5 problems

Let $X$ be a Fano manifold, so $K_X^{-1}$ is ample, and suppose that its tangent bundle $TX$ is nef. An algebraic cellular decomposition is a chain of Zariski closed subsets whose…

Webster–Yakimov's conjecture. The totally nonnegative double flag variety admits a cellular decomposition.

Lusztig's conjecture. The totally nonnegative part of a partial flag variety is a regular CW complex.

Cellular tree normal form conjecture. The root $\alpha$ admits a cellular tree normal form, with $c_i$ cells of dimension $i$.

Kac's cellular decomposition conjecture. This set admits a cellular decomposition by locally closed subvarieties isomorphic to affine spaces, with $c_i$ equal to the number of cell…