Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces

Fields, Strings, Matrices and Symmetric Products

“…This is actually proved for the Hilbert scheme of an algebraic surface in [17,18]. Although we will only be in a position to understand this well in the next section, we can also determine the full partition function that encodes the quantum mechanics on S N X.…”

“…Actually, the relevant concept will turn out to be the orbifold Euler number. Using this concept there is a beautiful formula that was first discovered by Göttsche [16] (see also [17,18]) in the context of Hilbert schemes of algebraic surfaces, but which is much more generally valid in the context of orbifolds, as was pointed out by Hirzebruch and Höfer [19]. First some notation.…”

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…This explicit form was given by L. Göttsche and W. Soergel in [90] as an application of M. Saito's theorem [156]. Since S (n) is the quotient of the nonsingular variety S n by the finite group S n , its rational cohomology H i (S (n) , Q) is just the S n -invariant part of H i (S n , Q).…”

Algebraic structures behind Hilbert schemes and wreath products

“…Now let us get back to our setup when X, Y are complex surfaces, and τ : X → Y /Γ is a resolution of singularities. The Euler and Hodge numbers of Hilbert schemes were calculated in [Got1,Got2,GS] (also cf. [Che]).…”

“…[DHVW,Zas,DMVV,Bat,BKR,Wa2,CR] and references therein. The Euler numbers and Hodge numbers of Hilbert schemes have been computed in [Got1,GS]. The Hilbert-Chow morphism X…”