UC Berkeley

The Knop Conjecture, which was proven by Losev in [Los09a], states that smooth affinespherical varieties are classified up to equivariant isomorphism by their weight monoids. This is in contrast with the standard classification of spherical varieties, which involves combinatorial invariants related to divisors and valuations. In this thesis, we prove that some of these combinatorial invariants are also determined by weight monoids in the smooth projective case. This results in certain partial analogs of the Knop Conjecture for smooth projective spherical varieties. We provide counterexamples to demonstrate that these partial analogs are relatively optimal.

Our results indicate that weight monoids of smooth projective spherical varieties are closelyrelated to the data of certain divisors on these varieties. In analogy with the total coordinate ring discussed in [Bri07], we develop methods for comparing the data of weight monoids with the data of divisors, even without smoothness hypotheses. We then show that, under mild hypotheses, the data provided by weight monoids is equivalent to the data provided by divisors on projective spherical varieties.