Abstract: Working on previous work and ideas of T. Zhang, I will consider geographical problems regarding the notion of the slope of a fibred variety over a surface. I will apply recent stability results and Chern degree functions on surfaces to propose a refined invariant: the positive second Chern character. As an application, I will present slope inequalities for fibrations of varieties of maximal Albanese dimension over surfaces. This is a joint work in progress with M. Lahoz and A. Rojas.

Abstract: 设𝐺是紧黎曼曲面 𝐶 的一个有限自同构群。20世纪30年代，Chevalley和Weil利用𝐺-作用的固定点描述了多典范层ωC⊗𝑛的整体截面空间 𝐻^0(𝐶,ω_C^⊗𝑛) 作为𝐺-表示的不可约分解。1980年，Ellingsgrund和Lønsted 将该公式推广为关于 𝐺-等变局部自由层ℰ的欧拉（虚拟）𝐺-模的𝜒_𝐺(𝐶,ℰ)≔[𝐻^0(𝐶,ℰ)]−[𝐻^1(𝐶,ℰ)]∈𝑅(𝐺)的一个表达式。在此基础上，人们从各个角度细化了光滑射影代数曲线上的Chevalley-Weil公式，但没有人尝试过给出2维及以上的紧复流形/光滑射影簇上的相应公式。基于Atiyah-Singer的全纯Lefschetz不动点公式，我将在这个报告中定义固定轨迹的（虚拟）分歧𝐺-模，并由此给出Chevalley-Weil 公式的一个高维推广。这是与吕人杰的合作工作。

Abstract: The effective adjunction conjecture of Prokhorov–Shokurov predicts that the moduli part of an lc-trivial fibration is effectively base-point-free. Li proposed a broader version, the Γ-effective adjunction conjecture, better adapted to moduli theory. While it was known that the former implies a weaker form of the latter, the precise relationship between the two conjectures remained unclear.
    In this talk, I will explain a recent result showing that the two conjectures are in fact equivalent in all relative dimensions. The proof relies on a uniform rational polytope for canonical bundle formulas, building on recent progress in the minimal model program for algebraically integrable foliations. As an application, we deduce Li’s conjecture in relative dimension one.