Topology and geometry of moduli spaces of semistable sheaves on bielliptic surfaces
We study the topology and low-degree Hodge theory of moduli spaces of semistable sheaves on bielliptic surfaces. For primitive rank-zero Mukai vectors $\mathbf{v}=(0,c_1(L),\chi)$ satisfying the positivity condition $\operatorname{nt}(L)\geq3$, the distinguished fixed-determinant component $M_{H,S}(\mathbf{v},L)$ admits a support morphism to $|L|$ and is interpreted as a relative compactified Jacobian. Using this fibration, Lefschetz-type properties of positive linear systems, monodromy, and mixed Hodge structures, we construct a surjective homomorphism $ \pi_1(S)\twoheadrightarrow\pi_1\bigl(M_{H,S}(\mathbf{v},L)\bigr) $ and compute the first two Betti numbers of an Albanese fiber $F$, $M_{H,S}(\mathbf{v},L)$, and the distinguished component $M^\circ_{H,S}(\mathbf{v})$. Fourier-Mukai transforms and Bridgeland wall crossing extend these computations to primitive admissible Mukai vectors of positive rank. We further prove that $H^2(F,\mathbb{C})$ is of pure Hodge type $(1,1)$. If $\lambda_S=\ell(\mathbf{v})=1,$ then $F$ is a strict irreducible Calabi-Yau variety up to a finite quasi-\'etale cover. Under the additional genericity assumption on the pullback polarization, $M^\circ_{H,S}(\mathbf{v})$ is smooth and is noncanonically birational to $\operatorname{Pic}^0(S)\times\operatorname{Hilb}^{\mathbf{v}^2/2}(S)$.