Volumes of complex analytic subvarieties of Hermitian symmetric spaces

Abstract

We give lower bounds of volumes of k -dimensional complex analytic subvarieties of certain naturally defined domains in n -dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n , the lower bounds are sharp in the sense that these bounds are attained by k -dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k -dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.

Bi-algebraic geometry and the André-Oort conjecture

“…We also have at our disposal a lower bound for the volume of complex-analytic subvariety of X + due to Hwang and To [HwTo02]. Let us denote by Vol C the area on C for the restriction of the metric g X to C. For a positive real number R we denote by B(x 0 , R) the geodesic ball of X + of center x 0 and radius R.…”

The Kodaira dimension of complex hyperbolic manifolds with cusps

“…Proposition 3.3 is analogous to the multiplicity bound proven by Hwang-To [HT02] for an interior point x of a quotient of a bounded symmetric domain. They show for a k-dimensional subvariety that vol(Y ∩ B(x, r)) ≥ vol(D(r)) k · mult x Y where B(x, r) is an isometrically embedded hyperbolic ball around x of radius r and D(r) is the volume in B(x, r) of a complex geodesic through x.…”

“…In § 3 we prove the volume bounds on the multiplicity of curves along the boundary, and use it to conclude Theorem A. These bounds are the boundary analogs of those proven by Hwang and To [HT02] for interior points of locally symmetric varieties. We provide an independent algebraic proof of Theorem C in § 4.…”

The Ax–Schanuel conjecture for variations of Hodge structures

“…Second, the proof of Theorem 1.1 requires a volume bound on Griffiths transverse 1 subvarieties X ⊂ D analogous to those proven by Hwang-To for hermitian symmetric domains [HT02]. We prove this in §2 and the result is as follows:…”