Nonabelian Jacobian of Smooth Projective Surfaces - A Survey

The nonabelian Jacobian $\JA$ of a smooth projective surface $X$ is inspired by the classical theory of Jacobian of curves. It is built as a natural scheme interpolating between the Hilbert scheme $\XD$ of subschemes of length $d$ of $X$ and the stack ${\bf M}_X (2,L,d)$ of torsion free sheaves of rank 2 on $X$ having the determinant $\OO_X (L)$ and the second Chern class (= number) $d$. It relates to such influential ideas as variations of Hodge structures, period maps, nonabelian Hodge theory, Homological mirror symmetry, perverse sheave, geometric Langlands program. These relations manifest themselves by the appearance of the following structures on $\JA$: 1) a sheaf of reductive Lie algebras, 2) (singular) Fano toric varieties whose hyperplane sections are (singular) Calabi-Yau varieties, 3) trivalent graphs. This is an expository paper giving an account of most of the main properties of $\JA$ uncovered in [R1] and [R2].