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Let $H(A,B)$ be a hypergeometric family of degree $d$ and weight $w$ with defining parameters $A=[a_1,\ldots,a_m]$ and $B=[b_1,\ldots,b_n]$. The Hodge vector $(h^{0,w},\dots,h^{w,0})$ of $H(A,B)$ records the dimensions $h^{p,q}$ (with $p+q=w$) of the Hodge subspaces associated to the family.

The Hodge vector depends on how the indices $a_j$ and $b_k$ and intertwine on the circle $\R/\Z$, and we have $d = \sum_{p=0}^{w} h^{p,w-p}$. At the extreme of complete intertwining, the Hodge vector is $(h^{0,0}) = (d)$. At the extreme of complete separation, $d = w+1$ and the Hodge vector is $(h^{0,w},\dots,h^{w,0}) = (1,\dots,1)$.

For $t \in \Q$ with $t \neq 0,1$, the motive $H(A,B,t)$ has the same Hodge vector as the family. For $t=1$, the motive $H(A,B,t)$ has degree $d-1$ or $d-2$, according to whether $w$ is even or odd: if $w$ is even, then the middle Hodge number $h^{w/2,w/2}$ drops by $1$, whereas if $w$ is odd, then the two middle Hodge numbers drop by $1$.

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  • Last edited by John Voight on 2019-08-23 14:22:12
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