Kaehler-Einstein metrics on toric manifolds

I had a wonderfully unexpected intersection of interests with my officemate who works with PDEs a few weeks ago. He works on optimal transport problems; a motivation for which is finding Kaehler-Einstein metrics on Fano manifolds (or smooth Fano varieties). Tian, one of the pioneers of the theory, wrote that Kaehler-Einstein metrics are designed to be ‘canonical’ choices of metric on certain types of manifold and, to that end, are very well motivated. The issue, of course, is that they don’t always exist. The statement my officemate shared that intrigued me was the following…

If a toric Fano manifold admits a Kaehler-Einstein metric, then the barycentre of its moment polytope is the origin.

A good reason for this is that the barycentre turns out to be a geometric invariant of the manifold called the ‘Futaki character’ that measures the obstruction to the existence of a Kaehler-Einstein metric. There’s an interesting and, to me at least, not yet fully explored feature of the combinatorics at play in which both the algebra (from toric algebraic geometry) and the analysis (from the Einstein PDE governing the metric) fix a particular translate of the moment polytope, which is only defined a priori up to translation. Otherwise the ‘barycentre’ doesn’t even make sense!

Anyway, here are my notes elaborating on the claim above and the nearby theory based on a paper of Mabuchi.

There’s also an element of Lie theory brought in by the fact that manifolds with a Kaehler-Einstein metric should have that the identity component of their automorphism group is reductive, which is interestingly complementary to discussion I’ve posted in the past of root systems associated to the ‘discrete’ part of the automorphism groups del Pezzo surfaces. More on this soon.

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