Let $X$ be a complex abelian fourfold of Mumford-type and let $ V=H^{1}(X,\Q) \, .$ The complex Mumford-Tate group of $X$ is isogenous to $\gr.\,$ We recover information about the Hodge structure of $X$ using representations of the Lie algebras $\alg \,$ and $ \mathfrak{sp} (8) $ acting on $ V \otimes_{\scriptstyle {\Q}} \C .\, $ Using these techniques we show that there is a Kuga-Satake variety $A$ associated to $X$ in such a way that $A$ is isogenous to $X ^{32}.$
Abelian fourfold of Mumford-type and Kuga-Satake varieties
GALLUZZI, Federica
2000-01-01
Abstract
Let $X$ be a complex abelian fourfold of Mumford-type and let $ V=H^{1}(X,\Q) \, .$ The complex Mumford-Tate group of $X$ is isogenous to $\gr.\,$ We recover information about the Hodge structure of $X$ using representations of the Lie algebras $\alg \,$ and $ \mathfrak{sp} (8) $ acting on $ V \otimes_{\scriptstyle {\Q}} \C .\, $ Using these techniques we show that there is a Kuga-Satake variety $A$ associated to $X$ in such a way that $A$ is isogenous to $X ^{32}.$
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