Yes, that does indeed sound like something I might have said :)
I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:
- The cohomology groups of automorphic vector bundles on toroidal compactifications of Shimura varieties are isomorphic to certain spaces of (usually non-holomorphic) automorphic forms. (These vector bundles are coherent sheaves, hence "coherent cohomology".)
- Via this isomorphism, the Serre duality pairing on coherent cohomology groups is given by integration of automorphic forms.
The general results are all set out in Harris' paper "Automorphic forms of $\overline{\partial}$-cohomology type as coherent cohomology classes" (J. Diff. Geom. 1990); the relation between Serre duality and integrals is Proposition 3.8.
To get some feel for how this is applied in practice I recommend reading some other papers of Harris from around the same time, e.g. this one with Kudla: Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2).