Some curvature estimates of Kähler-Ricci flow

In this worK, first we will obtain some local curvature estimates for Kähler-Ricci flow on Kähler manifolds with initial metrics of nonnegative bisectional curvature. As a corollary, we prove that if g(t) is a complete solution of the Kähler Ricci flow which satisfies |Rm(g(t))| ≤ at^−θ for some 0 < θ < 2, a > 0 and g(0) has nonnegative bisectional curvature, then g(t) also has nonnegative bisectional curvature. This generalizes results in [Amer. J. Math. 140 (2018), pp. 189–220] and [J. Differential Geom. 45 (1997), pp. 94–220]. Using the local curvature estimate, we prove that for a complete solution g(t) of the Kähler-Ricci flow with g(0) to have nonnegative bisectional curvature, to be noncollapsing, and sup_M×[τ,T] |Rm(g(t))| < +∞ for all τ > 0, then the curvature of g(t) is in fact bounded by at⁻¹ for some a > 0. In particular, g(t) has nonnegative bisectional curvature for t > 0. This result is similar to a result by Simon and Topping in the Kähler category.