The existence/nonexistence boundary for complete Kähler–Einstein metrics on complements of exceptional divisors and higher-codimension subvarieties

Gao–Yau–Zhou (Math. Ann. 2017) proved that the complement $M\setminus N$ of a subvariety $N$ of codimension at least two in a compact Kähler manifold $M$ admits no complete Kähler–Einstein metric, and they extended this nonexistence to complements $M\setminus E$ of exceptional divisors whose blow-down $X$ has only canonical or terminal singularities and a nonempty codimension-two singular stratum, and to klt pairs via log resolutions. The open problem is the completion of this nonexistence picture along the three directions the authors themselves flagged: whether the conclusion survives when the codimension-two singular stratum is empty (isolated or higher-codimension singular loci) in the generality of arbitrary birational morphisms of normal varieties; which singularity types and exceptional-locus configurations permit or preclude complete Kähler–Einstein metrics beyond the treated Du Val and small-quotient local models; and where the sharp positivity threshold between the Tian–Yau–Wu existence theorems and these nonexistence results lies. Later work has mapped adjacent territory — complete negative Kähler–Einstein metrics near isolated log-canonical (non-log-terminal) singularities (Datar–Fu–Song; Fang–Fu), constant-scalar-curvature Kähler and conformally Kähler analogues where the obstruction fails (Y. Feng; Z. Feng; Fu–Yau–Zhou), and merely 'almost-complete' metrics in the nef-and-big regime (Dang–Vu) — but the codimension-two-free nonexistence statement, the classification of obstructing singularities, and the ampleness threshold remain open.

Background

A Kähler–Einstein (KE) metric is a Kähler metric $\omega$ on a complex manifold whose Ricci tensor satisfies $\mathrm{Ric}(\omega)=\lambda\omega$ for a constant $\lambda$; on a noncompact manifold only $\lambda\le 0$ can occur, since $\lambda>0$ would force compactness by the Bonnet–Myers theorem. 'Complete' means geodesically complete as a Riemannian metric. After Yau's resolution of the Calabi conjecture settled the compact case, the noncompact theory developed along two lines. Cheng–Yau and Mok–Yau proved that a bounded pseudoconvex domain carries a complete KE metric of negative scalar curvature precisely when it is a domain of holomorphy (a domain on which some holomorphic function cannot be extended across any boundary point); and Tian–Yau constructed complete Ricci-flat Kähler metrics on quasi-projective manifolds — a complex manifold biholomorphic to $\bar M\setminus D$ with $\bar M$ compact Kähler and $D$ a divisor — when the removed divisor is, roughly, a positive multiple of the anticanonical divisor, while the negative-Ricci analogue was later extended by Damin Wu to general quasi-projective manifolds under ampleness of the adjoint class $K_{\bar M}+D$ (a line bundle or $\mathbb{Q}$-divisor class is ample if it admits a Hermitian metric with positive curvature form).

Yau raised the complementary question, recorded as Problem 1.1 in the source paper: find canonical metrics on $M\setminus N$ when $N$ is a subvariety of codimension at least two — a regime where the Tian–Yau machine does not apply. Gao–Yau–Zhou answered the Kähler–Einstein version negatively and completely at this level: (Theorem 1.3) $M\setminus N$ admits no complete Kähler metric with $-\lambda\le\mathrm{Ric}\le 0$ for any constant $\lambda\ge 0$. The proof is local: the volume density $\det g_{i\bar j}$ is plurisubharmonic (because $-\mathrm{Ric}=\sqrt{-1}\partial\bar\partial\log\det g_{i\bar j}\ge 0$ up to sign conventions), a maximum-principle argument bounds it near $N$, while a Chern–Lu inequality (a classical curvature-comparison estimate relating the volume densities of two Kähler metrics) combined with Yau's Schwarz lemma type estimate forces a lower bound on it, and the two bounds contradict completeness along a path approaching $N$.

The subtler question concerns exceptional divisors. Let $f:M\to X$ be a birational morphism (a holomorphic map between complex varieties that is an isomorphism over dense open sets) of normal varieties, and let $E$ be the sum of its exceptional divisors — the divisors of $M$ that $f$ maps to sets of smaller dimension. Then $M\setminus E$ is a manifold minus a codimension-one set, which Theorem 1.3 does not address, and any obstruction must come from the singularities created by the contraction. Recall the birational classifications: for $X$ normal with $\mathbb{Q}$-Cartier canonical divisor $K_X$, the discrepancies $a_F\in\mathbb{Q}$ are defined by $K_Y=f^{*}K_X+\sum a_F F$ over all birational models $f:Y\to X$ and all exceptional divisors $F$; $X$ has canonical (resp. terminal) singularities if $a_F\ge 0$ (resp. $a_F>0$) for all such $F$, and a pair $(X,D)$ is Kawamata log terminal (klt) if all discrepancies of the pair exceed $-1$ and $\lfloor D\rfloor\le 0$, and log-canonical if all discrepancies are at least $-1$ — the largest of these classes; log-canonical pairs that are not klt are called strictly log-canonical. Du Val (or A–D–E) singularities are the canonical surface singularities, equivalently quotients $\mathbb{C}^{2}/G$ by finite $G\subset SL(2,\mathbb{C})$; compound-Du Val singularities are the Gorenstein (Cartier canonical class) threefold germs whose general hyperplane section is a Du Val surface singularity — the typical isolated canonical singularities of threefolds; a small quotient singularity is $\mathbb{C}^{n}/G$ for a finite $G\subset GL(n,\mathbb{C})$ acting freely in codimension one. A log resolution is a resolution $f:M\to X$ for which the preimage of the singular locus (with the boundary) is a divisor with simple normal crossing support.

Gao–Yau–Zhou then prove (Theorem 1.5) that if $X$ has only canonical or terminal singularities and its singular locus contains a codimension-two stratum, then $M\setminus E$ carries no complete KE metric; in dimension two this specializes (Corollary 1.6) to surfaces with A–D–E singularities. The mechanism: by a theorem of Reid, canonical singularities are analytically products of Du Val singularities with a smooth factor near a general point of any codimension-two stratum, and Du Val points are quotient singularities, near which a complete KE metric would restrict to one on punctured balls, contradicting the local argument of Theorem 1.3. For pairs, (Theorem 1.7) if $(X,D)$ is klt and $f:M\to X$ is a log resolution with a simple normal crossing condition, then $M\setminus E$ again carries no complete KE metric; here the reduction uses that klt singularities have quotient singularities outside a subset of codimension at least three. The authors state that it is difficult to relax the assumption on the codimension of the singular locus in Theorem 1.5, and they observe that in these birational configurations the adjoint bundle $K_M+E$ is generally not ample — exhibiting the ampleness hypothesis of the Tian–Yau–Wu existence theorems as necessary.

The surrounding literature has since developed on three fronts that border, but do not close, these questions. On the log-canonical side, Fang–Fu and Datar–Fu–Song constructed complete KE metrics of negative scalar curvature on punctured neighborhoods of isolated log-canonical (non-log-terminal) singularities — cones over abelian varieties (via uniformization by the complex ball) and Calabi–Yau manifolds (via the Calabi ansatz, Calabi's explicit one-variable reduction for constructing Kähler–Einstein metrics on disk bundles), and, in complex dimension two, general such singularities, whose complete negative KE metrics are asymptotic to hyperbolic models of Kobayashi–Nakamura — so the existence/nonexistence boundary runs somewhere inside the log-canonical class, at the klt interface. On the equation side, the obstruction is specific to the Einstein condition: Fu–Yau–Zhou produced complete constant-scalar-curvature Kähler (cscK) metrics with Poincaré–Mok–Yau asymptotics (asymptotic along the end to the Poincaré-type cusp model of Mok–Yau) on local models of conifold transitions (threefold degenerations in which a rational curve is contracted to an ordinary double point), Yueqing Feng glued complete cscK metrics of Poincaré type on complements of exceptional divisors of point blow-ups, and Zhiming Feng constructed complete conformally Kähler Einstein metrics on complements of codimension $\ge 2$ subvarieties. On the positivity side, the existence theory has been pushed only partway toward the ampleness boundary: Chengjie Yu constructed complete Ricci-flat metrics for certain deleted divisors without normal crossings, Wu–Yau established existence under covering and negative holomorphic-sectional-curvature hypotheses, and Dang–Vu proved that when $K_{\bar M}+D$ is big and nef — big meaning the class has positive volume, nef meaning it has nonnegative degree on every curve, and neither implying the positively curved metric that ampleness provides — the natural singular KE metric on $\bar M\setminus D$ is merely 'almost-complete' in the sense of Tian–Yau, a weak completeness property strictly weaker than geodesic completeness. A parallel line studies the singular KE metrics on singular varieties (in the sense of Eyssidieux–Guedj–Zeriahi): near isolated log-terminal singularities these metrics are conical under stable-degeneration hypotheses (Junsheng Zhang, following Hein–Sun) and their metric completion adds the singular point — i.e. they are incomplete there — but this concerns one distinguished singular metric, not the nonexistence of all complete KE metrics.

Problem Statement

The problem is to complete the boundary between existence and nonexistence of complete Kähler–Einstein metrics on noncompact Kähler manifolds obtained from compact ones by deleting exceptional or higher-codimension loci. The single objective — determining exactly which configurations admit such metrics — is made concrete by three linked components fixed by Gao–Yau–Zhou's own stated limitations; these components are independent of one another, in that settling one does not by itself settle the others.

(1) Codimension-two-free nonexistence. Let $f:M\to X$ be a birational morphism of normal complex varieties and $E$ the sum of its exceptional divisors, where $X$ has only canonical or terminal singularities (terminal being the stronger condition) and the singular locus of $X$ has no codimension-two stratum — it is empty, or every singular stratum has codimension at least three, the model case being isolated singularities such as isolated compound-Du Val points on threefolds. Determine whether $M\setminus E$ admits a complete Kähler–Einstein metric ($\mathrm{Ric}=\lambda\omega$, $\lambda\le 0$): either prove nonexistence in the generality of Gao–Yau–Zhou's Theorem 1.5 — arbitrary birational $f$ not required to be a log resolution, with $M\setminus E$ a complex manifold (equivalently, $M$ smooth away from $E$) — or exhibit a counterexample. The klt/log-resolution form of the statement (Theorem 1.7) is part of the source; completing and independently establishing the higher-codimension case of that theorem, whose published proof reduces tersely to codimension-two strata, is within scope.

(2) Classification of obstructing configurations. Beyond the two local models treated in the source — Du Val surface points and small quotient singularities — characterize which singularity types of $X$ and which configurations of the exceptional locus permit or preclude a complete Kähler–Einstein metric on $M\setminus E$; since the obstruction is local, obstructing configurations can be studied on punctured neighborhoods of the singularities of $X$. In particular: delimit where inside the log-canonical class existence begins, taking the known existence results for isolated log-canonical cone and surface singularities (cones over abelian varieties or Calabi–Yau manifolds; dimension-two log-canonical germs) as the current frontier, and extend the classification to the klt-but-not-canonical and strictly log-canonical germs and to exceptional-locus configurations beyond a single contracted divisor.

(3) Positivity threshold. Clarify the relationship between the ampleness hypotheses of the Tian–Yau–Wu existence theorems and the necessary conditions for nonexistence established by Gao–Yau–Zhou: determine the sharp positivity threshold — ample versus nef and big versus trivial — of the adjoint class on the relevant compactification that separates existence of complete Kähler–Einstein metrics on the complement from nonexistence, in one picture covering both the divisor-complement setting (where ampleness gives complete metrics and big-and-nef currently gives only almost-complete ones) and the birational exceptional-divisor setting (where the adjoint class is not ample and the metrics do not exist).

References

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