Let $\mathcal M$ be a moduli space of sheaves on a nice variety $X$, with fixed rank $r$ and Chern classes $c_i$, semi-stable with respect to an ample bundle $H$. Let $E$ be a "functorial" vector bundle on $\mathcal M$, for example, the tangent bundle or a bundle whose fiber over a point corresponding to a sheaf $F$ is $H^1(F)^{\oplus k}$. If the rank of $E$ is equal to the dimension of $\mathcal M$, then the top Chern class of $E$ is an integer, and I know three methods how to calculate it: rudely, they are to replace $\mathcal M$ by something larger, to replace it by something smaller or to vary it.

  1. A first method is to embed $\mathcal M$ into a large variety $\mathcal N$ whose cohomology ring is known. If $\mathcal M$ is the vanishing set of a section of a vector bundle $V$ on $\mathcal N$, and $E$ can be extended to $\mathcal N$, then the definition of Chern classes via degeneracy loci gives $$\int_{\mathcal M} c_{top}(\mathcal M, E)=\int_{\mathcal N} c_{top}(\mathcal N, E \oplus V).$$

For example, in my problem $X=\mathbb{CP^2}$, and one can embed $\mathcal M$ into the moduli space $\mathcal N$ of representations of a certain quiver, whose cohomology ring is computed in Franzen Chow Rings of Fine Quiver Moduli are Tautologically Presented, but is very difficult.

  1. A better method is to use Bott residue formula: on $X=\mathbb{CP}^2$ acts a torus $T=(\mathbb C^*)^2$, and its action descends to an action on $\mathcal M$. Then the top Chern class of $E$ may be calculated by restricting $E$ to the torus fixed points: $$\int_{\mathcal M} c_{top}(\mathcal M, E)=\sum_{\mathcal M_i \subset \mathcal M^T} \int_{\mathcal M_i} \frac{c_{top}^T (\mathcal M_i, E)}{c_{top}^T (\mathcal M_i, N_{\mathcal M_i} \mathcal M)},$$ where the sum on the right is taken over all components $\mathcal M_i$ of the torus fixed points, the Chern classes are computed in the equivariant cohomology $H^*_T(\mathcal M_i)$, and $N_{\mathcal M_i} \mathcal M$ is the normal bundle.

This method in the case when the torus fixed points are isolated is nicely explained in Ellingsrud, Strømme Bott's formula and enumerative geometry. It actually allows to calculate the top Chern class $a_{c_2}$ of a vector bundle with a fiber $H^1(F)^{\oplus 2r}$ on $\mathcal M=\mathcal M(\mathbb{CP}^2; r, c_1, c_2; \mathcal O_{\mathbb{CP}^2}(1))$, but the combinatorics which enumerates the components of the torus fixed points is quite complicated, so one can not find the generating function $\sum_{c_2 \in \mathbb Z} a_{c_2} t^{c_2}$.

  1. The last method is to change $X$ and to vary an ample bundle $H$. For example, one can consider a blow-up $\pi: \hat X \to X$ and find some connections between the moduli spaces $$\mathcal M(X; r, c_1, c_2; H) \text{ and } \mathcal M(\hat X; r, \pi^* c_1-mE, c_2; \pi^* H-mE),$$ where $X$ is a surface, $E$ is the exceptional divisor, and $m$ is an integer. If $X=\mathbb{CP}^2$, then $\hat X=\mathbb P(\mathcal O_{\mathbb{CP}^1} \oplus \mathcal O_{\mathbb{CP}^1}(1))$ is the first Hirzebruch surface. Similarly, one can make use of the fact that $\hat X$ is a ruled surface deforming $H'=\pi^*H-mE$ to a divisor of a fiber and replacing a moduli space again.

This is done in the case $E=T\mathcal M$, so that the top Chern class is the Euler characteristic, in Mozgovoy Invariants of moduli spaces of stable sheaves on ruled surfaces. A big advantage of this reduction is that it naturally deals with a generating function $\sum_{c_2 \in \mathbb Z} a_{c_2} t^{c_2}$ instead of its terms by themselves. Unfortunately, the formulas are quite difficult and I do not understand them yet, so I would not repeat them here.

Maybe, there is another interesting method how to calculate the top Chern class of a "functorial" vector bundle on a moduli space of sheaves?


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