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Let $X$ be a regular scheme, flat and of finite type over Z. let be a sequence of vector bundles on $X$, which is exact on the generic fiber of $X$. We endow each $E_i$ , $i = 0, 1, 2$ with an hermitian metric. We give a formula for \widehat{ch}(E_0) + \widehat{ch}(E_2) - \widehat{ch}(E_1), where $\widehat{ch}$ is the arithmetic Chern character, as a sum of secondary characteristic classes. Next, we compute more explicitly these secondary characteristic classes in a situation encountered by the second author when proving a “ Kodaira vanishing theorem ” on arithmetic surfaces. |