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 Mosc. Math. J., 2016, Volume 16, Number 2, Pages 237–273 (Mi mmj599)

Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $\mathbb CP^5$

Victor M. Buchstabera, Svjetlana Terzićb

a Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street 8, 119991 Moscow, Russia
b Faculty of Science, University of Montenegro, Dzordza Vasingtona bb, 81000 Podgorica, Montenegro

Abstract: We consider the canonical action of the compact torus $T^4$ on the complex Grassmann manifold $G_{4,2}$ and prove that the orbit space $G_{4,2}/T^4$ is homeomorphic to the sphere $S^5$. We prove that the induced map from $G_{4,2}$ to the sphere $S^5$ is not smooth and describe its smooth and singular points. We also consider the action of $T^4$ on $\mathbb CP^5$ induced by the composition of the second symmetric power representation of $T^4$ in $T^6$ and the standard action of $T^6$ on $\mathbb CP^5$ and prove that the orbit space $\mathbb CP^5/T^4$ is homeomorphic to the join $\mathbb CP^2\ast S^2$. The Plücker embedding $G_{4,2}\subset\mathbb CP^5$ is equivariant for these actions and induces the embedding $\mathbb CP^1\ast S^2\subset\mathbb CP^2\ast S^2$ for the standard embedding $\mathbb CP^1\subset\mathbb CP^2$.
All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian $G_{4,2}(\mathbb R)$ and the real projective space $\mathbb RP^5$ for the action of the group $\mathbb Z_2^4$. We prove that the orbit space $G_{4,2}(\mathbb R)/\mathbb Z_2^4$ is homeomorphic to the sphere $S^4$ and that the orbit space $\mathbb RP^5/\mathbb Z_2^4$ is homeomorphic to the join $\mathbb RP^2\ast S^2$.

Key words and phrases: torus action, orbit, space, Grassmann manifold, complex projective space.

DOI: https://doi.org/10.17323/1609-4514-2016-16-2-237-273

Full text: http://www.mathjournals.org/.../2016-016-002-002.html
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Bibliographic databases:

ArXiv: 1410.2482
MSC: 57S25, 57N65, 53D20, 53B20, 14M25, 52B11
Received: April 29, 2015; in revised form October 21, 2015
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Citation: Victor M. Buchstaber, Svjetlana Terzić, “Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $\mathbb CP^5$”, Mosc. Math. J., 16:2 (2016), 237–273

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Simonetta Abenda, Petr G. Grinevich, “Real soliton lattices of the Kadomtsev–Petviashvili II equation and desingularization of spectral curves: the $\mathrm {Gr^{ \scriptscriptstyle TP}}(2,4)$ case”, Proc. Steklov Inst. Math., 302 (2018), 1–15
2. Anton A. Ayzenberg, “Torus actions of complexity 1 and their local properties”, Proc. Steklov Inst. Math., 302 (2018), 16–32
3. V. M. Buchstaber, S. Terzić, “The foundations of $(2n,k)$-manifolds”, Sb. Math., 210:4 (2019), 508–549
4. D. V. Millionshchikov, R. Jimenez, “Geometry of Central Extensions of Nilpotent Lie Algebras”, Proc. Steklov Inst. Math., 305 (2019), 209–231
5. Masashi Noji, Kazuaki Ogiwara, “The Smooth Torus Orbit Closures in the Grassmannians”, Proc. Steklov Inst. Math., 305 (2019), 251–261
6. V. M. Bukhshtaber, S. Terzić, “Toric topology of the complex Grassmann manifolds”, Mosc. Math. J., 19:3 (2019), 397–463
7. S. Abenda, P. G. Grinevich, “Reducible m-curves for le-networks in the totally-nonnegative grassmannian and kp-ii multiline solitons”, Sel. Math.-New Ser., 25:3 (2019), UNSP 43