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We consider certain families of Hecke characters $\phi$ over a quadraticimaginary field $F$. The order of vanishing of the $L$-function $L(\phi,s)$ atthe central point $s=-1$, according to the Beilinson conjectures, should beequal to the dimension of the space of extensions of the Tate motive$\mathbb{Q}(1)$ by the motive associated with $\phi$. In this article,candidates for the corresponding extensions of Hodge structure are constructed,under the assumptions that the sign of the functional equation of $L(\phi,s)$is $-1$ and that $L^\prime(\phi,-1) \neq 0$. This is achieved by means of thecohomology of variations of Hodge structures over Picard modular surfacesattached to $F$ and Harder's theory of Eisenstein cohomology. Moreover, weprovide a criterion for the non-triviality of these extensions, based on thenon-vanishing of a height pairing that we define and study.

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces