Vertical sheaves and Fourier-Mukai transform on elliptic Calabi-Yau threefolds (Q505016)

The paper under review studies the important question that whether the Fourier-Mukai transform preserves the stability of sheaves, or more specifically, if it yields isomorphism of the moduli spaces of semistable sheaves. There are already affirmative answers to these questions for certain open subsets of the moduli spaces, however the isomorphism of proper moduli spaces are much harder to achieve. \textit{K. Yoshioka} [Nagoya Math. J. 154, 73--102 (1999; Zbl 0955.14032)] has proven the isomorphism of the proper moduli spaces of semistable one dimensional sheaves on an elliptic surface with the moduli spaces of semistable torsion free sheaves on the dual surface. The paper under review study this similar question for the elliptic threefolds. Let \(p:X\to B\) be a smooth projective Weierstrass model with \(K_X=\mathcal O_X\) over a smooth Fano surface \(B\). Let \(\widehat{X}\) be the Mukai dual realized as a fine moduli space of rank 1 degree 0 torsion free sheaves on the fibers of \(p\). It is known that for sufficiently generic \(X\), there is a canonical isomorphism \(X\cong \widehat X\), and an equivalence of derived categories \(\Phi: D^b(X)\to D^b(\widehat X)\), where \(\Phi\) is the Fourier-Mukai transform with the kernel is a Poincaré universal sheaf \(\mathcal P\) over \(X\times \widehat X\). Let \(\Theta\) be the image of the canonical section \(\sigma:B \to X\). For the real numbers \(s>t>0\), \(\omega:=t\Theta-\mathrm{sp}^*K_B\) is a Kähler class on \(X\). For the Kähler class \(\hat \omega:=\Theta -\hat{\mathrm{sp}}^*K_B\) with \(s>1\), and a fixed suitable topological invariant \(\hat \gamma\) of a 1-dimensional torsion sheaf on \(\widehat X\), let \(\mathcal M_{\hat \omega}(\widehat X,\hat{\gamma})\) be the moduli stack of Simpson \(\hat \omega\)-semistable sheaves with Chern character \(\hat \gamma\). The main result of the paper under review proves that one can find the real number \(t>0\) such that \(\Phi\) yields an isomorphism of the moduli stacks \(\mathcal M_{\hat \omega}(\widehat X,\hat{\gamma})\cong \mathcal M_{\omega}^{\text{ad}}(X,\phi(\hat{\gamma}))\), where \(\omega=t\Theta-\mathrm{sp}^*K_B\) and \(\phi\) is the map on topological invariants induced by \(\Phi\), and \(\mathcal M_{\omega}^{\text{ad}}(X,\phi(\hat{\gamma}))\) is a certain open and closed substack of the moduli stack of Simpson \(\omega\)-semistable vertical 2-dimensional sheaves in class \(\phi(\hat{\gamma})\). Moreover, this substack is the only component of the moduli stack provided that \(B\) is the Hirzebruch surface \(\mathbb F_a\) for \(0\leq a\leq 1\). This result has applications in Donaldson-Thomas theory and studying the modularity conjecture of the invariants of 2-dimensional sheaves, as discussed in the paper under review.