This paper finds new numerical relations that link a Calabi–Yau threefold’s basic numbers to the geometry of its tangent spaces. The authors use the first jet bundle (a tool that records first-order infinitesimal data of sections), projective duality, and mixed intersection theory to produce a quadratic inequality that relates the projective degree d = H^3 and the Chern numbers of a polarized Calabi–Yau threefold. They then combine that inequality with estimates coming from hyperplane sections and tangent varieties to sharpen bounds on Hodge numbers.

At a high level, the authors study the incidence of tangent spaces to the embedded threefold and how often those tangent spaces pass through a general point of projective space. A classical identity links the Chern data of the jet bundle to the degree of the tangent variety and to a map that records tangent spaces through points. By computing mixed intersections on the projectivized dual of the first jet bundle and applying the Khovanskii–Teissier inequality, they get a nonlinear (quadratic) constraint on the tangent degree that improves earlier bounds in many cases. They also show that equality in one key inequality is achieved by the quintic threefold.

One concrete outcome is an improved uniform bound on the difference of Hodge numbers h^{1,1}(X) − h^{2,1}(X). The authors prove the double inequality −4d − 80 ≤ h^{1,1}(X) − h^{2,1}(X) ≤ (173/66) d, which is a substantial tightening of previously known linear bounds. In low codimension, where the ambient projective space has small dimension, they recover known classifications (for example the quintic) and in the critical case of embeddings in projective 6-space (P^6) they obtain sharper results: the projective degree d of a nondegenerate Calabi–Yau threefold in P^6 is at most 39 (improving a prior bound of 41), and the tangent degree is given exactly by the quadratic polynomial d^2 − 17d + 84.