Ben Tighe: Extending differential forms across singularities
Channel: D. Zack Garza
Date: 2021-07-04

Abstract: Given a smooth projective variety 𝑋, there are many contexts for which one can study "differentials" on 𝑋.  When the ground field is 𝐂, we may define "holomorphic differentials" on 𝑋 which often encapsulate the geometry of 𝑋 and show that these agree with the algebraic differentials that one encounters in a first−year algebraic geometry course in a very precise way.    

When 𝑋 has singularities, it no longer makes sense to define holomorphic differentials.  Moreover, even though algebraic differentials exist for all varieties, they do not capture the geometry of 𝑋 in the same way holomorphic differentials do in the smooth case.  The aim of this talk will be to describe a suitable replacement in this setting.  The outline of the talk then will be as follows.  

We will discuss how holomorphic differentials show up in complex algebraic geometry (Serre duality, Kodaira vanishing, Hodge decomposition theorem, Serre GAGA).  We will then look at different kinds of differentials on singular spaces and see how good they are at replacing holomorphic differentials.  In particular, we will look closely at reflexive differentials, which inherit many properties that we see in the smooth case.  The last part of the talk will concern the "extension problem" for reflexive differentials.  Time permitting, we will look at applications of the extension problem to recent work.