239]. In particular if $E$ has rank one, then $E$ is semi-homogeneous.

\[semi:homog:index\] Let $E$ be a non-degenerate simple semi-homogeneous vector bundle on $X$. Then $E$ satisfies the index condition. In addition, if $A$ is an ample line bundle on $X$, then the roots of the polynomial $\chi(A^N \otimes E)$ are real and ${\operatorname{i}}(E)$ equals the number of positive roots counted with multiplicity.

By [@Muk78 Theorem 5.8, p. 260], $E \cong f_*(L)$ for some isogeny $f \colon Y \rightarrow X$, and some line bundle $L$ on $Y$. Let $A$ be an ample line bundle on $X$ and let $N$ be an integer. Using the push-pull formula and the Leray spectral sequence we check that $$\label{semi:homog:index:1}  {\mathrm{H}}^i(X, A^{\otimes N} \otimes E) \cong {\mathrm{H}}^i(Y, f^*(A)^{\otimes N} \otimes L) \text{, for all $i$.}$$ Using the isomorphisms , we conclude that $$\label{semi:homog:index:2} \chi(A^{\otimes N} \otimes E) = \chi(f^*(A)^{\otimes N} \otimes L)  \text{.}$$

Setting $N = 0$, and using and , we conclude that $L$ is non-degenerate and that ${\mathrm{H}}^j(X, E) = 0$ for all $j \not = {\operatorname{i}}(L)$. Hence, $E$ satisfies the index condition and ${\operatorname{i}}(E) = {\operatorname{i}}(L)$.

On the other hand, if $N$ is positive then $f^*(A)^{\otimes N}$ is an ample line bundle 