| 概要 | The nonabelian Jacobian J(X;L,d) of a smooth projective surface X is
inspired by the classical theory of Jacobian of curves. As its classical
counterpart it is related on the one hand to the Hilbert schemes of
points on X and on the other hand to the vector bundles ( of rank 2 this
time - here resides the nonablian aspect of the theory) on X. But it
also relates to such influential ideas as variations of Hodge
structures, period maps, nonabelian Hodge theory, Homological mirror
symmetry, perverse sheaves, geometric Langlands program. These relations
manifest themselves by the appearance of the following structures on
J(X;L,d):
1) a sheaf of reductive Lie algebras,
2) (singular) Fano toric varieties whose hyperplane sections are
(singular) Calabi-Yau varieties,
3) trivalent graphs.
In my talk I will explain the appearance of those structures and give
some illustrative examples. I will also discuss how the nonabelian
Jacobian might be used to address the problems of algebraic cycles and
its relation to the geometric Langlands program for surfaces.
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