Let $\mathscr{F}^\bullet$ be a complex of coherent sheaves on a smooth projective variety $X$. Suppose that the support of $\mathscr{F}^\bullet$ (the union of the supports of the cohomology sheaves, if I understand correctly) has dimension $k$. Why does it follow that the Chern character $\mathrm{ch}(\mathscr{F}^\bullet)$ is zero in codimensions 0 to $k - 1$? (The Chern character of a complex is defined to be the Chern character of its Euler characteristic.)
Edit: this is supposed to be true for a complex $\mathscr{F}^\bullet$ on the product $X \times Y$ of two n-dimensional varieties which is supported in dimension n. Maybe the above generalization is incorrect.
Let $\newcommand{\F}{\mathscr{F}}\F$ be a coherent sheaf on a smooth variety $X$. Let $Z = \mathrm{Supp}(\F) \subset X$ be the support of $\F$, considered as a closed subscheme with the ``annihilator'' scheme structure. Then one has the sheaf $i^*(\F)$ on $Z$, which is still coherent with support $Z$, and further there is a canonical isomorphism $\F \stackrel{\sim}{\longrightarrow} i_*(i^*(\F))$. Now by the Grothendieck-Riemann-Roch theorem the cycle $\newcommand{\ch}{\mathrm{ch}}\ch(\F) = \ch(i_*(i^*(\F)))$ lies in the image of the homomorphism $i_* : A_*(Z) \to A_*(X)$. The latter maps cycles of codimension $j$ to cycles of codimension $n-m+j$, where $m = \dim(Z)$ and $n = \dim(X)$. In particular the Chern character $\ch(\F)$ has no components in codimension less than $n - m$.