Given a bounded constructible complex of sheaves \(\mathcal {F}\) on a complex Abelian variety, we prove an equality relating the cohomology jump loci of \(\mathcal {F}\) and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety X; one from Green-Lazarsfeld’s cohomology jump loci and one from the Kashiwara’s estimates for singular supports. This result is related to Kotschick’s conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if X has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients. In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of \(H^0(X,\Omega _X^1)\) . We show that this is indeed the case for forms admitting zero locus of codimension 1.
