Lectures on the geometry of ag varieties

Michel Brion

Description

Niveau: Supérieur, Doctorat, Bac+8
Lectures on the geometry of ag varieties Michel Brion Introduction In these notes, we present some fundamental results concerning ag varieties and their Schubert varieties. By a ag variety, we mean a complex projective algebraic variety X, homogeneous under a complex linear algebraic group. The orbits of a Borel subgroup form a stratication of X into Schubert cells. These are isomorphic to ane spaces; their closures in X are the Schubert varieties, generally singular. The classes of the Schubert varieties form an additive basis of the cohomology ring H (X), and one easily shows that the structure constants of H (X) in this basis are all non-negative. Our main goal is to prove a related, but more hidden, statement in the Grothendieck ring K(X) of coherent sheaves on X. This ring admits an additive basis formed of structure sheaves of Schubert varieties, and the corresponding structure constants turn out to have alternating signs. These structure constants admit combinatorial expressions in the case of Grassmanni- ans: those of H (X) (the Littlewood-Richardson coecients) have been known for many years, whereas those of K(X) were only recently determined by Buch [10]. This displayed their alternation of signs, and Buch conjectured that this property extends to all ag va- rieties.

• complex geometry


• algebro-geometric methods


• geometric explanations


• group


• projective algebraic variety


• littlewood-richardson coecients


• ishing theorems


• schubert varieties

Sujets

• Complex geometry
• Algebraic variety
• Schubert variety

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 28
• Langue: English

Extrait

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