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Xiuxiong Chen, Simon Donaldson, Song Sun

September 19, 2013

http://www2.imperial.ac.uk/~skdona/KEDEVELOPMENTS-9-19-2013.PDF

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Gang Tian has made claims to credit for these results. The purpose of this document is to rebut these claims on the grounds of originality, priority

and correctness of the mathematical arguments. We acknowledge Tian's many

contributions to this field in the past and, partly for this reason, we have avoided raising our objections publicly over the last 15 months, but it seems now that this is the course we have to take in order to document the facts. In addition, this seems to us the responsible action to take and one we owe to our colleagues, especially those affected by these developments.

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In sum, our fundamental objections to Tian’s claim over the partial C^0 estimate are:

– It seems to us highly improbable that Tian independently came on the proof, involving exactly the same ideas, in the short time interval (roughly April-June 2012) in question. Here we have in mind that, as noted above, the techniques which underpin the proof have been available for ten years or more.

– Even given that it is not impossible that such a coincidence occurred, we have clear priority in the presentation of both outline and detailed proofs.

– Even after 15 months from the appearance of Donaldson and Sun's paper [2] to the date of this writing, Tian has not produced a convincing complete proof of this result.

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Our fundamental objections to Tian's claims with respect to Yau's conjecture

are:

• that we feel that there is no evidence that Tian was in possession of anything approaching a complete proof at the time of his announcement [6] in Stony Brook;

• that both arXiv versions [11], [12] of his paper have serious gaps and mistakes;

• that, insofar as these gaps and mistakes have been partially filled and corrected (in comparing [11], [12], [13]), many of the changes and additions made reproduce ideas and techniques that we had previously introduced in our publicly available work [7], [8], [9], 10], without any kind of acknowledgement. We will not attempt to take up every single gap and mistake that we see in Tian's proposed proofs (including the necessity of checking carefully the relevant results of Jeffres, Mazzeo and Rubinstein, noted above), but concentrate on three points in the subsections 3.1,3.2,3.3 below.

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These assertions are blatant copying without attribution. This is almost half a year since the appearance of our third paper [10], in which the detailed proof of the reductivity is provided, based on the uniqueness theorems proved by Berndtsson and Berman-Boucksom-Essydieux-Guedj-Zeriahi, and the technical difficulty in extending the usual proof of the Matsushima theorem is pointed out.