Positivity in K\"ahler-Einstein theory

Tian initiated the study of incomplete K\"ahler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2\pi(1-\alpha)$ for $\alpha\in (0, 1)$. In this paper we study how the existence of such K\"ahler-Einstein metrics depends on $\alpha$. We show that in the negative scalar curvature case, if such K\"ahler-Einstein metrics exist for all small cone-angles then they exist for every $\alpha\in(\frac{n+1}{n+2}, 1)$, where $n$ is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge K\"ahler-Einstein metrics with cone angle close to $2\pi$. Again if these metrics exist for all cone-angles close to $2\pi$, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.Comment: Some changes according the comments of the referee and references update

Effective results for complex hyperbolic manifolds

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of cusps.Comment: Some changes according the comments of the referee and references update

Exceptional collections and the bicanonical map of Keum's fake projective planes

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.Comment: Section 3 is expanded. Added acknowledgements. To appear in Commun. Contemp. Mat

On Seshadri constants of varieties with large fundamental group

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $\pi^{alg}_{1}(X)$ is large, we show that the Seshadri constant $\epsilon(p^{*}L)$ can be made arbitrarily large by passing to a finite \'etale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when $\pi_{1}(X)$ is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle $L$ on $X$ and a positive number $N>0$, we show that there exists a finite \'etale cover $p: X'\rightarrow X$ such that the Seshadri constant $\epsilon(p^{*}L; x)\geq N$ for any $x\notin p^{*}\textbf{B}_{+}(L)=\textbf{B}_{+}(p^{*}L)$, where $\textbf{B}_{+}(L)$ is the augmented base locus of $L$.Comment: Final versio

Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + \alpha D$ for $\alpha \in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.Comment: To appear in Trans. Amer. Math. Soc., 21 pages, 3 figures, and 1 tabl

Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}\pi^{2}$, i.e., they attain all possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of the two families have all $2$-cusps, so that we can saturate the entire volume spectrum with $2$-cusped manifolds. Finally, we show that the associated neat lattices have infinite abelianization and finitely generated commutator subgroup. These appear to be the first known nonuniform lattices in $\mathrm{PU}(2,1)$, and the first infinite tower, with this property.Comment: To appear in Ann. Inst. Fourie

Multiple realizations of varieties as ball quotient compactifications

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.Comment: Minor changes and references updated. To appear in Michigan Math.

Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.Comment: Some changes and typos corrected following referees' reports. To appear in Comm. Anal. Geom., 31 page

Extended Graph 4-Manifolds, and Einstein Metrics

We show that extended graph 4-manifolds (as defined by Frigerio-Lafont-Sisto in [FLS15]) do not support Einstein metrics.Comment: Typos in the bibliography corrected, 12 page

On the Boundary Injectivity Radius of Buser-Colbois-Dodziuk-Margulis Tubes

We give a lower bound on the boundary injectivity radius of the Margulis tubes with smooth boundary constructed by Buser, Colbois, and Dodziuk. This estimate depends on the dimension and a curvature bound only.Comment: Some changes and typos corrected following referees' reports. To appear in Ann. Global Anal. Geom., 12 page