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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?

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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.

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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

See here

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