Geometry and Topology 2026

 I will explain the main results of my paper (arXiv:2602.17435).
Khovanov homology is a categorification of the Jones polynomial, introduced by M. Khovanov. A persistent theme in the subject is that adding extra structures on Khovanov homology strengthens the invariant, and often detects phenomena invisible at the level of polynomials or bigraded vector spaces.
Motivated by the y-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the sl_2-action constructed by Gorsky, Hogancamp and Mellit, we construct a y-ification of Khovanov homology and define an action of the element e in sl_2 on these y-ifications. Our construction is compatible with the previous ones via Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology; yet our approach is more elementary and suited to diagrammatic and algorithmic computations. As an application, we show that the additional structure can distinguish knots with identical Khovanov homology and identical HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.

Positivity of compact Kahler varieties admitting an int-amplified endomorphism

In this talk, I will discuss compact Kahler varieties admitting int-amplified endomorphisms. Here, an endomorphism is called int-amplified if the pullback of some Kahler class dominates the class itself in a suitable sense. I will explain that the tangent sheaf of such a variety is pseudo-effective in a strong sense. This provides a new link between complex dynamics and positivity theory.
As an application, we prove a Kahler analogue of Yoshikawa's structure theorem: for a compact Kahler manifold admitting an int-amplified endomorphism, its Albanese map is a smooth fibration over a complex torus, and its periodic fibers are of Fano type.This is joint work with Guolei Zhong (East China Normal University).