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Tr. Mat. Inst. Steklova, 2014, Volume 286, Pages 219–230 (Mi tm3570)  

This article is cited in 4 scientific papers (total in 4 papers)

Geometry of compact complex manifolds with maximal torus action

Yu. M. Ustinovsky

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

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 $\Sigma$ and a complex subgroup $H\subset T_\mathbb C=(\mathbb C^*)^k$. On every manifold $M$ we define a canonical holomorphic foliation $\mathcal F$ and, under additional restrictions on the combinatorial data, construct a transverse Kähler form $\omega _\mathcal F$. As an application of these constructions, we extend some results on the geometry of moment–angle manifolds to the case of manifolds $M$.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-91151-GFEN_a
Dynasty Foundation
The work was supported by the Russian Foundation for Basic Research (project no. 13-01-91151-GFEN_a) and by the Dynasty Foundation.


DOI: https://doi.org/10.1134/S0371968514030108

Full text: PDF file (238 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 286, 198–208

Bibliographic databases:

UDC: 514.763.42
Received in March 2014

Citation: Yu. M. Ustinovsky, “Geometry of compact complex manifolds with maximal torus action”, Algebraic topology, convex polytopes, and related topics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 286, MAIK Nauka/Interperiodica, Moscow, 2014, 219–230; Proc. Steklov Inst. Math., 286 (2014), 198–208

Citation in format AMSBIB
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\by Yu.~M.~Ustinovsky
\paper Geometry of compact complex manifolds with maximal torus action
\inbook Algebraic topology, convex polytopes, and related topics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 70th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2014
\vol 286
\pages 219--230
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968514030108}
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\jour Proc. Steklov Inst. Math.
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\pages 198--208
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. F. Battaglia, D. Zaffran, “Foliations modeling nonrational simplicial toric varieties”, Int. Math. Res. Notices, 2015, no. 22, 11785–11815  crossref  mathscinet  zmath  isi  elib  scopus
    2. H. Ishida, “Torus invariant transverse Kähler foliations”, Trans. Am. Math. Soc., 369:7 (2017), 5137–5155  crossref  mathscinet  zmath  isi  scopus
    3. F. Galaz-Garcia, M. Kerin, M. Radeschi, M. Wiemeler, “Torus orbifolds, slice-maximal torus actions, and rational ellipticity”, Int. Math. Res. Notices, 2018, no. 18, 5786–5822  crossref  mathscinet  zmath  isi  scopus
    4. Battaglia F., Prato E., Zaffran D., “Hirzebruch Surfaces in a One-Parameter Family”, Boll. Unione Mat. Ital., 12:1-2 (2019), 293–305  crossref  mathscinet  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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