This talk reports on joint work with Timothy Chow.

Suppose $f:X\to Y$ is a proper morphism of smooth complex varieties. The local invariant cycle theorem states that the cohomology of the (possibly singular) fiber $X_s$ of $f$ over a point $s\in Y$ surjects onto the local monodromy invariants of the nearby smooth fibers. This theorem is fairly direct consequence of the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber (BBDG), and, in fact, it is one of the last results in their book on perverse sheaves.

The main topic of my talk will be a theorem stating that the surjection of BBDG is actually an isomorphism if and only if the fiber $X_s$ has palindromic cohomology. Chow and I used this to prove a conjecture of Shareshian and Wachs relating Tymoczko's dot action on the cohomology of Hessenberg varieties to a combinatorial object called the chromatic quasisymmetric function. I will explain this application, and also say a few words about an independent proof of the Shareshian-Wachs conjecture given by Guay-Paquet.