Heights and Geometric Invariant Theory

Abstract

Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified representation $\T$, a hermitian tensor bundle $\E_T$. We can prove then that there exists a lower bound for the heights of points $x\in\P(\E_T)$ with $SL_N(K)$--semistable generic fibre in terms of the degree of $\E$ and some universal constants depending only on the compactified representation. We give then three applications: a universal lower bound for general flag varieties, an application to the adjoint representation of $SL_N(K)$ and a construction of a height on the moduli space of semistable vector bundles over algebraic curves.