Speaker: Jiajun Hu (Tsinghua University)

Title: On the kernels of Lefschetz operators

Time: 10:00–11:00, November 4, 2026 (Wednesday)

Inviter: Prof. Weizhe Zheng (MCM, CAS)

Venue: MCM110

Abstract: We study the kernels of Lefschetz operators arising from complete intersections of nef classes. When the classes are semiample, we provide a complete characterization of mixed Lefschetz properties below a fixed Hodge coniveau – all obstructions are detected by irreducible subvarieties of relevant codimensions. Whenever these properties hold, we further show that each kernel, viewed as a sub-Hodge structure, is supported on a Zariski closed subset of the corresponding codimension, as predicted by the generalized Hodge conjecture. In particular, the diagonal kernels admit bases consisting of fundamental classes of irreducible subvarieties annihilated by the Lefschetz operator. We also establish a kernel decoupling theorem for Lefschetz modules. As an application, we obtain a kernel characterization for supercritical collections of nef classes, thereby proving a conjecture of Shenfeld-van Handel and Hu-Xiao.