  We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a
Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary
$F_\sigma$ ideal, or is any finite or countably iterated Fubini product of
$F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD
families, and assuming Projective Determinacy there are no infinite projective
$\mathcal J$-MAD families; and under the full Axiom of Determinacy +
$V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families.
These results apply in particular when $\mathcal J$ is the ideal of finite sets
$\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The
proofs combine ideas from invariant descriptive set theory and forcing.
