7 ReviewsCited in 146 Documents MSC: 53C43 Differential geometric aspects of harmonic maps 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58D27 Moduli problems for differential geometric structures Keywords:differential three-forms; self-duality of 3-forms; moduli space; pseudo-Kähler structure PDFBibTeX XMLCite \textit{N. Hitchin}, J. Differ. Geom. 55, No. 3, 547--576 (2000; Zbl 1036.53042) 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. · Zbl 0732.53056 [2] D. S. Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999) 31–52. · Zbl 0940.53040 [3] R. Hartshorne, Stable vector bundles and instantons, Comm. Math. Phys. 59 (1978) 1–15. · Zbl 0383.14006 [4] N. J. Hitchin, The moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999) 77–92. · Zbl 0958.53057 [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. · Zbl 0581.32012 [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. · Zbl 0854.32016 [8] S. Merkulov & L. Schwachhöfer, Classification of irreducible holonomies of torsionfree affine connections, Ann. of Math. 150 (1999) 77–149, 1177–1179 (addendum). · Zbl 0992.53038 [9] A. Nijenhuis & W. B. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. 77 (1963) 424–489. · Zbl 0115.16103 [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. · Zbl 0696.53040 [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. · Zbl 0688.53030 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.