Recently, "Cohomological Milnor formula and Saito's conjecture on characteristic classes" by Zhao Yigeng, E Fund Endowed Assistant Professor, Westlake University, and Yang Enlin, Assistant Professor, Peking University was published online in Inventiones Mathematicae, a leading international journal of mathematics. They have made significant progress in the study of characteristic classes in geometric ramification theory.
One of the central issues in the research of geometric ramification theory is creating characteristic classes of constructible étale sheaves. Three different definitions of characteristic classes have appeared in the literature: Abbes and Saito proposed a cohomological version of characteristic classes in 2007; Kato and Saito defined the Swan classes in 2008; and Saito developed geometric characteristic classes in 2017 using Beilinson's theory of singular support. Although the methods for creating these three characteristic classes differ, Saito conjectured that they are essentially equivalent. The new study by Zhao and Yang confirms this, under the assumption that the algebraic variety in question is quasi-projective.
For the first time, the study introduces a new cohomological class, which Prof. Saito suggested calling the non-acyclicity class. Based on this characteristic class, the study establishes a fibration formula for cohomological characteristic classes and proposes cohomological analogs of three important formulas in geometric ramification theory: the Grothendieck-Ogg-Shafarevich formula, the Milnor formula, and the Bloch's conductor formula. Notably, Saito's conjecture is actually a direct corollary of the fibration formula and the cohomological version of the Milnor formula.
Yigeng Zhao completed his bachelor's program at Qingdao University. He received his master degree in pure mathematics from Capital Normal University in 2011 and doctorate at the University of Regensburg in 2016. After that, he was a postdoc there. He is currently the E Fund Endowed Assistant Professor.
Link：https://link.springer.com/article/10.1007/s00222-025-01319-y
