The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms

We classify CR maps from the hyperquadric of signature $l>0$ in ${\mathbb{C}}^n$, $n\ge 3$, to the local model for the tube over the null cone of a symmetric form in ${\mathbb{C}}^{n+1}$, up to CR automorphisms of the source and target. In contrast to the setting of the Heisenberg hypersurface in ${\mathbb{C}}^3$ (i.e., the case $l=0$), studied earlier in Reiter–Son [27], our analysis uncovers two new equivalence classes of CR maps of geometric rank one and one new class of geometric rank two in the case $n=3$. In the case $n\ge 4$, we establish that all maps extend to local isometries of certain indefinite Kähler metrics. We further derive a classification of (local) proper holomorphic maps from the generalized unit ball ${\mathbb{B}}_l^n$ into a generalized version of the Lie ball $D_{m,l}^{\mathrm{I}\mathrm{V}}$ (the generalized classical domain of type~IV).