Ricci curvature boundedness near singular fibers of the Iitaka fibration

Establish that, for a compact Kähler manifold X with semiample canonical bundle and intermediate Kodaira dimension and for the immortal solution ω•(t) of the normalized Kähler–Ricci flow, the Ricci curvature of ω•(t) remains uniformly bounded also in neighborhoods of the singular fibers of the Iitaka fibration f: X → B (i.e., near the subset S consisting of the preimage of the singular values of f together with the singular set of B), uniformly for all times t ≥ 0. This extends the uniform Ricci curvature bounds proven on compact subsets of X \ S to the regions approaching the singular fibers.

Background

The paper proves that for compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, the immortal Kähler–Ricci flow collapses smoothly to a canonical metric on the base of the Iitaka fibration away from singular fibers, and it establishes uniform Ricci curvature bounds on compact subsets disjoint from the singular fibers. These results resolve previously stated conjectures of Song and Tian concerning smooth convergence and Ricci boundedness on X \ S.

In Remark 1.4, the authors raise a further conjecture that seeks to extend the Ricci curvature boundedness to neighborhoods of the singular fibers. Here, S denotes the preimage under f of the singular values together with the singular set of the base B. The suggested approaches involve developing parabolic analogs of established gluing methods (Gross–Wilson for elliptic surfaces and Li’s gluing for certain 3-folds), underscoring both the difficulty and potential avenues toward resolving the conjecture.