Yanir Rubinstein (University of Maryland, USA) will speak in the geometry seminar on Monday 25 February, at 1.30pm in the Salle de Profs. Yanir’s title is Differential, algebraic, and convex geometry arising from asymptotic positivity and his abstract is below.
A general theme in geometry is the classification of algebraic/differential geometric structures which satisfy a positivity property. I will describe an “asymptotic” version of this theme based on joint work with Cheltsov, Martinez-Garcia, and Zhang. On the algebraic side, we introduce the class of asymptotically log Fano varieties and state a classification theorem in dimension 2, generalizing the classical efforts of the 19th century Italian school. The novelty here is the use of a convex optimization theorem that reduce the asymptotic positivity to determining intersection properties of high-dimensional convex bodies. On the differential side, I will give a conjectural picture for existence of singular Kahler-Einstein metrics and explain progress towards this conjecture making use of symmetry, log canonical thresholds, test configurations, and Fujita-Odaka’s basis type invariant. Time permitting, I will also touch on relations to singular Kahler-Ricci solitons, mention some conjectures and results about the `small angle limit’ when the angle tends to zero, and tie this picture to non-compact Calabi-Yau fibrations, steady Ricci solitons, and recent work of Liu on wall-crossings in moduli space.