A geometric realization of Catalan functions

We construct a smooth projective variety ${\mathcal{X}}_{\mathrm{\Psi }}$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{G}\mathrm{L}(n)$, determined by a root ideal $\mathrm{\Psi }$. A natural family of line bundles on ${\mathcal{X}}_{\mathrm{\Psi }}$ yields the Catalan functions – symmetric functions introduced by Chen–Haiman and studied further by Blasiak–Morse–Pun–Summers. By analyzing the geometry of ${\mathcal{X}}_{\mathrm{\Psi }}$, 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.