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# Bound on the (anticanonical) degree of somesmoothtoric Fano varieties

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ is at most $C^n (n+1)^n$?

This would follow from the weakening of Ehrhart's conjecture that I proposed in http://mathoverflow.net/questions/88153/reference-request-ehrharts-conjecture-on-the-geometry-of-numbers (or at least this is what I understand from reading page 6 of the paper of Nill and Paffenholtz http://front.math.ucdavis.edu/0905.2054).

I wonder:

1. Is this known?

2. Is this interesting?

My knowledge of algebraic geometry is pitiful so please do not be offended if the question is really naive. I'm just trying to see what possible interesting consequences could have the weakened conjecture could have.

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# Bound on the (anticanonical) degree of some smooth Fano varieties

Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ is at most $C^n (n+1)^n$?

This would follow from the weakening of Ehrhart's conjecture that I proposed in http://mathoverflow.net/questions/88153/reference-request-ehrharts-conjecture-on-the-geometry-of-numbers (or at least this is what I understand from reading page 6 of the paper of Nill and Paffenholtz http://front.math.ucdavis.edu/0905.2054).

I wonder:

1. Is this known?

2. Is this interesting?

My knowledge of algebraic geometry is pitiful so please do not be offended if the question is really naive. I'm just trying to see what possible interesting consequences could have the weakened conjecture.