Abstract
We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S 1)k. We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ⊂ T ℂ = (ℂ*)k. On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.
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References
V. I. Danilov, “The geometry of toric varieties,” Usp. Mat. Nauk 33(2), 85–134 (1978) [Russ. Math. Surv. 33 (2), 97–154 (1978)].
W. Fulton, Introduction to Toric Varieties (Princeton Univ. Press, Princeton, NJ, 1993).
D. A. Cox, “The homogeneous coordinate ring of a toric variety,” J. Algebr. Geom. 4, 17–50 (1995).
R. P. Stanley, “The number of faces of a simplicial convex polytope,” Adv. Math. 35, 236–238 (1980).
J. E. Pommersheim, “Toric varieties, lattice points and Dedekind sums,” Math. Ann. 295, 1–24 (1993).
N. C. Leung and V. Reiner, “The signature of a toric variety,” Duke Math. J. 111(2), 253–286 (2002).
L. Meersseman, “A new geometric construction of compact complex manifolds in any dimension,” Math. Ann. 317(1), 79–115 (2000).
F. Bosio and L. Meersseman, “Real quadrics in ℂn, complex manifolds and convex polytopes,” Acta Math. 197(1), 53–127 (2006).
J. Tambour, “LVMB manifolds and simplicial spheres,” Ann. Inst. Fourier 62(4), 1289–1317 (2012).
L. Meersseman and A. Verjovsky, “Holomorphic principal bundles over projective toric varieties,” J. Reine Angew. Math. 572, 57–96 (2004).
T. Panov, Yu. Ustinovsky, and M. Verbitsky, “Complex geometry of moment-angle manifolds,” arXiv: 1308.2818 [math.CV].
F. Battaglia and D. Zaffran, “Foliations modelling nonrational simplicial toric varieties,” arXiv: 1108.1637 [math.CV].
T. Panov and Yu. Ustinovsky, “Complex-analytic structures on moment-angle manifolds,” Moscow Math. J. 12(1), 149–172 (2012).
H. Ishida, “Complex manifolds with maximal torus actions,” arXiv: 1302.0633 [math.CV].
G. E. Bredon, Introduction to Compact Transformation Groups (Academic, New York, 1972).
T. Delzant, “Hamiltoniens périodiques et images convexes de l’application moment,” Bull. Soc. Math. France 116(3), 315–339 (1988).
V. M. Buchstaber and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian].
J.-P. Demailly, “Complex analytic and differential geometry,” http://www-fourier.ujf-grenoble.fr/demailly/documents.html
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 286, pp. 219–230.
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Ustinovsky, Y.M. Geometry of compact complex manifolds with maximal torus action. Proc. Steklov Inst. Math. 286, 198–208 (2014). https://doi.org/10.1134/S0081543814060108
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DOI: https://doi.org/10.1134/S0081543814060108