# Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?

Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry

In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale universal cover $\tilde X$, and its classification of solutions corresponds to Beauville-Bogomolov type decomposition.

The Beauville-Bogomolov decomposition theorem states that given a compact Kähler manifold $X$ with zero real first Chern class, the universal cover $\tilde X$ of $X$ splits holomorphically (and isometrically, once a Ricci-flat Kähler metric is chosen) into the product of a flat factor $\mathbb C^q$ and simply connected compact Kähler manifolds with special unitary or compact symplectic holonomy i.e., Calabi-Yau or hyper-Kähler manifolds.

We have analogue of the Beauville-Bogomolov decomposition theorem when anti-canonical bundle $-K_X$ is nef.

We have the following conjecture. In fact when $X$ is projective then it has been solved . See this paepr.

Conjecture: Let $X$ be a compact Kähler manifold with nef anticanonical class i.e $-K_X$ is nef. Then the universal cover $\tilde X$ of $X$ decomposes as a product $$\tilde X ≃\mathbb C^q ×\prod_j Y_j ×\prod_k S_k × Z,$$ where $Y_j$ are irreducible Calabi-Yau manifolds, $S_k$ are irreducible hyper-kähler manifolds, and $Z$ is a rationally connected manifold

Note that, $X$ is rationally connected if and only if for every invertible subsheaf $\mathcal L⊆\Omega^p_X$, $1≤p≤n$, $\mathcal L$ is not pseudo-effective;

We have the same result when $-K_X\geq 0$ or $T_X$ is nef. See Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider. Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom., 3(2):295– 345, 1994

• Note that the "The Beauville-Bogomolov decomposition theorem" first solved by Calabi(by using Berger theorem) several years before of the papers of Beauville, and Bogomolov, see E. Calabi; On Kahler manifolds with vanishing canonical class. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 78–89. Princeton University Press, 1955 – user21574 Oct 22 '17 at 10:39
• Jean-Pierre Bourguignon, also mentiond this fact in his nice survey paper published in Séminaire Bourbaki, see p.6 line 3 numdam.org/article/SB_1977-1978__20__1_0.pdf – user21574 Oct 22 '17 at 10:42