## Maximal slope of tensor product of Hermitian vector bundles.(English)Zbl 1170.14013

The author gives 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. He applies Minkowski’s First Theorem, and so he has to find an estimation for 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 the author uses an explicit version of a result of Ramanan and Ramanathan to reduce the general case to the former one.

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14G40 Arithmetic varieties and schemes; Arakelov theory; heights

### Keywords:

hermitian vector bundles; maximal slope; tensor product
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### References:

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