Adelic Toric Varieties and Loop Groups

Explore the intricate world of adelic toric varieties and loop groups in this 47-minute lecture by Alberto Verjovsky from the Universidad Nacional Autónoma de México. Delve into the description of proalgebraic spaces as inverse limits of finite branched covers over normal toric varieties. Examine the completions of adelic abelian algebraic groups and their relation to profinite completions of algebraic tori. Investigate the vector bundle category of proalgebraic toric completions and its connection to finite toric variety coverings. Focus on the complex projective line case, introducing the adelic projective line P and its holomorphic vector bundles. Learn about smooth, Sobolev, and Wiener adelic loop groups, their corresponding Grassmannians, and the Birkhoff factorization theorem. Discover the isomorphism between the adelic Picard group of holomorphic line bundles and the rationals, and explore the Birkhoff-Grothendieck splitting theorem for higher-rank holomorphic bundles over P. For further reading, refer to the related paper available on arXiv: arXiv:2001.07997v2 [math.AG].