2017/09/27 by Nicolas Ressayre, Ressayre, N
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1709.09406
openalex publication_date 2017/09/27 · openalex created_date 2021/02/15 · openalex updated_date 2026/07/28
Let G be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product odot_0 on thecohomology group H^*(G/P,\mathbb C) of any projective G-homogeneousspace G/P.Their definition uses the notion of Levi-movability for triples ofSchubert varieties in G/P.In this article, we introduce a family of G-equivariant subbundlesof the tangent bundle of G/P and the associated filtration of the DeRham complex of G/P viewed as a manifold. As a consequence one gets a filtration of the ring H^*(G/P,\mathbb C)and proves that \odot_0 is the associated graded product.One of the aim of this more intrinsic construction of \odot_0 isthat there is a natural notion of fundamental class[Y]_\odot_0∈(H^*(G/P),\odot_0) for any irreducible subvariety Y of G/P.Given two Schubert classes σ_u and σ_v inH^*(G/P), we define a subvariety Σ_uv of G/P. This variety should play the role of the Richardson variety; moreprecisely, we conjecture that[Σ_uv]_\odot_0=σ_u\odot_0σ_v.We give some evidence for this conjecture, and prove special cases.Finally, we use the subbundles of TG/P to give a geometriccharacterization of the G-homogeneous locus of any Schubertsubvariety of G/P.