Positivity of the Cotangent Bundle of Complex Hyperbolic Manifolds with Cusps

Let $\bar{X}$ be the toroidal compactification of a cusped complex hyperbolic manifold $X={\mathbb{B}}^n\mathrm{/}\mathrm{\Gamma }$ with the boundary divisor $D=\bar{X}\setminus X$. The main goal of this paper is to find the positivity properties of ${\mathrm{\Omega }}_{\bar{X}}^1$ and ${\mathrm{\Omega }}_{\bar{X}}^1(\log (D))$ depending intrinsically on $X$. We prove that ${\mathrm{\Omega }}_{\bar{X}}^1(\log (D))⟨-rD⟩$ is ample for all sufficiently small rational numbers $r>0$, and ${\mathrm{\Omega }}_{\bar{X}}^1(\log (D))$ is ample modulo $D\mathrm{.}$ Further, we conclude that if the cusps of $X$ have uniform depth greater than $4\pi$, then ${\mathrm{\Omega }}_{\bar{X}}^1$ is semi-ample and is ample modulo $D$, all subvarieties of $X$ are of general type, and every smooth subvariety $V\subset \bar{X}$ intersecting $\bar{X}$ has ample $K_V$. Finally, we show that the minimum volume of subvarieties of $\bar{X}$ intersecting both $X$ and $D$ tends to infinity in towers of normal covering of $X\mathrm{.}$