Let \(X\) be a smooth projective variety of dimension \(n\ge 1\) and let \(\mathcal{D}\) be an arrangement of pairwise distinct smooth and irreducible divisors on \(X\) with simple normal crossings. A \(\mathcal{D}\)-logarithmic co-Higgs sheaf on \(X\) is defined as a pair \(\left(\mathcal{E},\Phi \right)\), where \(\mathcal{E}\) is a torsion-free coherent sheaf on \(X\) and \(\Phi :\mathcal{E}\to\mathcal{E}\otimes\mathcal{T}_{\mathcal{D}}\) is a morphism with \(\Phi \wedge \Phi =0\); \(\mathcal{T}_{\mathcal{D}}\) here denotes the logarithmic tangent bundle on \(X\) associated to \(\mathcal{D}\) assuming negative degree \(\deg\mathcal{T}_{\mathcal{D}}\).
In the first part of this article the authors primarily investigate a numeric criterion on the rank and degree of the successive quotients in the Harder-Narasimhan filtration of a torsion-free coherent sheaf \(\mathcal{E}\) in order to admit a non-trivial co-Higgs field \(\Phi \). The existence of such a co-Higgs field is induced by explicit usage of positive elementary transformations and the affirmative answer to the Lange conjecture, which in turn provides the existence of an exact sequence of stable vector bundles on \(X\) with prescribed ranks and degrees.
The second part of the article deals with extending the notion of the Segre invariant to the setting of logarithmic co-Higgs structures. Namely, for \(\left(\mathcal{E},\Phi \right)\) a logarithlic co-Higgs sheaf with \(r:=\text{rk}\mathcal{E}\ge 2\) and for \(\mathcal{S}\left( k,\mathcal{E},\Phi \right)\) the set of all subsheaves \(\mathcal{A}\subset\mathcal{E}\) of fixed rank \(k\in \left\{ 1,\ldots ,r-1 \right\}\) such that \(\Phi \left(\mathcal{A}\right)\subseteq\mathcal{A}\otimes \mathcal{T}_{\mathcal{D}}\), the \(k\)-th Segre invariant is defined by \[ s_k\left(\mathcal{E},\Phi \right):=k\deg\mathcal{E}-\max_{\mathcal{A}\in\mathcal{S}\left(k,\mathcal{E},\Phi \right)}r\deg \mathcal{A}. \] It is shown that this invariant is well-defined and some of its basic properties are being studied.