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We study the local curvature estimates of long-time solutions to thenormalized K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-amplecanonical line bundles. Using these estimates, we prove that on such amanifold, the set of singular fibers of the semi-ample fibration on which theRiemann curvature blows up at time-infinity is independent of the choice of theinitial K\"ahler metric. Moreover, when the regular fiber of the semi-amplefibration is not a finite quotient of a torus, we determine the exact curvatureblow-up rate of the K\"ahler-Ricci flow near the regular fiber.

Local curvature estimates of long-time solutions to the K\"ahler-Ricci flow