Venue: Room 204, Hai Na Yuan #2, Zijingang Campus

Time: 10:30 - 11:50, December 29

Speaker: Trung Nghiem (Université Claude Bernard Lyon 1)

• Pretalk: Introduction to toric Calabi--Yau cones

• Abstract: A toric variety is a normal algebraic variety that contains an algebraic torus as a dense set, whose action extends to the whole variety. Since their conception, the varieties have provided many insightful examples for important conjectures in algebraic geometry. This pretalk aims to introduce the concept of complex affine toric varieties with Gorenstein singularities; their classification in terms of rational polytopes; and their equivalent metric characterization as toric Calabi--Yau cones (i.e. Ricci-flat Kähler cone metrics with toric isometry).

• Research talk: An effective construction of asymptotically conical Calabi--Yau manifolds

• Abstract: An asymptotically conical Calabi--Yau manifold is a Ricci-flat Kähler manifold whose shape, when zoomed out towards infinity, looks like a Calabi--Yau cone. A recent work of Conlon--Hein shows that an AC Calabi--Yau manifold is obtained either by algebraic deformations or crepant resolution in a reversible and exhaustive process. In terms of the metric on the cone, the behavior of the AC Calabi--Yau metric is said to be quasi-regular or irregular. Examples of the latter are notoriously rare in the literature: in fact the only such example before our work was built by Conlon--Hein using ad-hoc computations; but so far there has been no explicit way to obtain them, and an open question in their paper was whether there exist more metrics of the same kind. In my research talk, I'll present an effective strategy to construct irregular AC Calabi--Yau manifolds via Altmann's deformation theory of isolated toric Gorenstein singularities (i.e. toric Calabi--Yau cones by the previous talk). This is a joint work with Ronan Conlon (University of Texas, Dallas).