Boundary smoothness in the Higgs-bundle characterization of quasi-projective manifolds uniformized by the unit ball (the exceptional simple normal crossing case of Deng–Cadorel's Theorem 4.7)

Simpson (1988) characterized compact projective manifolds uniformized by Hermitian symmetric spaces via slope polystability of the tautological Hodge Higgs bundle $(\Omega_X^1\oplus\mathcal O_X,\theta)$ together with equality in the Bogomolov–Gieseker (Miyaoka–Yau) inequality. Deng and Cadorel (2022) extended this to non-compact ball quotients: if $X$ is a smooth projective $n$-fold, $D$ a smooth divisor, and the log Hodge bundle $(\Omega_X^1(\log D)\oplus\mathcal O_X,\theta)$ is slope-polystable with equality in the logarithmic Miyaoka–Yau inequality, then $X-D\simeq\mathbb B^n/\Gamma$ is a ball quotient and $X$ is its toroidal compactification. When $D$ is only simple normal crossing (snc) and not smooth, their Theorem 4.7(ii) shows that the universal cover of $X-D$ is not the ball, although the period map is still locally biholomorphic; the authors expect (Remark 4.9) that this exceptional case cannot occur, and prove this for surfaces. The open problem is to settle this for dimension $n\ge 3$: prove that slope polystability plus the Chern-class equality forces an snc boundary divisor to be smooth — thereby extending the characterization theorem to snc boundaries — or construct an explicit example of the exceptional case, which would be a quasi-projective manifold whose universal cover admits a locally biholomorphic map onto the ball that is not a biholomorphism.

Background

A classical theme in complex geometry links Chern-class inequalities to uniformization. On a smooth complex projective variety $X$ of dimension $n$ with ample canonical bundle, Yau's solution of the Calabi conjecture produces a Kähler–Einstein metric and implies the Miyaoka–Yau inequality $c_1(\Omega_X^1)^2\cdot c_1(L)^{n-2}\le\frac{2(n+1)}{n}c_2(\Omega_X^1)\cdot c_1(L)^{n-2}$ for every ample line bundle $L$, with equality if and only if $X$ is a quotient of the complex unit ball $\mathbb B^n$ by a torsion-free lattice $\Gamma\subset PU(n,1)$ (a lattice is a discrete subgroup of finite covolume; torsion-free means it acts without fixed points); the ball arises here as the Hermitian symmetric space $PU(n,1)/U(n)$. Cheng–Yau, Tian–Yau, Tsuji, and Kobayashi extended such differential-geometric characterizations to open and orbifold settings, but their statements assume metric positivity (for example $K_X+D$ ample modulo the boundary divisor $D$) rather than purely algebro-geometric data.

Simpson (1988) gave a different, algebro-geometric characterization in the language of Higgs bundles. A Higgs bundle on a complex manifold is a holomorphic vector bundle $E$ together with an $\mathcal O_X$-linear field $\theta:E\to E\otimes\Omega_X^1$ satisfying $\theta\wedge\theta=0$; it is slope-(poly)stable with respect to an ample line bundle $L$ if every proper $\theta$-invariant subsheaf has (strictly) smaller normalized first Chern class, polystable meaning a direct sum of stable summands of equal slope. The Hitchin–Simpson (Kobayashi–Hitchin) correspondence equates slope-polystability of degree-zero Higgs bundles with the existence of a harmonic metric; a graded Higgs bundle (a system of Hodge bundles, i.e. $E=\bigoplus_{p+q=w}E^{p,q}$ with $\theta:E^{p,q}\to E^{p-1,q+1}\otimes\Omega_X^1$) carrying such a metric is equivalent to a complex variation of Hodge structure, whose period map takes values in a quotient $G/K$ that, for the tautological system $(\Omega_X^1\oplus\mathcal O_X,\theta)$ with $\theta(a,b)=(0,a)$, is the ball $PU(n,1)/U(n)\simeq\mathbb B^n$. Simpson proved that, for $X$ projective, this system is polystable with $\big(2(n+1)c_2(\Omega_X^1)-nc_1(\Omega_X^1)^2\big)\cdot c_1(L)^{n-2}=0$ exactly when $X$ is uniformized by a Hermitian symmetric space, giving the ball case by equality in the Miyaoka–Yau inequality.

For quasi-projective manifolds $X-D$, where $D$ is a simple normal crossing divisor (snc: locally a union of coordinate hyperplanes), the analogous objects are log Higgs bundles (Higgs fields with at most logarithmic poles along $D$) and systems of log Hodge bundles; the analytic backbone is Mochizuki's extension of the correspondence to this setting: a slope-polystable log Higgs bundle with vanishing characteristic numbers carries a harmonic metric of at most logarithmic growth, adapted to the log structure (a tame harmonic bundle). On the geometric side, a non-compact ball quotient $\mathbb B^n/\Gamma$ by a torsion-free lattice $\Gamma\subset PU(n,1)$ admits a canonical projective toroidal compactification whose boundary is a disjoint union of quotients of abelian varieties by finite groups acting freely (Mumford, Ash–Rapoport–Tai, Mok). Deng and Cadorel combined these inputs into the non-compact analogue of Simpson's theorem: if $X$ is a smooth projective $n$-fold, $D$ a smooth divisor, the log Hodge bundle $(\Omega_X^1(\log D)\oplus\mathcal O_X,\theta)$ is slope-polystable with respect to an ample $L$, and

$\Big(2(n+1)\,c_2\big(\Omega_X^1(\log D)\big)-n\,c_1\big(\Omega_X^1(\log D)\big)^2\Big)\cdot c_1(L)^{n-2}=0,$

then $X-D\simeq\mathbb B^n/\Gamma$ for a torsion-free lattice $\Gamma$, and $X$ is the toroidal compactification with $K_X+D$ big, nef and ample over $X-D$ (Theorem A = Theorem 4.7(i) of their paper); here nef, big and ample over $X-D$ are the standard positivity conditions on the line bundle $K_X+D$, namely nef means non-negative degree on every curve, big means maximal growth of the space of holomorphic sections, and ample over $X-D$ means positive exactly on the curves not contained in $D$. The smoothness of $D$ is essential to the argument: Simpson's completeness step for the period metric works only across a smooth boundary. If $D$ is snc but not smooth, their Theorem 4.7(ii) proves that the period map $\widetilde{X-D}\to\mathbb B^n$ from the universal cover is locally biholomorphic yet the universal cover of $X-D$ is not biholomorphic to $\mathbb B^n$; in Remark 4.9 the authors state that they expect this exceptional situation cannot occur at all — they verify this in dimension two by combining their theorem with Kobayashi's Miyaoka–Yau uniformization of open orbifold surfaces, but not in higher dimensions. Naive attempts to build the exceptional case by blowing up a smooth toroidal compactification fail: a single blow-up changes the relevant Chern combination by a positive amount (on surfaces $3c_2-c_1^2=1$ after one blow-up), destroying the required equality. The question is thus the exact boundary regularity forced by stability plus the Chern equality: whether the equality case of this log-geometric Simpson criterion detects precisely the toroidal ball quotients with smooth boundary, as the authors conjecture, or whether there exists a genuinely new class of quasi-projective manifolds whose universal covers admit non-covering local biholomorphisms onto the ball.

Problem Statement

Let $X$ be a smooth complex projective variety of dimension $n\ge 3$, let $D$ be a simple normal crossing divisor on $X$ which is not smooth, and let $L$ be an ample line bundle on $X$. Consider the system of log Hodge bundles

$E=\Omega_X^1(\log D)\oplus\mathcal O_X,\qquad \theta(a,b)=(0,a),$

i.e. the Higgs field is the tautological morphism $\Omega_X^1(\log D)\to\mathcal O_X\otimes\Omega_X^1(\log D)$. Assume that $(E,\theta)$ is slope-polystable with respect to $L$ and that equality holds in the logarithmic Miyaoka–Yau (Bogomolov–Gieseker) inequality:

$\Big(2(n+1)\,c_2\big(\Omega_X^1(\log D)\big)-n\,c_1\big(\Omega_X^1(\log D)\big)^2\Big)\cdot c_1(L)^{n-2}=0.$

Can such a triple $(X,D,L)$ exist?

If no such triple exists, then whenever $(\Omega_X^1(\log D)\oplus\mathcal O_X,\theta)$ is slope-polystable with respect to an ample line bundle and the above equality holds, the snc divisor $D$ is necessarily smooth. Combined with Deng–Cadorel's Theorem A this removes the smoothness hypothesis from the characterization of non-compact ball quotients: $X-D$ is then a ball quotient $\mathbb B^n/\Gamma$ with $X$ its toroidal compactification and $K_X+D$ big, nef and ample over $X-D$.

If such a triple exists, then by Deng–Cadorel's Theorem 4.7(ii) it carries a locally biholomorphic, non-bijective period map $\widetilde{X-D}\to\mathbb B^n$, exhibiting a quasi-projective manifold whose universal cover is not the ball although equality holds in the Chern-class inequality.

References

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