## 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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### References:

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