2024/04/25 by Xin Lü, Lü, Xin, Tan, Shengli · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2404.16293
openalex publication_date 2024/04/25 · openalex created_date 2024/04/27 · openalex updated_date 2026/07/28
Let \mathcal F be a foliation on a smooth projective surface S over the complex number ℂ. We introduce three birational non-negative invariants c12(\mathcal F), c2(\mathcal F) and χ(\mathcal F), called the Chern numbers. If the foliation \mathcal F is not of general type, the first Chern number c12(\mathcal F)=0, and c2(\mathcal F)=χ(\mathcal F)=0 except when \mathcal F is induced by a non-isotrivial fibration of genus g=1. If \mathcal F is of general type, we obtain a slope inequality when \mathcal F is algebraically integral. As a corollary, \mathcal F is always transcendental if the slope is less than 2. On the other hand, we also prove three sharp Noether type inequalities if \mathcal 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.