Let $X$ be a smooth projective Calabi-Yau threefold over $\mathbb{C}$. Let $M_{X}(r,c_1,c_2)$ denote the moduli space of Gieseker-stable sheaves on $X$ with Mukai vector $(r,c_1,c_2)$.

Is the following true:

There exists a natural virtual fundamental class $[M_{X}(r,c_1,c_2)]^{\text{vir}}$ in the Chow group $A_*(M_{X}(r,c_1,c_2))$ with the following properties:

1. The virtual fundamental class is invariant under deformations of the complex structure of $X$.

2. The virtual Euler characteristic $$\chi^{\text{vir}}(M_{X}(r,c_1,c_2)) = \int_{[M_{X}(r,c_1,c_2)]^{\text{vir}}} 1$$ is an integer.

3. The virtual Euler characteristic is equal to the Euler characteristic of the derived category of coherent sheaves on $X$ with the same Mukai vector: $$\chi^{\text{vir}}(M_{X}(r,c_1,c_2)) = \chi(\mathcal{D}^b(X), (r,c_1,c_2)).$$

The challenge, as I see it, lies in the fact that the moduli space $M_{X}(r,c_1,c_2)$ is often singular and non-compact, which makes constructing a virtual fundamental class quite difficult.

It seems natural to try and extend the conjecture to include K3 surfaces or even higher-dimensional Calabi-Yau varieties.