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Sibirsk. Mat. Zh., 2010, Volume 51, Number 5, Pages 974–989 (Mi smj2139)  

This article is cited in 1 scientific paper (total in 1 paper)

On isotopies and homologies of subvarieties of toric varieties

N. A. Bushueva

Siberian Federal University, Krasnoyarsk, Russia

Abstract: In $\mathbb C^n$ we consider an algebraic surface $Y$ and a finite collection of hypersurfaces $\{S_i\}$. Froissart's theorem states that if $Y$ and $\{S_i\}$ are in general position in the projective compactification of $\mathbb C^n$ together with the hyperplane at infinity then for the homologies of $Y\setminus\bigcup S_i$ we have a special decomposition in terms of the homology of $Y$ and all possible intersections of $S_i$ in $Y$. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of $\mathbb C^n$ in which $Y$ and $\{S_i\}$ are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in $Y$ leaving invariant all hypersurfaces $Y\cap S_k$ with the exception of one $Y\cap S_i$, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface $Y$ the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.

Keywords: homology group, toric variety, coboundary operator.

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English version:
Siberian Mathematical Journal, 2010, 51:5, 776–788

Bibliographic databases:

UDC: 517.55+512.761
Received: 07.07.2009
Revised: 25.02.2010

Citation: N. A. Bushueva, “On isotopies and homologies of subvarieties of toric varieties”, Sibirsk. Mat. Zh., 51:5 (2010), 974–989; Siberian Math. J., 51:5 (2010), 776–788

Citation in format AMSBIB
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\by N.~A.~Bushueva
\paper On isotopies and homologies of subvarieties of toric varieties
\jour Sibirsk. Mat. Zh.
\yr 2010
\vol 51
\issue 5
\pages 974--989
\mathnet{http://mi.mathnet.ru/smj2139}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2757922}
\transl
\jour Siberian Math. J.
\yr 2010
\vol 51
\issue 5
\pages 776--788
\crossref{https://doi.org/10.1007/s11202-010-0078-4}
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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. Natalia A. Bushueva, Konstantin V. Kuzvesov, Avgust K. Tsikh, “On the asymptotic of homological solutions to linear multidimensional difference equations”, Zhurn. SFU. Ser. Matem. i fiz., 7:4 (2014), 417–430  mathnet
  • Сибирский математический журнал Siberian Mathematical Journal
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