Nonabelian Jacobian of smooth projective surfaces -- a survey (Q1935714)

The author in [``Nonabelian Jacobian of smooth projective varieties'', Science China, Vol. 56, No. 1, 1--42 (2013)] is a survey of the work done in [J. Differ. Geom. 74, No. 3, 425--505 (2006; Zbl 1106.14030)] as well as in [{I. Reider}, ``Nonabelian Jacobian of smooth projective surfaces and representation theory'', \url{arXiv:1103.4749}]. Central in this work is the construction of a nonabelian Jacobian \(J(X;L,d)\) of a smooth projective surface \(X\), a scheme over the Hilbert scheme \(X^{[d]}\) of subschemes of length \(d\) in \(X\), with a morphism to the stack of torsion free sheaves of rank 2 on \(X\) with determinant \(\mathcal{O}_X(L)\) and second Chern class \(d\). Among the various constructions covered in this survey, the existence of a sheaf of reductive Lie algebras on \(J(X;L,d)\) takes center stage. It originates from a well-chosen filtration on \(X^{[d]}\). This sheaf, which is the object of the second part of the paper, paves the way for the use of representation theoretic methods in the study of projective surfaces. There is some interesting work covered in the survey, though what makes it alluring are the possible applications. The seasoned algebraic geometer will no doubt appreciate the expansive coverage done in this work; worthy of further investigation are the connections with quantum gravity, homological mirror symmetry, the geometric Langlands program as well as quiver representations/Gromov-Witten invariants.