Realization conjecture for homology triplets by complexes of coherent sheaves

Let $(B,H,C)$ be a homology triplet of type $n$, with degrees of $B$ given by

$$h=d_0<d_1<d_2<\cdots<d_t=\overline{c},$$

and strand starts of $H$ and $C$ given respectively by

$$h=h_0<h_1<\cdots,\qquad c=c_0<c_1<\cdots.$$

A complex of coherent sheaves $\mathcal{E}^{\bullet}$ on some projective space $\mathbb{P}(W)$ should have hypercohomology nonzero exactly at the specified Betti degrees, homology modules nonzero exactly for the nonempty strands of $H$, and dual cohomology modules nonzero exactly for the nonempty strands of $C$, with the regularity, dimension, and least nonzero cohomology degree specified in the source. Realization conjecture. There is such a complex $\mathcal{E}^{\bullet}$ satisfying all three conditions. This is the paper’s main conjecture, proposing realization of every homology triplet by a complex of coherent sheaves with precisely prescribed Betti degrees and homology and cohomology strands. The supplied context gives no resolution status.