Lefschetz Without Positivity: An Overview

Karim Alexander Adiprasito

The Lefschetz property is central in the theory of projective varieties, detailing a fundamental property of their Chow rings, essentially saying that the multiplication with a geometrically motivated class is of full rank.   We drop the keyword projective, and replace it with the word "generic". I will describe how, for instance, a complete, quasismooth toric and NON-projective variety still satisfies the Lefschetz property, provided the torus action is sufficiently generic. I will survey the theory and the techniques (including an intriguing relation to Parseval-Rayleigh identities in positive characteristic), as well as some of the motivations and applications to quantitative topology, combinatorics, discrete geometry and more. I will end with some cool conjectures.

https://www.ias.edu/video/lefschetz-without-positivity-overview