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Accusonus ERA Bundle Pro 4.1.10

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Accusonus Era Pro Audio Plugins Bundle 6.0.4 for Windows.
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Accusonus ERA Bundle Pro 6.1.1. Reasoner’s.
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Accusonus ERA Bundle Pro 6.1.1.Q:

On the Hodge spectrum of a smooth rationally connected variety

Hi.
Let $X$ be a smooth projective variety over $\mathbb C$ with ample canonical bundle $K_X$ and let $\mathcal L$ be an ample invertible sheaf. The Hodge spectrum $\mathcal M_H(X,\mathcal L)$ of the pair $(X,\mathcal L)$ is a closed subcone of the Hodge fan $\mathcal F(X)$. Is it true that $\mathcal M_H(X,\mathcal L)$ has full dimension? I know of the example of an elliptic curve over $\mathbb C$ with a non-trivial polarization, so I guess that this is not true in full generality. But what about smooth rationally connected varieties?
Note that in this case we have that the Hodge spectrum $\mathcal M_H(X)$ of the polarized variety $(X,\mathcal L)$ is actually a closed subset of a face of $\mathcal F(X)$.

A:

No. For example, for any $g\ge 0$, the Hodge spectrum has dimension $\le 2g+2$. However, it is not hard to prove using Kawamata’s theorem that on a smooth projective rationally connected variety $X$, the Hodge spectrum $\mathcal M_H(X,\mathcal O_X(1))$ of the invertible sheaf $\mathcal O_X(1)$ is a full-dimensional convex polyhedral cone (i.e. $\mathcal M_H(X,\mathcal O_X(1))$ is a convex polyhedron) in $\operat

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