Nonabelian Jacobian of smooth projective surface : a survey

TitreNonabelian Jacobian of smooth projective surface : a survey
Type de publicationArticle de revue
AuteurReider, Igor
EditeurSpringer Verlag
TypeArticle scientifique dans une revue à comité de lecture
Titre de la revueScience China Mathematics
Mots-clésHilbert scheme, Jacobian, Vector bundle
Résumé en anglais

The nonabelian Jacobian J(X; L, d) 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 X [d] of subschemes of length d of X and the stack M X (2, L, d) of torsion free sheaves of rank 2 on X having the determinant O 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 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.

This is an expository paper giving an account of most of the main properties of J(X; L, d) uncovered in Reider 2006 and ArXiv:1103.4794v1.

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Titre abrégéSci. China Math.