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The Jacobian of a smooth projective curve is undoubtedly one of the
most remarkable and beautiful objects in algebraic geometry. This
work is an attempt to develop an analogous theory for smooth
projective surfaces - a theory of the nonabelian Jacobian of smooth
projective surfaces. Just like its classical counterpart, our
nonabelian Jacobian relates to vector bundles (of rank 2) on a
surface as well as its Hilbert scheme of points. But it also comes
equipped with the variation of Hodge-like structures, which
produces a sheaf of reductive Lie algebras naturally attached to
our Jacobian. This constitutes a nonabelian analogue of the
(abelian) Lie algebra structure of the classical Jacobian. This
feature naturally relates geometry of surfaces with the
representation theory of reductive Lie algebras/groups. This work's
main focus is on providing an in-depth study of various aspects of
this relation. It presents a substantial body of evidence that the
sheaf of Lie algebras on the nonabelian Jacobian is an efficient
tool for using the representation theory to systematically address
various algebro-geometric problems. It also shows how to construct
new invariants of representation theoretic origin on smooth
projective surfaces.
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