A geometric realization of Catalan functions

We construct a smooth projective variety $\mathscr X_\Psi$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $\Psi$. A natural family of line bundles on $\mathscr X_\Psi$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of $\mathscr X_\Psi$, we prove the vanishing conjecture of Chen--Haiman, confirm the tame case of the vanishing conjecture of Blasiak--Morse--Pun, and establish the monotonicity conjectures of Shimozono--Weyman.