SUCCESSIVE MINIMA ON ARITHMETIC VARIETIES

Mumford's theory of stability, when applied to varieties over number f ields, has interesting consequences, as was shown in recent years by s everal authors [12], [5], [13], [3], [16], [20]. In this paper, we use it to get informations on the successive minima of the lattice of sec tions of bundles on arithmetic varieties. More precisely, let E be a p rojective module of rank N over the ring of integers in a number field K, and E(K)(boolean OR) = Hom(E, K) Consider a closed subvariety X(K) subset of P(E(K)(boolean OR)) in the projective space of lines in E(K )(boolean OR). Fix a hermitian metric on E x(Z) C. Bost proved in [3] that Chow semi-stability of X(K) in P(E(K)(boolean OR)) implies a lowe r bound for the height of X(K) (see 3.1 below). By a different method we show that the proof that X(K) is semi-stable gives, in some cases, a stronger inequality (see however the remark in 3.1.2) which involves the successive minima of E. Our general result, Theorem 1, can be app lied to surfaces of general type, Theorem 3, using the work of Gieseke r [7], and to line bundles on smooth curves, Theorem 4, using the work of Morrison [14]. A variant of Theorem 1 gives results for rank two s table bundles on curves, Theorem 5, by using the work of Gieseker and Morrison [8]. Finally, we derive another inequality for successive min ima on arithmetic surfaces, Theorem 6, from the vanishing theorem prov ed in [16]. I thank J.-B. Best, J.-E Burnol, I. Morrison, I. Reider, E . Ullmo and S. Zhang for helpful discussions, and the referee for poin ting out several inaccuracies