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

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

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 286, 198–208

Bibliographic databases:

UDC: 514.763.42

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
\Bibitem{Ust14} \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 \mathnet{http://mi.mathnet.ru/tm3570} \crossref{https://doi.org/10.1134/S0371968514030108} \elib{http://elibrary.ru/item.asp?id=22020639} \transl \jour Proc. Steklov Inst. Math. \yr 2014 \vol 286 \pages 198--208 \crossref{https://doi.org/10.1134/S0081543814060108} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000343605900010} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84919790066} 

• http://mi.mathnet.ru/eng/tm3570
• https://doi.org/10.1134/S0371968514030108
• http://mi.mathnet.ru/eng/tm/v286/p219

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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
2. H. Ishida, “Torus invariant transverse Kähler foliations”, Trans. Am. Math. Soc., 369:7 (2017), 5137–5155
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
4. Battaglia F., Prato E., Zaffran D., “Hirzebruch Surfaces in a One-Parameter Family”, Boll. Unione Mat. Ital., 12:1-2 (2019), 293–305
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