  We consider generalizations of the Springer resolution of the nilpotent cone
of a simple Lie algebra by replacing the cotangent bundle with certain other
vector bundles over the flag variety. We show that the analogue of the Springer
sheaf has as direct summands only intersection cohomology sheaves that arise in
the Springer correspondence. The fibers of these general maps are nilpotent
Hessenberg varieties, and we build on techniques established by De Concini,
Lusztig, and Procesi to study their geometry. For example, we show that these
fibers have vanishing cohomology in odd degrees. This leads to several
implications for the dual picture, where we consider maps that generalize the
Grothendieck-Springer resolution of the whole Lie algebra. In particular we are
able to prove a conjecture of Brosnan.
  As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg
varieties often satisfy a relation we call the geometric modular law, which
also has origins in the work on De Concini, Lusztig, and Procesi. We connect
this relation in type $A$ with a combinatorial modular law defined by
Guay-Paquet that is satisfied by certain symmetric functions and deduce some
consequences of that connection.
