Supergeometry of Π-projective spaces

JOURNAL OF GEOMETRY AND PHYSICS

卷:124

Article

Noja, Simone¹ 

[1] Univ Milan, Dipartimento Matemat, Via Saldini 50, I-20133 Milan, Italy

关键词: Supergeometry;    Pi-projective geometry;    Calabi-Yau supermanifolds;    Non-projected supermanifolds;   

DOI  :  10.1016/j.geomphys.2017.11.010

来源: Elsevier

【 摘 要 】

In this paper we prove that Pi-projective spaces P-Pi(n) arise naturally in supergeometry upon considering a non-projected thickening of P-n related to the cotangent sheaf Omega(1)(pn). In particular, we prove that for n >= 2 the Pi-projective space P-Pi(n) can be constructed as the non-projected supermanifold determined by three elements (P-n, Omega(1)(pn), lambda), where P-n is the ordinary complex projective space, Omega(1)(pn) is its cotangent sheaf and lambda is a non-zero complex number, representative of the fundamental obstruction class omega is an element of H-1(T-Pn circle times Lambda(2)Omega(1)(Pn)) congruent to C. Likewise, in the case n = 1 the Pi-projective line P-Pi(1) is the split supermanifold determined by the pair (P-1, Omega(1)(P1) congruent to O-p1(-2)). Moreover we show that in any dimension Pi-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also Pi-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 Pi-geometry. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free