Net logarithmic tangent sheaves of complete intersections
Sukmoon Huh, Min-gyo Jeong
TL;DR
The paper defines the net logarithmic tangent sheaf 𝕋_X(Y;ℙ^N) as a refinement of the classical logarithmic tangent sheaf for pairs of complete intersections and analyzes its fundamental properties, including torsion-free behavior, rank, and natural exact sequences tied to the gradient map and the Poincaré residue framework. It establishes μ-stability results for net tangent sheaves in the hypersurface setting and specializes to the cubic and quadric surfaces, proving Torelli-type injectivity results that identify hyperplanes with stable rank-2 bundles on these surfaces. In the cubic surface case, it shows the moduli space of stable sheaves with given Chern classes is generically smooth of dimension 33 and that the association H↦𝕋_S(H;ℙ^3) is injective, yielding a universal family and linking the geometry to classical invariants such as the nine inflection points of the associated plane cubic. Overall, the work provides a robust framework for net logarithmic tangent sheaves, their stability, and their connections to moduli spaces and Torelli-type phenomena on complete intersections, with concrete geometric implications for cubic surfaces.
Abstract
The main purpose of this paper is to define the {\it net logarithmic tangent sheaf}, as a generalization of the logarithmic tangent sheaf introduced by P.~Deligne, over the field of complex numbers, and prove some basic properties and give some applications. The generalization is valid for the pairs of the smooth complete intersection variety and its complete intersecting subvariety. As applications, we investigate the locus of net logarithmic tangent sheaves on a smooth cubic surface in the corresponding moduli space of semistable sheaves.