# Cohomology of the flag variety under PBW degenerations

## TL;DR

This paper investigates the cohomology of PBW degenerations of flag varieties, demonstrating surjectivity onto the original cohomology in both type A and symplectic cases, with implications for algebraic geometry.

## Contribution

It establishes the surjectivity of cohomology maps from PBW degenerations to original flag varieties, including in equivariant and symplectic settings, advancing understanding of degenerations.

## Key findings

- Cohomology of PBW degenerations surjects onto that of the original flag variety
- Surjectivity holds in equivariant setting for type A flag varieties
- Surjectivity also applies to Feigin's linear degeneration of symplectic flag varieties

## Abstract

PBW degenerations are a particularly nice family of flat degenerations of type A flag varieties. We show that the cohomology of any PBW degeneration of the flag variety surjects onto the cohomology of the original flag variety, and that this holds in an equivariant setting too. We also prove that the same is true in the symplectic setting when considering Feigin's linear degeneration of the symplectic flag variety.