Orbifoldes \`a premi\ere classe de Chern nulle

TL;DR

This paper extends the Bogomolov decomposition theorem to orbifolds with quotient singularities, showing that such spaces with zero first Chern class decompose into simpler components, confirming conjectures about K3 surfaces and threefolds.

Contribution

It provides an orbifold version of the Bogomolov decomposition theorem for Kähler spaces with quotient singularities and zero first Chern class, using Ricci-flat metrics and splitting theorems.

Findings

Normal K3 surfaces are orbifold-uniformized by either K3 surfaces or complex tori.

Supports conjectures on the structure of special threefolds.

Extends classical decomposition results to orbifold settings.

Abstract

An orbifold version of Bogomolov decomposition theorem is established for compact K\"ahler spaces with quotient singularities and first Chern class zero.The proof is a direct adaptation of the classical smooth case, using Ricci-flat K\"ahler metrics, and Cheeger-Gromoll splitting theorem. It implies that normal K3 surfaces are uniformised in the orbifold sense either by normal K3 surfaces or by a complex torus, as conjectured by D.Q. Zhang. It also implies some conjectures on "special" threefolds raised in math.AG/0110051