L²-vanishing theorem and a conjecture of Kollár

2024/09/17 by Deng, Ya, Wang, Botong · 2 citations
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.11399

Abstract

In 1995, Kollár conjectured that a smooth complex projective n-fold X with generically large fundamental group has Euler characteristic χ(X, K_X)≥ 0. In this paper, we prove the conjecture assuming X has linear fundamental group, i.e., there exists a representation π₁(X)→ \rm GL_N(ℂ) with finite kernel. We deduce the conjecture by proving a stronger L² vanishing theorem: for the universal cover \widetildeX of such X, its L²-Dolbeault cohomology H₍₂₎^n,q(\widetildeX)=0 for q≠ 0. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.

Cited by

• Topology and Euler characteristics of tropical varieties
• Euler characteristics of Kollár-hyperbolic varieties

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