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In Scattone " Compactification of Moduli Spaces of Algebraic K3 Surfaces" the author cites a correspondance between primitive isotropic sublattices of some lattice L and the rational boundary components of the Baily Borel compactification of a locally symmetric space.

Question: what is meant by a isotropic sublattice?


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up vote 1 down vote accepted

The concept of isotropic sublattices only makes sense with respect to a given bilinear form. Let $E$ be one on a lattice $\Lambda$, i.e. we have a bilinear map $$E\colon \Lambda\times\Lambda \to \mathbb{R}$$ (or to $\mathbb{Z}$ or whatever you're dealing with). A sublattice $\Gamma$ of $\Lambda$ is called isotropic for $E$ if the restriction of $E$ to $\Gamma$ is zero, i.e. $E|_{\Gamma\times\Gamma} = 0$.

Of course there's also the concept of isotropic vectors ($E(v,v) = 0$) and isotropic vector subspaces for a bilinear form defined on a vector space.

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