= {\operatorname{A}}_{g - \ell}(X)$ of codimension $\ell$-cycles modulo rational equivalence. We let $\operatorname{ch}_g(E)$ denote the component of $\operatorname{ch}(E)$ contained in ${\operatorname{A}}^g(X)_\QQ$. In this notation, the Hirzebruch-Riemann-Roch theorem reads $$\chi(E) =  \int_X \operatorname{ch}_g(E)$$ where if $\eta = \sum_p n_p [p] \in {\operatorname{A}}^g(X)_\QQ$, then $\int_X \eta$ denotes the number $\sum_p n_p$. We refer to [@Fulton] for more details regarding intersection theory.

Let ${\operatorname{E}}:= \{{\operatorname{E}}^{p,q}_r, d^{p,q}_r \}_{p,q \in \ZZ, r \in \NN}$ be a spectral sequence arising from a filtered complex of $\mathbb{K}$-vector spaces. Let $\{M^n\}_{n \in \ZZ}$ be a collection of $\mathbb{K}$-vector spaces with the property that each $M^n$ admits a descending filtration $$\dotsc  \supseteq F^{p-1} M^n \supseteq F^p M^n \supseteq F^{p+1} M^n \supseteq \dotsc \text{,}$$ with $F^q M^n = M^n$, for all sufficiently small $q$, and $F^q M^n = 0$, for all sufficiently large $q$. We use the notation $${\operatorname{E}}_r^{p,q} \Rightarrow M^{p + q}$$ to mean that, for all sufficiently large integers $s$, and all integers $p$ and $q$, we have specified (split) short exact sequences $$0 \rightarrow F^{p+1} M^{p + q} \rightarrow F^{p} M^{p + q} \rightarrow {\operatorname{E}}^{p,q}_s  \rightarrow 0 \text{.}$$

Index conditions {#pair:index:condition:section}
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Let $X$ be an abelian variety defined over an algebraically closed field. If $x \in 