Maximal slope of tensor product of Hermitian vector bundles

Huayi Chen (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle $\overline M$ of the tensor product. In the case where the generic fiber of $M$ is semistable in the sense of geometric invariant theory, the estimation is established by constructing, through the classical invariant theory, a special polynomial which does not vanish on the generic fibre of $M$. Otherwise we use an explicte version of a result of Ramanan and Ramanathan to reduce the general case to the former one.

Mots clés: algebraic geometry; Arakelov geometry; maximal slope; semistability

Domaines: Géométrie algébrique [math.AG]

Huayi Chen. Maximal slope of tensor product of Hermitian vector bundles. Journal of Algebraic Geometry, 2009, 18 (3), pp.575-603. ⟨10.1090/S1056-3911-08-00513-4⟩. ⟨hal-00151961v2⟩