Kähler-Ricci Flow with Degenerate Initial Class

In [2], the weak Kähler-Ricci flow was introduced for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler. The convergence as t → 0 is of great importance to study for this topic. 1 Motivation and Set-up Kähler-Ricci flow, the complex version of Ricci flow, has been under intensive study over the past twenty some years. In [10] and more recently [9], G. Tian proposed the intriguing program of constructing globally existing (weak) KählerRicci flow with canonical (singular) limit at infinity and applying it to the study of general algebraic manifold. Generally speaking, one should expect the classic Kähler-Ricci flow to encounter singularity at some finite time which is completely decided by cohomology information according to the optimal existence in [11]. Just as what people wanted to do and have had successes in some cases for Ricci flow, surgery on the underlying manifold should be expected. For Kähler-Ricci flow, we naturally want the surgery to have more flavor in algebraic geometry. For surface of general type, we only need the blow-down of (−1)-curves to apply the construction in [2] to push the flow through finite time singularities. The degenerate class at the singularity time would become Kähler for the new manifold because the (−1)-curves causing the cohomology degeneration have been crushed to points. Things can get significantly more complicated for higher dimensional manifold. In (complex) dimension 3, flips are involved. Simply speaking, one needs to blow up the manifold and then blow down. Naturally, we could expect the transformation of the degenerate class is still not Kähler. In this note, we want to say this is not a problem if formally the Kähler-Ricci flow is instantly taking the class into the K”ahler cone of the new manifold. As in [2], short time existence is the topic. In the following, the precise problem under consideration is stated ∗Research supported in part by National Science Foundation grants DMS-0904760.