Generalized Bergman metrics, tautological line bundle negativity over the moduli of Riemann surfaces, and the general existence problem for the Hull–Strominger system (open problems from Yau's 2010 survey 'Metrics on complex manifolds')

Open problems posed in the closing section of Shing-Tung Yau's 2010 survey 'Metrics on complex manifolds'. The survey introduces, for each integer $m\ge 1$ and each compact complex manifold, a generalized Bergman metric: a complex Finsler metric defined variationally from holomorphic sections of the $m$-th power $K^m$ of the canonical bundle, normalized by $\int|s|^{2/m}\le 1$. Three linked questions from that construction remain open. (A) Develop the theory of these metrics: their structure on compact Riemann surfaces, the relationship between the metrics for different $m$, whether the Kähler–Einstein metric is recovered from their asymptotics as $m\to\infty$, and their behavior near the singularities of Kähler–Einstein metrics on manifolds of general type. (B) Over the moduli space of Riemann surfaces, determine whether the Chern form of the tautological line bundle over the projectivized bundle of $m$-pluricanonical sections is negative, and identify the canonical forms obtained by pushing it forward to moduli. (C) Establish the general existence theory for the Hull–Strominger system on compact complex threefolds with trivial canonical bundle — Yau's conjecture that a balanced class, vanishing degree, equality of second Chern characters, and slope-polystability suffice for solvability — beyond the perturbative large-scaling regime of Li–Yau, beyond the Fu–Yau class of two-torus bundles over K3 surfaces, and without the smallness hypotheses on the Fu–Yau constant $A$.

Background

A compact complex manifold $M$ carries intrinsic metrics that depend only on its complex structure. The most classical is the Bergman metric. Let $K$ denote the canonical line bundle of $M$, the line bundle of holomorphic complex $n$-forms, where $n=\dim_{\mathbb{C}}M$. The finitely many holomorphic sections of $K$ that are square-integrable against a fixed volume form constitute a finite-dimensional Hilbert space; the Bergman kernel $\sum_j|s_j|^2$ of an orthonormal basis $\{s_j\}$ defines a Hermitian metric through the second-order complex differential operator $i\partial\bar\partial$, as $i\partial\bar\partial\log\sum_j|s_j|^2$. The Carathéodory metric instead measures a tangent vector by the supremum of derivatives of holomorphic maps from $M$ into the unit disk, and a classical Schwarz-lemma argument shows that the Bergman metric dominates it. When $K$ is ample — for a compact Riemann surface this means genus at least $2$, and in general it places $M$ among the manifolds of general type, those whose birational geometry is governed by the canonical bundle — a fundamental existence theorem of the 1970s produces a unique Kähler–Einstein metric: a Kähler metric, meaning one whose associated $(1,1)$-form is closed, whose Ricci curvature equals minus its own Kähler form. On singular models of general type this metric degenerates, and its singularities encode the geometry of the model.

Yau's survey proposes, for each integer $m\ge 1$, a generalized Bergman metric on $M$. Writing a holomorphic section of the $m$-th pluricanonical power $K^m$ locally as $s=f\,(dz^1\wedge\cdots\wedge dz^n)^m$, the survey imposes the non-Hilbert constraint $\int_M|s|^{2/m}\le 1$; the quantity $(\int_M|s|^{2/m})^{m/2}$ is a norm on the space of $m$-pluricanonical sections when $m\le 2$ and a quasinorm when $m\ge 3$, since the exponent $2/m$ then drops below $1$, and is induced by an inner product only when $m=1$. At a point $z_0\in M$, let $f_0$ maximize $|f(z_0)|^2$ under this constraint; the generalized Bergman metric is the complex Finsler norm on tangent vectors — a norm on each tangent line, allowed to depend on the point — defined by

$$\|T\|^2_{z_0}\;=\;\max\Big\{\,\frac{|T(f)(z_0)|^2}{|f_0(z_0)|^2}\;:\;\int_M|f|^{2/m}\le 1,\ f(z_0)=0\,\Big\},$$

together with the distributional $(1,1)$-form $i\partial\bar\partial\log|f_0|$ as a candidate Hermitian counterpart. The survey notes that the classical Schwarz-type argument that bounds the Bergman metric below by the Carathéodory metric extends to these generalized metrics as well, and that, because pluricanonical sections are unchanged by birational modification, the metrics depend only on the birational geometry of $M$. The survey asks for a theory of these metrics — their structure, especially on Riemann surfaces; the relationship among the metrics for different exponents $m$; the recovery of the Kähler–Einstein metric as $m\to\infty$; and their asymptotics near Kähler–Einstein singularities — while establishing the foundational properties (existence of the extremal sections for $m>1$, the character of the distributional form, and the comparison bound) only in outline.

The second question lives on the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the space that parametrizes all such curves. Over $\mathcal{M}_g$ sits the Hodge bundle, the vector bundle whose fiber over a curve $C$ is the space $H^0(C,K_C)$ of holomorphic one-forms; more generally, the spaces $H^0(C,K_C^m)$ of $m$-pluricanonical sections assemble into a bundle $\mathbb{E}_m$ over $\mathcal{M}_g$, and over its projectivization $\mathbb{P}(\mathbb{E}_m)$ — the manifold of one-dimensional subspaces of the fibers — sits the tautological line bundle, whose fiber over a line $\ell\subset H^0(C,K_C^m)$ is $\ell$ itself. For $m=1$, with the $L^2$ norm on the fibers, curvature computations for this structure recover the Weil–Petersson form on moduli, the symplectic form governing the hyperbolic geometry of the varying curves; Liu–Sun–Yau studied the metric comparisons on this space in depth, proving the equivalence of the classical Bergman, Carathéodory, and Kähler–Einstein metrics there and constructing further canonical metrics, including the perturbed Ricci metric. The survey asks for the $m$-pluricanonical analogue: equip the fibers of $\mathbb{E}_m$ with the norm of the previous paragraph and determine whether the Chern form — the curvature form — of the tautological line bundle over $\mathbb{P}(\mathbb{E}_m)$ is negative, and which canonical forms on $\mathcal{M}_g$ arise by pushing this Chern form forward to the moduli space.

The third question concerns the Hull–Strominger system, a system of partial differential equations introduced by Strominger in 1986 in the study of supersymmetric compactifications of heterotic string theory with nonzero torsion. On a compact complex threefold $X$ carrying a nowhere-vanishing holomorphic volume form $\Omega$ (equivalently, trivial canonical bundle) and a holomorphic vector bundle $V$ over $X$, the system seeks a Hermitian metric $g$ on $X$ with fundamental $(1,1)$-form $\omega$, a Hermitian metric $h$ on $V$, and a unitary connection $\nabla$ on the tangent bundle of $X$ — a first-order differentiation rule compatible with both the Hermitian metric and the complex structure, the Chern connection being the canonical such choice determined by $g$ — satisfying, in one standard convention,

$$F_h\wedge\omega^2=0,\qquad d\big(\|\Omega\|_\omega\,\omega^2\big)=0,\qquad \sqrt{-1}\,\partial\bar\partial\,\omega=\alpha'\big(\mathrm{tr}\,R_\nabla\wedge R_\nabla-\mathrm{tr}\,F_h\wedge F_h\big),$$

where $F_h$ and $R_\nabla$ are the curvature forms of $h$ and $\nabla$, $\|\Omega\|_\omega$ is the pointwise norm of $\Omega$ against $\omega$, and $\alpha'>0$ is a physical parameter; the placement of the factors $\|\Omega\|_\omega$ and $\alpha'$ varies across the literature. The first equation is the Hermitian–Yang–Mills condition on $h$ (vanishing of the trace component of its curvature against $\omega$); the second says that $\omega$ is conformally balanced — a positive function multiple of $\omega^2$ is closed, the natural non-Kähler generalization of the Kähler condition, and when the multiple can be taken constant $\omega$ is called balanced; the third is the anomaly equation, a quadratic curvature identity inherited from the Bianchi identity of the underlying physical theory. Solvability forces algebraic-topological necessary conditions: the class $b=[\|\Omega\|_\omega\,\omega^2]$ must be a closed balanced class, the degree $\deg_b(V)=c_1(V)\cdot b$ must vanish, the second Chern characters must agree, $\mathrm{ch}_2(V)=\mathrm{ch}_2(X)$, in Bott–Chern cohomology (the cohomology of forms modulo $i\partial\bar\partial$-exact ones, the correct receptacle on manifolds where the Kähler-type $\partial\bar\partial$-lemma can fail), and $V$ must be slope-polystable with respect to $b$: writing the slope of a bundle as its degree per unit rank, $V$ must be a direct sum of stable pieces — pieces admitting no coherent subsheaf of strictly larger slope — each of zero slope. Yau conjectured — originally in a 2006 talk, and in the survey with $\nabla$ specified as the Chern connection of $g$ — that these necessary conditions are also sufficient: given a balanced class $b$ and a holomorphic bundle $V$ satisfying the degree, second-Chern-character, and polystability conditions, the system should admit a solution. The survey reviews the two existence theorems known at the time, both far from this conjecture. Li–Yau construct solutions by perturbing the reducible solution on a Calabi–Yau threefold carrying the bundle $E=T_X\oplus\mathbb{C}^{\oplus r}$: solutions exist for a sufficiently large scaling parameter $c$ and for deformations of the holomorphic structure whose extension classes are nondegenerate, with the Fermat quintic threefold as an example. Fu–Yau solve the system on compact non-Kähler threefolds that are two-torus bundles over a K3 surface $S$, constructed from two anti-self-dual integral $(1,1)$-forms $\omega_1,\omega_2$ on $S$ (a two-form is anti-self-dual if it pairs to zero with the Kähler form and equals minus its Hodge star), under the topological constraint that the survey labels (6.2),

$$2\big(24-c_2(E)\big)=-\Big(Q\big(\tfrac{\omega_1}{2\pi}\big)+Q\big(\tfrac{\omega_2}{2\pi}\big)\Big),$$

where $Q$ is the intersection pairing on $H^2(S,\mathbb{Z})$ and $c_2(E)$ is the second Chern number of the bundle; their continuity-method proof reduces the system to a Monge–Ampère-type equation on the base and requires the constant $A=(\int_S e^{-4u}\,\omega_S^2/2!)^{1/4}$, where $u$ is the unknown of that reduced equation, to be sufficiently small. The survey's closing questions ask precisely for what lies beyond these two results.

Problem Statement

These are the open problems posed in the closing section of Yau's 2010 survey 'Metrics on complex manifolds'. Their common objective is to extend the classical dictionary between canonical metrics and canonical-bundle geometry beyond its original Hilbert-space, Kähler, and fixed-class settings. The components are linked by this objective but logically independent.

(A) Generalized Bergman metrics. For a compact complex manifold $M$ with canonical bundle $K$ and an integer $m\ge 1$ such that at every point of $M$ some holomorphic section of $K^m$ does not vanish (the survey's standing assumption, under which the extremal function below is defined), writing a section locally as $s=f\,\sigma^m$ with $\sigma$ a local nonvanishing holomorphic $n$-form and $n=\dim_{\mathbb{C}}M$, normalize by $\int_M|s|^{2/m}\le 1$ and define the generalized Bergman metric by the variational formula of the Background. Develop the theory of these metrics, in particular: (A1) their structure on compact Riemann surfaces, including explicit or effectively computable descriptions; (A2) the relationship between the metrics (equivalently, between the extremal functions) for different exponents $m$; (A3) whether, in what sense (pointwise, $C^k$ for fixed $k$, or Gromov–Hausdorff convergence of metric spaces), and at what rate these metrics converge as $m\to\infty$ to the Kähler–Einstein metric of the Background — the unique one whose Ricci form equals minus its own Kähler form, which exists when $K$ is ample; (A4) their asymptotic behavior near the singularities that the Kähler–Einstein metric develops on manifolds of general type. Establish, as part of the theory, the foundational properties that the survey asserts only in outline: existence of the extremal sections for $m>1$, the character (smoothness, positivity) of the associated distributional form $i\partial\bar\partial\log|f_0|$, and a rigorous comparison with the Carathéodory metric.

(B) Tautological negativity over moduli. Let $\mathcal{M}_g$ be the moduli space of compact Riemann surfaces of genus $g\ge 2$; over it sits the bundle $\mathbb{E}_m$ whose fiber over a curve $C$ is the space $H^0(C,K_C^m)$ of $m$-pluricanonical sections, carrying the fiberwise norm $(\int_C|s|^{2/m})^{m/2}$. Over the projectivization $\mathbb{P}(\mathbb{E}_m)$ sits the tautological line bundle $\mathcal{O}(-1)$. Determine: (B1) whether the Chern form of $\mathcal{O}(-1)$, for the Hermitian structure induced by this norm, is negative or semi-negative; (B2) which canonical forms on $\mathcal{M}_g$ are obtained by pushing this Chern form forward to the moduli space — pushforward along the projection $\mathbb{P}(\mathbb{E}_m)\to\mathcal{M}_g$, that is, integration over its fibers, applied to the Chern form and its powers — and how these forms and their relations behave for different $m$ (for $m=1$ the construction is expected to recover, up to normalization, the Weil–Petersson form).

(C) General existence for the Hull–Strominger system. (C1) Yau's conjecture: let $(X,\Omega)$ be a compact complex threefold with nowhere-vanishing holomorphic volume form $\Omega$, let $b$ be a balanced class on $X$, and let $V$ be a holomorphic vector bundle with $\deg_b(V)=0$ and $\mathrm{ch}_2(V)=\mathrm{ch}_2(X)$ in Bott–Chern cohomology. If $V$ is slope-polystable with respect to $b$, does $(X,\Omega,V)$ admit a solution of the Hull–Strominger system, in the survey's formulation — in which the tangent-bundle connection $\nabla$ in the anomaly equation is the Chern connection of $g$ — or in the formulation in which $\nabla$ is left unspecified, the non-Kähler case being the essential one, and how does the prescribed class $b$ relate to the balanced class of the solution? (C2) Perturbative regime: establish existence of solutions for arbitrarily small scaling parameter $c$, and for holomorphic bundles beyond small nondegenerate deformations of $T_X\oplus\mathbb{C}^{\oplus r}$. (C3) Beyond the Fu–Yau class: construct solutions on compact non-Kähler threefolds other than the two-torus bundles over K3 surfaces subject to the constraint (6.2) of the Background; and remove the smallness hypotheses on the constant $A=(\int_S e^{-4u}\,\omega_S^2/2!)^{1/4}$, equivalently the large-radius or large-scaling conditions in the later reformulations of the problem.

References

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