Chern bounds and tangent geometry of polarized Calabi-Yau threefolds

Abstract

We study the numerical geography and tangent geometry of very amply polarized Calabi--Yau threefolds $(X,H)$ through the positivity of the first jet bundle $J^1(H)$. Writing $d={\int }_X{H}^3$, $c={\int }_X{c}_2(X)H$, and $e={\int }_X{c}_3(X)$, we exploit two different positivity properties of this single bundle. Mixed intersections on $\mathbb{P}(J^1(H{)}^{\ast })$ give $e\ge -5d-c-c^2/(4d)$, while a volume estimate for a perturbed tautological class gives $e\ge 40d\left[(1-\frac{c}{10d}{)}^{3/2}-1\right]$; in particular $e+6c\ge 0$, improving Sun's inequality $e+10c\ge 0$. As consequences, we obtain the uniform Hodge bounds $-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le 173d/66$, the lower bound $\mathrm{deg}{X}^{\vee }\ge 78$ for the dual hypersurface, and, in the critical case $X\subset {\mathbb{P}}^6$, the upper bound $d\le 34$. We also prove that, for every $m\ge 2$, the tangent-incidence morphism associated with $|mH|$ is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings $X\subset {\mathbb{P}}^N$ with $N\ge 7$, and verify it for several families.