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Successive minima on arithmetic varieties. (English) Zbl 0871.14019

Given a hermitian vector bundle \((E,h)\) over the spectrum \(S\) of the ring of integers of a number field \(K\), one can define just as for classical lattices so-called successive minima. The \(i\)-th minimum is the infimum of all numbers such that \(i\) independent sections of norm at most that number exist. The bundle \(E\) gives a vector space \(E_K\) over \(K\) with dual vector space \(E_K^\vee\). A closed subvariety \(X_K\) in the projectivation of \(E_K^\vee\) then has a Faltings height \(h(X_K)\), a degree \(\deg(X_K)\) and a closure \(X\) over \(S\). The present paper bounds a combination of \(h(X_K)\) and \(\deg(X_K)\) in terms of the successive minima of \((E,h)\). The arithmeticity of the variety \(X\) of course plays an important role in the proof of this. Several applications are discussed, for instance to rank 2 vector bundles on curves of genus at least 2, and to pluricanonical images of surfaces of general type.
Reviewer: J.Top (Groningen)

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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