Beauville--Bogomolov--Yau decomposition for K\"ahler generalized pairs

Abstract

In this paper, we establish the Beauville--Bogomolov--Yau decomposition for generalized pairs in the K\"ahler setting, thereby extending this structure theorem to the natural framework of the K\"ahler generalized Minimal Model Program. More precisely, we prove that, after passing to a finite quasi-\'etale cover, a K\"ahler generalized klt pair $(X,\Delta,\boldsymbol{\beta})$ of Calabi--Yau type admits a locally constant fibration $(X,\Delta)\to Y$ over a Calabi--Yau variety $Y$ whose fiber $(F,\Delta|_{F})$ is rationally connected. Equivalently, after base change to the universal cover of $Y$, the fibration becomes a product, and its global structure is determined by a monodromy action preserving the pair $(F,\Delta|_{F})$. As a principal application, when the nef b-part $\boldsymbol{\beta}$ vanishes, we show that the monodromy can be eliminated after a further finite quasi-\'etale cover, yielding a product decomposition into a rationally connected pair, strict Calabi--Yau varieties, irreducible holomorphic symplectic varieties, and complex tori. The proof introduces new analytic methods involving relative projectivity, localized positivity and flatness of direct image sheaves, generalized pairs in the analytic Minimal Model Program, and foliations on singular K\"ahler varieties.