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We construct K\"ahler groups with arbitrary finiteness properties by mappingproducts of closed Riemann surfaces holomorphically onto an elliptic curve: foreach $r\geq 3$, we obtain large classes of K\"ahler groups that haveclassifying spaces with finite $(r-1)$-skeleton but do not have classifyingspaces with finitely many $r$-cells. We describe invariants which distinguishmany of these groups. Our construction is inspired by examples of Dimca,Papadima and Suciu.

Branched covers of elliptic curves and K\"ahler groups with exotic finiteness properties