Beijing Institute of Mathematical Sciences and Applications

In gauge theory, the prominent examples of enumerative invariants are Donaldson polynomials and Seiberg-Witten invariants, which help to distinguish different smooth structures on 4 dimensional manifolds. In recent years, other 4-manifold invariants have been introduced by changing the gauge theory (the PDE’s and the “counting” problem) or, by changing the dimension, similar gauge theory invariants were defined on higher-dimensional manifolds. Notable examples include Donaldson-Thomas (DT) invariants for six-dimensional, Calabi-Yau, manifolds. The study of enumerative geometry (counting of algebraic subspaces) of complex surfaces and threefolds proved to be deeply related to physical structures, e.g. around Gopakumar-Vafa invariants (GV); Gromov-Witten invariants, DT, as well as Pandharipande-Thomas (PT) invariants; and their “motivic lifts". On the other hand, physical dualities in Gauge and String theory, such as Montonen-Olive duality and heterotic/Type II duality have also been a rich source of spectacular predictions about these counting invariants. An example of this is the modularity properties of GW or DT invariants which is proved mathematically in some cases, as suggested by the heterotic/Type II duality. Furthermore, in recent years, there has been enormous amount of activity to generalize these constructions to complex 4 folds and 5 folds. This course is aiming at setting background on geometry of moduli spaces of sheaves over algebraic surfaces, 3 folds, 4 folds and 5 folds. The course gives brief introduction to gauge theoretic foundations behind each enumerative theory in each case, and discusses, in detail, the techniques of enumerative algebraic geometry such as, construction of suitable obstruction theories, degeneration techniques, localization technique, wall-crossing in master space developed by Mochizuki, as well as motivic wall-crossing developed by Kontsevich-Soibelman and Joyce-Song in each dimension. Many examples will be discussed during the course and many comparisons between DT theory and GW theory will be discussed. This course can be regarded as part 2 of the earlier course on Gromov-Witten theory and Donaldson-Thomas theory last semester. We will review some of the background material on geometry of moduli spaces of sheaves, as well as topics covered from the last semester in the “GW and DT theory” course, then we spend an extensive amount of time on the book by Mochizuki on “Donaldson-Thomas type invariants for algebraic surfaces”, and further into the semester discuss how to generalize these constructions and use DT theory in order to count curves and surfaces in 3 folds, 4 folds and 5 folds.