BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ben Tighe (University of Illinois at Chicago) DTSTART:20210630T190000Z DTEND:20210630T200000Z DTSTAMP:20260423T024516Z UID:GOSS2021/6 DESCRIPTION:Title: Extending differential forms across singularities\nby Ben Tighe (Univ ersity of Illinois at Chicago) as part of Graduate Online Seminar Series ( GOSS)\n\nLecture held in TBA.\n\nAbstract\nGiven a smooth projective varie ty $X$\, there are many contexts for which one can study "differentials" o n $X$. When the ground field is $\\mathbf C$\, we may define "holomorphic differentials" on $X$ which often encapsulate the geometry of $X$ and sho w that these agree with the algebraic differentials that one encounters in a first-year algebraic geometry course in a very precise way. \n\nWhen $ X$ has singularities\, it no longer makes sense to define holomorphic diff erentials. Moreover\, even though algebraic differentials exist for all v arieties\, they do not capture the geometry of $X$ in the same way holomor phic differentials do in the smooth case. The aim of this talk will be to describe a suitable replacement in this setting.\n\nThe outline of the ta lk 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 differe nt 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 p roblem" for reflexive differentials. Time permitting\, we will look at ap plications of the extension problem to recent work.\n LOCATION:https://researchseminars.org/talk/GOSS2021/6/ END:VEVENT END:VCALENDAR