Douglas–Reinbacher–Yau conjecture on stable reflexive sheaves

Let $X$ be a simply connected Calabi–Yau threefold. Let $c_i\in H^{2i}(X,\mathbb Z)$ for $i=2,3$, and let $r>1$ be an integer. Suppose there is an ample class $H\in H^2(X,\mathbb Q)$ such that

$$c_2=r\left(H^2+\frac{1}{24}c_2(X)\right),\qquad c_3<\frac{2^{9/2}}{3}rH^3.$$

Douglas–Reinbacher–Yau conjecture. There exists a rank-$r$ stable reflexive sheaf $\mathcal E$ on $X$ with respect to some polarization such that $c_1(\mathcal E)=0$, $c_2(\mathcal E)=c_2$, and $c_3(\mathcal E)=c_3$.

This conjecture proposes a general existence result for stable reflexive sheaves with prescribed Chern classes on simply connected Calabi–Yau threefolds. The source introduces it as a conjecture of Douglas, Reinbacher, and Yau; its resolution status is not established by the supplied text.