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This article is cited in 2 scientific papers (total in 2 papers)
Research Papers
The complex shade of a real space and its applications
T. Ekholm Department of Mathematics, Uppsala University, Uppsala, Sweden
Abstract:
A natural oriented $(2k+2)$-chain in $\mathbb{C}P^{2k+1}$ with boundary twice $\mathbb{R}P^{2k+1}$, the complex shade of $\mathbb{R}P^{2k+1}$, is constructed. The intersection numbers with the shade make it possible to introduce a new invariant, the shade number, of a $k$-dimensional subvariety $W$ with a normal vector field $n$ along the real set. If $W$ is an even-dimensional real variety, then the shade number and the Euler number of the complement of $n$ in the real normal bundle of its real part agree. If $W$ is an odd-dimensional orientable real variety, a linear combination of the shade number and the wrapping number (self-linking number) of its real part does not depend on $n$ and equals the encomplexed writhe as defined by Viro [V]. The shade numbers of varieties without real points and the encomplexed writhes of odd-dimensional real varieties are, in a sense, Vassiliev invariants of degree 1.
The complex shades of odd-dimensional spheres are constructed. The shade numbers of real subvarieties in spheres have properties similar to those of their projective counterparts.
Keywords:
algebraic variety, complexification, real algebraic knot, rigid isotopy, isotopy, linking number.
Received: 19.09.2001
Citation:
T. Ekholm, “The complex shade of a real space and its applications”, Algebra i Analiz, 14:2 (2002), 56–91; St. Petersburg Math. J., 14:2 (2003), 223–250
Linking options:
https://www.mathnet.ru/eng/aa841 https://www.mathnet.ru/eng/aa/v14/i2/p56
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Abstract page: | 238 | Full-text PDF : | 92 | First page: | 1 |
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