The nonabelian Kodaira-Spencer map is explicitly the composition of the usual Kodaira-Spencer map of the family with a natural cohomology map induced by the universal Higgs field.
Algebraic Geometry Santa Cruz 1995, Part 2
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Nonabelian Kodaira-Spencer maps
The nonabelian Kodaira-Spencer map is explicitly the composition of the usual Kodaira-Spencer map of the family with a natural cohomology map induced by the universal Higgs field.