Speaker：Xin Lv (East China Normal University)

Time：2024-12-5, 9:30

Location：Conference Room S204 at Experiment Building at Haiyun Campus

Abstract:

Let F be a foliation on a smooth projective surface S over the complex numbers. We introduce three birational non-negative invariants c_1^2(F), c_2(F) and chi(F), called the Chern numbers. If the foliation F is not of general type, the first Chern number c_1^2(F)=0, and c_2(F)=chi(F)=0 except when F is induced by a non-isotrivial fibration of genus g=1. If F is of general type, we obtain a slope inequality when F is algebraically integrable. As a corollary, F is always transcendental if the slope is less than 2. On the other hand, we also prove three sharp Noether type inequalities if F is of general type. As applications, we obtain a criterion for foliations to be transcendental using Noether type inequalities, and we also give a partial positive answer to the question on the lower bound on the volume of a foliation of general type. This is a joint work with Professor Shengli Tan.