D-equivalence conjecture for hyperkähler varieties of generalized Kummer type

We show that birationally equivalent hyperkähler varieties of generalized Kummer type are derived equivalent, confirming the D-equivalence conjecture in this case. Combined with recent results of other authors, this settles the conjecture for all known deformation types of hyperkähler varieties. The kernel is built from an equivariant Fourier-Mukai kernel on the sum-zero loci of powers of an abelian surface. We deform it along diagonal twistor paths, using the hyperholomorphic endomorphism algebra. Comparing wall hyperplanes gives a common twisted derived category for two adjacent birational models, and a finite sequence of wall crossings then gives the equivalence.