On Borel mappings and σ-ideals generated by closed sets

We obtain some results about Borel maps with meager fibers on Polish spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy.We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable spaces X, every Borel map on a Borel set in X not in I, either has a fiber not in I or else it is injective on a Borel set not in I. This is the case for the σ-ideal generated by finite-dimensional closed sets in the Hilbert cube, which provides an answer to a question asked by M. Elekes.