Jost and Yau (Acta Mathematica 170, 1993) introduced Hermitian harmonic maps — solutions of a non-variational elliptic system that extends the harmonic map equation from Kähler to arbitrary compact Hermitian domains — and proved, under the astheno-Kähler condition ∂∂ˉωn−2=0\partial\bar\partial\omega^{n-2}=0 on the domain, a Siu-type rigidity theorem for maps into compact locally Hermitian symmetric spaces of noncompact type and strongly negatively curved compact Kähler manifolds. Among the applications they proved that a connected sum X=M#NX=M\#N of a compact complex surface NN carrying a strongly negatively curved Kähler metric with any compact smooth 4-manifold MM satisfying π1(M)≠0\pi_1(M)\neq 0 admits no complex structure, and that in complex dimension n≥3n\geq 3, when NN is a compact quotient of the complex unit ball, XX admits no meromorphic map onto any compact complex manifold of dimension n−2n-2, so its algebraic dimension is at most n−3n-3, if it carries a complex structure at all. Whether such higher-dimensional connected sums carry any complex structure is the case Jost and Yau 1993 left open; no resolution is known in the literature to date. The real-dimension-4 case is settled in the original paper; the noncompact-domain part of the existence theory has since been substantially developed; the astheno-Kähler characterization question has become a separate active subfield.

Background

A harmonic map f:(M,g)→(N,h)f:(M,g)\to(N,h) between Riemannian manifolds is a critical point of the Dirichlet energy ∫M∣df∣2\int_M|df|^2; equivalently, it solves the Euler–Lagrange system τ(f)=tr⁡∇df=0\tau(f)=\operatorname{tr}\nabla df=0, where τ(f)\tau(f) is the tension field. Eells and Sampson (1964) proved that any continuous map from a compact Riemannian manifold to a complete manifold of nonpositive sectional curvature is homotopic to a harmonic map. When the domain is a compact Kähler manifold (a complex manifold whose associated metric (1,1)(1,1)-form ω\omega is closed), harmonic maps exhibit rigidity: Siu (1980) proved that a harmonic map from a compact Kähler manifold to a compact Kähler manifold whose metric has strongly negative curvature — in Siu's sense, the curvature tensor, regarded as a Hermitian quadratic form Q(σ,σˉ)Q(\sigma,\bar\sigma) on T1,0N⊗T0,1NT^{1,0}N\otimes T^{0,1}N, is negative on every nonzero element; this is stronger than negative sectional curvature and is satisfied for instance by compact quotients of the complex unit ball Bn\mathbb B^n, which satisfy the still stronger condition that QQ be negative on all of Λ2(TCN)\Lambda^2(T_{\mathbb C}N) — is holomorphic or antiholomorphic as soon as its real rank is at least 4 at some point, equivalently unless its rank is at most 3 everywhere (strong rigidity).

Jost and Yau asked what survives of this theory when the domain is merely Hermitian, i.e. a complex manifold with a compatible metric whose associated (1,1)(1,1)-form ω\omega need not be closed (a non-Kähler manifold). The natural replacement of the harmonic map system on a Hermitian domain is

γαβˉ(∂2fi∂zα ∂zˉβ+Γjki ∂fj∂zα ∂fk∂zˉβ)=0,\gamma^{\alpha\bar\beta}\left(\frac{\partial^2 f^i}{\partial z^\alpha\,\partial\bar z^\beta}+\Gamma^i_{jk}\,\frac{\partial f^j}{\partial z^\alpha}\,\frac{\partial f^k}{\partial\bar z^\beta}\right)=0,

where γαβˉ\gamma^{\alpha\bar\beta} is the inverse Hermitian metric tensor of the domain and Γjki\Gamma^i_{jk} are the Christoffel symbols of the target; solutions are called Hermitian harmonic maps. Holomorphic maps solve the system automatically, and on a Kähler domain the system reduces to the ordinary harmonic map system. The system is not of divergence type, so it is not the Euler–Lagrange equation of any known functional and the variational existence machinery does not apply. Jost and Yau nevertheless proved existence: a map from a compact Hermitian manifold to a compact manifold of sufficiently negative sectional curvature, not homotopic to a map onto a closed geodesic, admits a Hermitian harmonic representative in its homotopy class; an example involving Hopf manifolds (compact complex manifolds diffeomorphic to S1×S2m−1S^1\times S^{2m-1}, standard non-Kähler examples) shows that nonpositive curvature of the target alone does not suffice.

For their rigidity theorem they introduced the astheno-Kähler condition: a compact Hermitian manifold (X,ω)(X,\omega) of complex dimension n≥3n\geq 3 is astheno-Kähler if ∂∂ˉωn−2=0\partial\bar\partial\omega^{n-2}=0 (their original definition additionally requires that ωn\omega^n be a positive multiple of the volume form; the modern convention keeps only the differential condition). The condition is vacuous in complex dimension 2, is equivalent to the SKT (strong Kähler with torsion, also called pluriclosed) condition ∂∂ˉω=0\partial\bar\partial\omega=0 in dimension 3, and interpolates between Kähler and general Hermitian in general. Its key consequence in Jost–Yau is a lemma: every holomorphic 1-form on a compact astheno-Kähler manifold is closed. Building on this, they extended Siu's rigidity: a Hermitian harmonic map from a compact astheno-Kähler manifold to a compact locally Hermitian symmetric space of noncompact type without a unit-disc factor (i.e. a quotient of a bounded symmetric domain other than the disc itself) or to a compact Kähler manifold of strongly negative curvature is pluriharmonic — its restriction to every complex curve in the domain is harmonic — and is holomorphic or antiholomorphic as soon as its real rank is at least 4 at some point (with an analogous rank hypothesis for the symmetric-space targets).

The connected-sum applications are the source of the question. The connected sum M#NM\#N of two smooth 2n2n-manifolds is the manifold obtained by deleting a small open ball from each and gluing the results along the boundary spheres. Jost and Yau proved: (i) in real dimension 4, if NN is a compact complex surface carrying a Kähler metric of strongly negative curvature — for example a compact quotient of the complex unit ball B2\mathbb B^2 by a torsion-free lattice, i.e. a discrete subgroup of finite covolume in PU(2,1)PU(2,1) acting freely — and MM is any compact 4-manifold with π1(M)≠0\pi_1(M)\neq 0, then M#NM\#N admits no complex structure at all; (ii) in complex dimension n≥3n\geq 3, if NN is a compact quotient of Bn\mathbb B^n (with its Bergman metric, the canonical Kähler metric on the ball, of strongly negative curvature) and MM is a compact 2n2n-manifold with π1(M)≠0\pi_1(M)\neq 0, then M#NM\#N admits no meromorphic map onto any compact complex manifold of dimension n−2n-2; in particular its algebraic dimension — the transcendence degree over C\mathbb C of the field of meromorphic functions, a measure of how close a compact complex manifold is to being projective algebraic — is at most n−3n-3. Statement (ii) is conditional on M#NM\#N carrying a complex structure, and whether it ever does is exactly what the paper leaves open in complex dimension at least 3. In contrast with real dimension 4, where gauge theory provides obstructions to complex structures, no general technique of this kind is available in real dimension at least 6.

Problem Statement

Let n≥3n\geq 3. Let NN be a compact quotient of the complex unit ball Bn⊂Cn\mathbb B^n\subset\mathbb C^n by a torsion-free lattice (a compact complex nn-fold carrying the strongly negatively curved Bergman Kähler metric), and let MM be any compact smooth 2n2n-dimensional manifold with π1(M)≠0\pi_1(M)\neq 0. Let X=M#NX=M\#N be the smooth connected sum.

Question (the case Jost and Yau 1993 left open): does XX admit a complex structure for some choice of MM and NN as above — or is a complex structure impossible for every such choice?

Jost and Yau proved that if XX carries a complex structure, then XX admits no meromorphic map onto any compact complex manifold of dimension n−2n-2; in particular the algebraic dimension of XX satisfies a(X)≤n−3a(X)\leq n-3, and for n=3n=3 one has a(X)=0a(X)=0, so any such structure would be far from algebraic. The analogous question in real dimension 4 was settled completely by the same authors: if NN is a compact complex surface with a strongly negatively curved Kähler metric and π1(M)≠0\pi_1(M)\neq 0, then M#NM\#N carries no complex structure at all.

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