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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
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##### 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.
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