Abstract

This serves as a complement to Junliang Shens talk. l will begin with the classical theory of theBeauville decomposition, which provides a canonical, multiplicative splitting of the Leray filtration forabelian schemes. Then l will discuss the problem of extending the Beauville decomposition tocertain abelian fibrations with singular fibers. l will explain how the extension leads to themultiplicativity of the perverse (Leray) filtration for such fibrations, and eventually to a new proof ofthe P= W conjecture. Further results and open guestions will also be discussed. Joint work withYounghan Bae, Davesh Maulik, and Junliang Shen.