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Supergeometry of [formula omitted]-projective spaces.

Authors :
Noja, Simone
Source :
Journal of Geometry & Physics. Jan2018, Vol. 124, p286-299. 14p.
Publication Year :
2018

Abstract

In this paper we prove that Π -projective spaces P Π n arise naturally in supergeometry upon considering a non-projected thickening of P n related to the cotangent sheaf Ω P n 1 . In particular, we prove that for n ⩾ 2 the Π -projective space P Π n can be constructed as the non-projected supermanifold determined by three elements ( P n , Ω P n 1 , λ ) , where P n is the ordinary complex projective space, Ω P n 1 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω ∈ H 1 ( T P n ⊗ ⋀ 2 Ω P n 1 ) ≅ C . Likewise, in the case n = 1 the Π -projective line P Π 1 is the split supermanifold determined by the pair ( P 1 , Ω P 1 1 ≅ O P 1 ( − 2 ) ) . Moreover we show that in any dimension Π -projective spaces are Calabi–Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also Π -Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of Π -geometry. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
03930440
Volume :
124
Database :
Academic Search Index
Journal :
Journal of Geometry & Physics
Publication Type :
Academic Journal
Accession number :
127982392
Full Text :
https://doi.org/10.1016/j.geomphys.2017.11.010