Cohomology of the flag variety under PBW degenerations (Q2328180)

Degenerate flag varieties in type \texttt{A}, in terms of quiver Grassmannians, were studied to produce a cellularization or to study their singular locus. In types \texttt{A} and \texttt{C}, Feigin's degenerations of flag varieties are isomorphic to Schubert varieties in an appropriate partial flag variety, thus, giving an explanation for many of their good properties, such as normality and Cohen-Macaulay-ness. More general families of linear degenerations of type \texttt{A} flag varieties have been studied recently, which are known as PBW degenerations; they are a large class of proper flat degenerations of flag varieties, which are isomorphic to Schubert varieties. The authors show that all PBW degenerations of the flag variety \(\mathcal{F}l_n\) have the following good property: their cohomology, with integral coefficients, surjects onto the cohomology of \(\mathcal{F}l_n\) (Theorem 3, page 840). Moreover, both varieties are equipped with an action of an \((n-1)\)-dimensional complex algebraic torus, and the same surjectivity result holds for the equivariant cohomology groups with integral coefficients (Theorem 4, page 841). The technique to compare the cohomology of a complex algebraic variety \(X\) and that of a proper flat degeneration \(Y\) of \(X\) is as follows. Consider a proper flat family \(\pi:\widetilde{X}\to\mathbb{C}\) with \(X=\pi^{-1}(1)\) and \(Y=\pi^{-1}(0)\). Since \(\widetilde{X}\) is proper and flat, \(\widetilde{X}\) contracts to \(Y\). Thus there is a surjection \(g:H^*(Y,\mathbb{Z})\cong H^*(\widetilde{X},\mathbb{Z})\twoheadrightarrow H^*(X,\mathbb{Z})\) induced by the inclusion of \(X\) in \(\widetilde{X}\). The authors also prove surjectivity in the symplectic setting when considering Feigin's linear degeneration of the symplectic flag variety (Theorem 5, page 843).