Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

The Hodge standard conjecture remains wide open. The numerical version for abelian fourfolds has been proved recently by Ancona. In this talk, I will first explain a proof of the Hodge standard conjecture for squares of K3 surfaces based on our previous work. In the remaining time, I will discuss related results and application of Ancona's idea. Joint with Kazuhiro Ito and Tetsushi Ito.