Chern bounds and tangent geometry of polarized Calabi-Yau threefolds

We bring new insights into the numerical geography of polarized Calabi-Yau threefolds through the first jet bundle, tangent geometry and projective duality. Let $X$ be a Calabi-Yau threefold with a very ample polarization $H$. We use mixed intersections on the projectivized dual of the first jet bundle to prove a quadratic inequality relating the degree $d=H^3$ and Chern numbers. Combining this inequality with hyperplane-section and tangent-variety estimates yields improved uniform bounds $-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le \frac{173}{66}d$. For nondegenerate embeddings in ${\mathbb{P}}^6$, we improve the upper bound on the degree from $41$ to $39$ and express the tangent degree as a quadratic polynomial in $d$. We also prove that, for every $m\ge 2$, the tangent-incidence morphism associated with $\mathrm{\mid }mH\mathrm{\mid }$ is the normalization morphism of the tangent variety. For $m=1$, we conjecture tangent degree $1$ for complete embeddings in ${\mathbb{P}}^N$ with $N\ge 7$ and verify this for several families, including general intersections of four quadrics and general GPK${}^3$ threefolds.