ICM 2026 Invited Section Lecture - Benjamin Bakker

Mixed Non-Abelian Hodge Theory and the Linear Shafarevich Conjecture For a complex algebraic variety X, classical Hodge theory imposes extra structure on the singular cohomology Hk(X,C) of X using the theory of harmonic forms. Non-abelian Hodge theory similarly enriches the geometry of the algebraic variety of representations of the fundamental group (which is thought of as H1(X,GLr(C))) by using harmonic metrics on local systems. The theory in the "pure" case of smooth projective X was largely worked out in the 1990s beginning with the seminal work of Simpson and has seen many applications, for example the proof of Eyssidieux--Katzarkov--Pantev--Ramachandran of the linear Shafarevich conjecture for such X. In this talk, I will survey recent advances in the "mixed" case of arbitrary complex algebraic varieties X, and show how these tools can be used to prove a number of important results about the topology of algebraic varieties, as well as a general version of the linear Shafarevich conjecture. This is based on joint work with Y. Brunebarbe and J. Tsimerman. Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices