Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not preserved by Kahler Ricci flow? More precisely, how can I see it explicitly from the maximum principle argument?
I am assuming you are asking the question on a compact Kahler manifold. You can find argument in Mok's paper for the part of non-negativity of holomorphic bisectional curvature (HBSC) is preserved under Kahler Ricci flow.(http://projecteuclid.org/download/pdf_1/euclid.jdg/1214441778) Roughly speaking, we need to compute the evolution equation of HBSC, then we will get a quadratic term involving curvature besides the heat operator. If the quadratic term satisfies "null vector condition", then the maximal principle will get us the result.
This is the conjecture of Feldman: The answer is in the page 207 of this book: The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects