On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains

The Toledo invariant, and Seshadri constants of fake projective planes

“…It would be interesting to compare the estimates given by Hwang and To in [HT99] for (K X ), with the exact result given in Corollary 3.3. This comparison would require the computation of the injectivity radii of fake projective planes, which seems an interesting problem on its own.…”

“…In recent years, there has been a growing interest in the study of Seshadri constants on locally symmetric spaces via geometric and analytic techniques. For example, Hwang and To in [HT99] were able to estimate from below the Seshadri constants, at any given point, of the canonical line bundle of a compact complex hyperbolic space, in terms of the radius of largest geodesic ball centered at that point with respect to the locally symmetric Bergman metric. For many more results in this direction, we refer to [Bauer et al09] and the bibliography therein.…”

Hyperbolicity and specialness of symmetric powers

“…Example 4. -Let Y 0 = Γ\ Ω be a smooth compact quotient, and let Y 1 → Y 0 be the étale cover provided by [HT00]. Let m ∈ N * , and let q be an integer such that q > 4m(m − 1)/γ.…”

On Seshadri constants of varieties with large fundamental group

“…This approach has been successfully explored mainly when the underlying variety X admits Kähler metrics of non-positive sectional curvature. For example, if (X, ω B ) is a n-dimensional complex hyperbolic manifold equipped with the standard locally symmetric metric Bergman metric whose holomorphic sectional curvature is normalized to be −1, J.-M. Hwang and W.-K. To in [HT99] were able to prove that (K X ; x) ≥ (n + 1) sinh 2 (i x /2) where i x is the injectivity radius of ω B at x. Similarly, if (X, ω) is a n-dimensional compact Kähler manifold with non-positive sectional curvature, say −a 2 ≤ R ω ≤ 0, and L is an ample line bundle on X, then S.-K. Yeung in [Yeu00] shows that (L; x) ≥ δ(i x , a, L; n) where δ is an explicitly computable function of the injectivity radius i x , the curvature bounds, the curvature of L and the dimension.…”