Nonamenability versus boundedness of the universal cover for general-type projective varieties (the q-Bergman metric hypothesis in Treger's tower of coverings)

The source preprint (R. Treger, arXiv:1107.5713, 2011) studies the universal cover U of a nonsingular projective variety X of general type with large, residually finite fundamental group, through a tower of finite Galois covers converging to U. It defines a q-Bergman metric on U from square-integrable holomorphic q-pluricanonical forms and proves that, if the corresponding evaluation map embeds U ('U has the q-Bergman metric' — a hypothesis the author notes does not follow from the other standing assumptions), then this metric is the limit of the q-Bergman metrics of the finite covers, and consequently U is a bounded domain in C^n for q sufficiently large. The paper's Remark 2 conjectures that this auxiliary analytic hypothesis can be replaced by the purely group-theoretic condition that the fundamental group is nonamenable. This is the higher-dimensional case of a uniformization statement attributed to Kazhdan.

Background

By the classical uniformization theorem, a compact Riemann surface of genus at least $2$ has as universal covering space the Poincaré disk, a bounded domain in $\mathbb{C}$ ; equivalently, its fundamental group is nonamenable (a discrete group is amenable if it admits a finitely additive, left-translation-invariant probability measure on its subsets, equivalently a left-invariant mean; free groups, and surface groups of genus $\geq 2$ , are nonamenable). Whether an analogous statement holds in higher dimensions — must the universal cover of a compact complex manifold of general type with nonamenable fundamental group be a bounded domain in $\mathbb{C}^{n}$ ? — is a long-standing open problem in the higher-dimensional uniformization program, and the source preprint attributes such a statement to Kazhdan.

A main tool is the Bergman kernel. For a domain $D\subset\mathbb{C}^{n}$ , the Hilbert space of square-integrable holomorphic functions has a reproducing kernel, and pulling back the Fubini–Study geometry of the associated projective space through the evaluation map defines the Bergman metric. Kazhdan proved a tower version of this in dimension one: if a tower of finite Galois (unramified) covers of a compact Riemann surface of genus $\geq 2$ converges to the universal cover, then the canonical metrics of the finite covers, pulled back, converge uniformly to the hyperbolic metric of the disk (this is the form in Kazhdan's 1975 paper; the Bergman-kernel version of the statement, for towers of bounded symmetric domains, was communicated by Kazhdan in a letter to McMullen and is reproduced in the appendix of McMullen's paper). Later proofs and generalizations are due to Rhodes, McMullen, Ohsawa, and Baik–Shokrieh–Wu, who extended the theorem to arbitrary infinite Galois covers using Lück-type $L^{2}$ -approximation and von Neumann algebras.

The source preprint (Treger, arXiv:1107.5713v2, unrefereed) works in the following setting: $X\subset\mathbb{P}^{\ell}$ is a nonsingular connected projective $n$ -fold with ample canonical bundle $K_{X}$ (canonical polarization; in particular, of general type — ampleness of $K_{X}$ is a stronger hypothesis than general type), the fundamental group $\pi_{1}(X)$ is large (in Kollár's Shafarevich-map theory: for every positive-dimensional subvariety $Z\subset X$ the image of $\pi_{1}(Z)$ in $\pi_{1}(X)$ is infinite) and residually finite (every nontrivial element survives in some finite quotient, which guarantees enough finite covers to build a tower converging to the universal cover $U$ ), and $X$ has no general elliptic curvilinear sections, i.e., a general curvilinear section through any point of $X$ — a general complete-intersection curve section in the given projective embedding — is not an elliptic curve (a standing hypothesis of the source's framework, spelled out in the sibling preprint arXiv:1008.1745 and belonging to the same Shafarevich-map framework of Kollár's book). Fixing $m$ minimal such that $K_{X}^{m}$ is very ample, Treger defines the $q$ -Bergman metric on $U$ for $q\geq m$ as the metric induced by the evaluation map of the Hilbert space of square-integrable holomorphic $q$ -pluricanonical forms (holomorphic sections of the tensor power $K^{q}$ of the canonical bundle, pulled back to $U$ ), provided this map is an embedding. His Proposition states that, assuming such a metric exists, it is the limit of the pullbacks of the $q$ -Bergman metrics of the finite covers $X_{i}$ ; his Corollary then concludes that $U$ is a bounded domain in $\mathbb{C}^{n}$ provided $U$ has the $q$ -Bergman metric for some sufficiently large $q$ . The embedding hypothesis is exactly what does not follow from the other assumptions, and the corollary is the content. Remark 2 of the source conjectures that this embedding hypothesis can be replaced by the purely group-theoretic condition that $\pi_{1}(X)$ is nonamenable, and the source envisages a Bergman-type reproducing kernel of the Hilbert space of square-integrable harmonic forms, rather than of holomorphic $q$ -pluricanonical forms, as the relevant analytic object.

The surrounding literature has advanced rapidly but only on the convergence side: Chen and Fu proved stability of Bergman kernels on towers under Kähler-hyperbolicity-type conditions (a Gromov-type requirement that the Kähler form of the cover have a bounded primitive), and Yoo and Yum later proved unconditionally that any tower of Galois coverings of a compact Kähler manifold converging to an infinite Galois cover is Bergman stable, establishing a Kazhdan equality for $L^{2}$ -Hodge numbers (von Neumann dimensions — the dimension theory for Hilbert-space representations of the deck group — of spaces of harmonic forms) along the way; their immersion statement for pluricanonical sections is, however, conditional on the limit cover already admitting one. On the qualitative side, a bounded domain in $\mathbb{C}^{n}$ is Stein (a Stein manifold is holomorphically convex and separable by holomorphic functions, the natural class of non-compact complex manifolds admitting an abundance of holomorphic functions), so the conjecture implies a strong (Stein) form of holomorphic convexity of universal covers — stronger than the Shafarevich conjecture, which is known only for fundamental groups that are (virtually) linear, by work of Eyssidieux and related authors. Since residually finite nonamenable groups need not be linear, no known Shafarevich-type theorem applies, and no counterexample is known either.

Problem Statement

Let $X$ be a nonsingular connected projective variety of dimension $n\geq 2$ over $\mathbb{C}$ with ample canonical bundle $K_{X}$ (the case $n=1$ being settled by classical uniformization), whose fundamental group $\pi_{1}(X)$ is large and residually finite, and suppose $X$ has no general elliptic curvilinear sections (a general curvilinear section through any point of $X$ , i.e., a general complete-intersection curve section in the given projective embedding, is not an elliptic curve); let $U\to X$ be the universal covering, realized as the limit of a tower of finite Galois covers, and let $q\geq m$ where $m$ is the least integer with $K_{X}^{m}$ very ample. In the source preprint, $U$ is provably a bounded domain in $\mathbb{C}^{n}$ only under the auxiliary hypothesis that, for some sufficiently large $q\geq m$ , the evaluation map of the Hilbert space of square-integrable holomorphic $q$ -pluricanonical forms on $U$ embeds $U$ into a projective space (the $q$ -Bergman metric hypothesis), a hypothesis which does not follow from the other standing assumptions. The question, stated as a conjecture in Remark 2 of the source: does the same conclusion hold if this analytic hypothesis is replaced by the purely group-theoretic condition that $\pi_{1}(X)$ is nonamenable? A positive answer proves the higher-dimensional Kazhdan uniformization statement for this class of varieties; a negative answer must exhibit a variety satisfying all the standing hypotheses, with nonamenable fundamental group, whose universal cover is not biholomorphic to any bounded domain in $\mathbb{C}^{n}$ .

References

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