Mathematical skeleton. More Frog than Bird. Big chilling with my homies. Bad at math and worse at physics. Acolyte of the hypergeometric. Aviation enjoyer. I’m the cookie. 我非常喜欢数学。
![Lectures on the Geometry of Flag Varieties
Michel Brion
Introduction
In these notes, we present some fundamental results concerning flag varieties and their Schubert varieties. By a flag variety, we mean a complex projective algebraic variety X, homogeneous under a complex linear algebraic group. The orbits of a Borel subgroup form a stratification of X into Schubert cells. These are isomorphic to affine 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 Grassmannians: those of H*(X) (the Littlewood-Richardson coefficients) 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 the flag varieties. In this setting, the structure constants of the cohomology ring (a fortiori, those of the Grothendieck ring) are yet combinatorially elusive, and Buch's conjecture was proved in [6] by purely algebro-geometric methods.
Here we have endeavored to give a self-contained exposition of this proof. The main ingredients are geometric properties of Schubert varieties (e.g., their normal-ity), and vanishing theorems for cohomology of line bundles on these varieties (these are deduced from the Kawamata-Viehweg theorem, a powerful generalization of the Kodaira vanishing theorem in complex geometry). Of importance are also the intersections of Schubert varieties with opposite Schubert vari…](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:26ypzvcklxb4w5qnyrr7h5cu/bafkreig45wc2qcq2pmjvke4rgs3hpykcmqubymjb3ttjox6aa4u2222qhy)
![It is interesting to note that the entire genesis of this article comes from a certain beautiful special function identity that the second author learned a number of years ago due to Clingher, Doran, & Malmendier (2, Corollary 2.2], that relates Appell's F2 function to the Gauss hypergeometric function 2F1 as a two-parameter Euler integral transform:
[There is an equation depicting Appell’s F2 function as an Euler integral transform of the Gauss hypergeometric function 2F1.]
Specifics will be recalled in Lemma 5.1. This article can be thought of as an extended interpretation of and an attempt to deeply understand this identity.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:26ypzvcklxb4w5qnyrr7h5cu/bafkreifn2xkiqtl37khri44bszbgp2lwgivc22fcqcrqs6jiql2zvzs34u)
Mathematical skeleton. More Frog than Bird. Big chilling with my homies. Bad at math and worse at physics. Acolyte of the hypergeometric. Aviation enjoyer. I’m the cookie. 我非常喜欢数学。