Let $X$ be a projective variety and $A$ and $B$ are two vector bundles on $X$. Let $C_{\bullet}$ denote the complex of sheaves
$$
0\rightarrow A\rightarrow B\rightarrow 0
$$
Then we have a cup product in hypercohomology
$$
\mathbb H^i(C_{\bullet})\otimes \mathbb H^j(C_{\bullet})\rightarrow \mathbb H^{i+j}(C_{\bullet}\otimes C_{\bullet})
$$
Is it possible to describe the cup product in terms of the co-cycles? A modern reference will also be helpful.
Best Answer
Although I doubt this counts as a "modern" reference, chapter II section 6 of Godement's Topologie algébrique... gives the most detailed account that I know for cup products in sheaf cohomology. This includes explicit formulas in terms of Cech cocycles.He doesn't treat products in hypercohomology, but the formulas are easy to modify: Given a bounded complex of sheaves $C^\bullet$, choose a good open cover $\mathcal{U}$ of $X$. In your case "good" means affine. One has $$\mathbb{H}^i(C^\bullet) \cong H^i(\check{C}(\mathcal{U},C^\bullet))$$ where the thing on the right is the cohomology of the total complex of the double complex formed from the Cech complex of $C^\bullet$. You should be able to modify the formulas in Godement to define products in $\check{C}(\mathcal{U},C^\bullet)$. The formulas are a bit messy, otherwise I would write them here.