Florent Schaffhauser (Universidad de Los Andes, Bogotá): On the Harder-Narasimhan-Shatz stratification for moduli spaces of real bundles
Abstract: Atiyah and Bott have shown via gauge-theoretic methods that the moduli stack of holomorphic vector bundles of rank r and degree d has an equivariantly perfect stratification, which enabled them to compute the equivariant Poincaré series of the moduli stack of semistable such bundles. The stratification in question is indexed by the possible Harder-Narasimhan types of holomorphic vector bundles of rank r and degree d. In joint work with Melissa Liu, we studied the analogous situation over the field of real numbers, where a variety of difficulties arise. First, one needs to consider mod 2 coefficients to show equivariant perfection. Second, depending on the real topological invariants of the curve, the Harder-Narasimhan type of a real bundle will depend on more parameters than the Harder-Narasimhan type of the associated complex bundle. Nonetheless, in the vector bundle case, it is possible to sort out those difficulties, and arrive at a formula for the equivariant Poincaré series of moduli stacks of semistable real vector bundles. In this talk, after briefly reviewing the Atiyah-Bott approach, we will describe the Harder-Narasimhan-Shatz stratification in the real case, and explain how it leads to such a formula. Time permitting, we will show how to use the formula to compute Betti numbers of moduli stacks of semistable real vector bundles of small rank.