Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula

This is the second of series of papers studying moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up $p\colon \widehat{X}\to X$ of a projective surface $X$ at a point 0.

The main results of this paper are as follows:

(a)

We describe the wall-crossing between moduli spaces caused by the twisting of the line bundle $\mathcal{O}(C)$ associated with the exceptional divisor $C$.

(b)

We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves.

Moreover, we also give proofs which we observed in a special case in

[Perverse coherent sheaves on blow-up. I. A quiver description, preprint, arXiv:0802.3120] for the following:

(c)

The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to $[C]$.

(d)

The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by $\mathcal{O}(-mC)$ for sufficiently large $m$.

Therefore the usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.