Topology of the complex varieties $A\sbs\sp{(n)}$

Dibag, I.

doi:10.4310/jdg/1214433721

Cited by 1 publication

(6 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The cases when 2d ≤ n ≤ 2d + 2 are called the unstable cases (the terminology belongs to Dibag, see [3]). The theorem of the previous section holds verbatim in the unstable cases if we provide the correct definitions of the ∆ j 's.…”

Section: The Unstable Casesmentioning

confidence: 99%

“…; Z) where * denotes the cell dual, PD denotes Poincaré duality and t denotes homology duality (see [3]). Since boundary operations are zero, the cycles and cocycles constitute the homology and cohomology respectively.…”

Section: Obstruction Classes We First Define the Chern Classes Asmentioning

confidence: 99%

“…represents A (n+2) d+1 , the space of normalized 2-forms in R n+2 of rank 2(d + 1), on which the Stiefel bundle V n+2,2(d+1) of orthonormal 2(d + 1)-frames in R n+2 induces a principal U (d + 1)-bundle (see [2,3]). …”

mentioning

confidence: 99%

“…, d + 1, of the principal U (d + 1)-bundle V n+2,2(d+1) necessarily vanish. If E is not trivial then the above geometry is analyzed on a certain subbundle S ω of E, depending on ω, and its triviality is another necessary condition for the decomposability of ω (see [3]). …”

mentioning

confidence: 99%

“…For recent applications one can refer to [10,12,13,14]. For the existence and the decomposability of 2-forms see [2,3,8]. Finally, for 2-forms on spheres see [2,4].…”

mentioning

confidence: 99%

See 4 more Smart Citations

Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Sertöz

2001

Colloq. Math.

View full text Add to dashboard Cite

Abstract. We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d, Q n ), the space of [d]-planes in the smooth quadric Q n in P n+1 over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank. Introduction.In this article we study from a purely projective-geometric point of view the obstructions to globally decomposing a 2-form. It was shown by Dibag [3] that the vanishing of certain Chern classes is necessary for such a decomposition. We construct new classes whose nonvanishing implies the nonvanishing of the Chern classes. Moreover some vanishing patterns of these new classes imply the vanishing of the Chern class obstructions. This is achieved by studying the intersection structure of the integral homology generated by the Chern cells. Our methods are purely geometric and determine the required products up to a nonzero multiplicative constant. However this suffices for our purposes since we eventually check for vanishing of obstructions. In the case of maximal planes these coefficients can be explicitly calculated. This is done by Hiller and Boe [7] who consider the case of type B maximal isotropic Grassmannians. Type D (which is a consequence of the result in type B) appeared in [9]. The results of [7] are further reproved by Pragacz and Ratajski [11] by using divided differences. Recently similar calculations in type B were done by Sottile [14]. In the nonmaximal case these calculations are due to Pragacz and Ratajski (see [12]). However to adopt these general formulas for our cases would lead to complicated combinatorial formulas. By checking only nonvanishing conditions we are able to present a purely geometrical argument which suffices for our results.We denote by G(d, Q n ) the space of complex projective [d]-planes lying in the smooth quadric hypersurface Q n of P n+1 . Dibag has shown that G(d, Q n )

show abstract

Section: The Unstable Casesmentioning

confidence: 99%

Section: Obstruction Classes We First Define the Chern Classes Asmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…For recent applications one can refer to [10,12,13,14]. For the existence and the decomposability of 2-forms see [2,3,8]. Finally, for 2-forms on spheres see [2,4].…”

mentioning

confidence: 99%

See 3 more Smart Citations

Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Sertöz

2001

Colloq. Math.

View full text Add to dashboard Cite

show abstract

Topology of the complex varieties $A\sbs\sp{(n)}$

Cited by 1 publication

References 3 publications

Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Contact Info

Product

Resources

About