Energy of harmonic maps on the Kuranishi space of a compact Kähler–Einstein manifold of general type: smooth harmonic-map families, nonsmooth Kuranishi spaces, and energy asymptotics (the deferred problems of Cao–Sun–Yau–Zhang)

In the final section of 'On deformations of Fano manifolds' (Math. Ann. 383 (2022), 809–836), Cao–Sun–Yau–Zhang prove formally that for a compact Kähler–Einstein manifold of general type with unobstructed deformations, the energy of a formally assumed smooth family of harmonic maps (into a target of Hermitian nonpositive curvature, in a fixed homotopy class) is a plurisubharmonic function on the Kuranishi space, and they explicitly defer three linked questions to be 'discussed elsewhere': (a) the study of the nonsmoothness of the Kuranishi space, (b) the existence of a smooth family of harmonic maps in a fixed homotopy class, and (c) the asymptotic behavior of the energy function. The precise open core is the rigorous, non-formal foundation of the harmonic-map energy functional on the deformation spaces of canonically polarized Kähler–Einstein manifolds.

Background

A Fano manifold is a compact complex manifold whose anti-canonical line bundle $K_X^{-1}$ is ample; a Kähler metric $ω$ is Kähler–Einstein (KE) if its Ricci curvature is proportional to the metric itself. At the opposite end of the Kodaira classification sits the general type case: a compact complex manifold is of general type when its canonical bundle $K_X$ is big, and when $K_X$ is ample the manifold is called canonically polarized; by Yau's solution of the Calabi conjecture it then carries a unique KE metric with negative Einstein constant (Yau 1978). The deformation theory of such manifolds is organized by the Kuranishi family: a classical construction of deformation theory (Kuranishi 1962) produces a locally complete holomorphic family of complex structures $(X, B, π)$ with central fiber $X_0 = π^{-1}(0)$, parametrized by the Kuranishi space $B$; $B$ is in general only a possibly singular complex analytic set embedded in the vector space of infinitesimal deformations, and it is smooth precisely when the deformations are unobstructed. Deformations of Fano manifolds are always unobstructed (by Kodaira vanishing), whereas manifolds of general type — already algebraic surfaces — are frequently obstructed, so their Kuranishi spaces are generally singular.

Cao–Sun–Yau–Zhang (2022) studied small deformations of a Fano KE manifold $(X_0, ω_0)$. Their Theorem 1.1 proves that all fibers $X_t$ of the Kuranishi family admit KE metrics if and only if the dimension $h^0(X_t, T^{1,0}X_t)$ of the space of holomorphic vector fields is independent of $t$, if and only if the identity components of the automorphism groups satisfy $Aut^0(X_t) ≅ Aut^0(X_0)$; this refines Koiso's 1983 theorem (the case of discrete automorphism group) and complements earlier criteria of Székelyhidi in terms of K-polystability, the algebro-geometric stability condition (defined via special one-parameter degenerations called test configurations) that Chen–Donaldson–Sun proved equivalent to the existence of a KE metric on a Fano manifold. Their Theorem 1.2 shows that the Weil–Petersson metric — the natural $L^2$ metric on the deformation space induced by the fiberwise KE metrics and the harmonic representatives of the infinitesimal deformation classes $H^1(X_t, T^{1,0}X_t)$ — is approximated by the normalized Ricci curvatures of the $L^2$ metrics on the direct image bundles $R^0π_*K_{X/B}^{-k}$ (the holomorphic vector bundles over the base whose fiber at $t$ is the space of pluricanonical anti-canonical sections $H^0(X_t, K_{X_t}^{-k})$).

The final section of the paper turns to manifolds of general type and to harmonic maps. A harmonic map $f: (X, g) → (N, h)$ between Riemannian manifolds is a critical point of the Dirichlet energy $E(f) = ∫_X |df|^2 dV$; two maps are homotopic if one is continuously deformable into the other, and by the theorem of Eells and Sampson a harmonic representative exists in every homotopy class when the target has nonpositive sectional curvature, while Hartman proved that two homotopic harmonic maps into such a target are joined by a geodesic homotopy through harmonic maps, which yields uniqueness of the harmonic representative when the target has no flat factor along the image (harmonic maps into a flat factor, for instance maps of a circle to a flat circle, need not be unique). The target curvature is called Hermitian nonpositive if $R^N(u,v,u,v) ≤ 0$ for all complexified tangent vectors $u, v$; in this case the Siu–Sampson vanishing theorem asserts that a harmonic map from a Kähler domain satisfies $∇^{1,0}\bar∂f = 0$ together with the vanishing of a certain curvature contraction term. For a closed Riemann surface of genus at least two, Toledo proved in 2012 that the energy of the (unique) harmonic map in a fixed homotopy class into a target of Hermitian nonpositive curvature is a plurisubharmonic function on the Teichmüller space of the surface (the space of marked conformal structures; plurisubharmonic means subharmonic along every holomorphic disk in the base). Yau suggested that such energy functions should likewise produce plurisubharmonic functions on the higher-dimensional Teichmüller/Kuranishi spaces of KE manifolds; a first version of this higher-dimensional construction was carried out in the 2014 Lehigh thesis of Yingying Zhang (reference [37] of the source paper).

Cao–Sun–Yau–Zhang's Section 5 gives the higher-dimensional statement, but only formally. For a compact KE manifold $(X_0, ω_0)$ of general type whose deformation is unobstructed, with Kuranishi family $(X, B, π)$, target $(N, h)$ of nonpositive sectional curvature, and homotopy class $A$ of maps $X_0 → N$, they assume given a smooth map $F: X → N$ from the total space whose restrictions $f_t = F|_{X_t}$ are the harmonic maps in class $A$, and compute the first and second variations of $E(t) = E(f_t)$, using the deformation formulas for the fiberwise KE metrics established in Sun's 'Deformation of canonical metrics' theory, which introduce nonlocal elliptic correction terms of Green-operator type such as $(1-Δ_0)^{-1}|φ_1|^2$, where $φ_1$ is a harmonic Beltrami differential (a harmonic $(0,1)$-form with values in the holomorphic tangent bundle) and $(1-Δ_0)^{-1}$ inverts the positive elliptic operator $1-Δ_0$ on the central fiber. Their Theorem 5.2 concludes that when $N$ has Hermitian nonpositive curvature, the energy function $E$ is plurisubharmonic on the base $B$. The authors then explicitly defer the three analytic ingredients to future work: the nonsmoothness of the Kuranishi space (dropping the unobstructedness assumption), the existence of the smooth family of harmonic maps (dropping the formal assumption on $F$), and the asymptotic behavior of the energy function. The interest of the construction is that it would produce, for the first time beyond the surface case, natural plurisubharmonic functions on the deformation spaces of canonically polarized manifolds — the basic objects for the potential theory and global geometry of these moduli spaces.

Problem Statement

Let $(X_0, ω_0)$ be a compact Kähler–Einstein manifold of general type (so the canonical bundle $K_{X_0}$ is ample and $Ric(ω_0) = -ω_0$), let $(X, B, π)$ be its Kuranishi family, let $(N, h)$ be a Riemannian manifold of nonpositive sectional curvature, and of Hermitian nonpositive curvature whenever the plurisubharmonicity conclusion is invoked, and let $A$ be a homotopy class of maps $X_0 → N$. Cao–Sun–Yau–Zhang's Theorem 5.2 computes the variations of the energy $E(t)$ of the harmonic map in class $A$ on each fiber $X_t$ and concludes its plurisubharmonicity on the base $B$, but under two simplifying assumptions: the deformation of $X_0$ is unobstructed (so $B$ is smooth), and a smooth map $F: X → N$ with harmonic restrictions $f_t = F|_{X_t}$ is formally assumed to exist. The problem, whose study the authors defer to future work, is to complete the program by resolving the following three linked questions.

(1) Smooth harmonic-map families. Prove that the harmonic maps $f_t: X_t → N$ in the fixed homotopy class $A$ exist for all parameters $t$ (existence for each fixed $t$ is classical by Eells–Sampson; uniqueness in the homotopy class, when it holds, follows from Hartman's geodesic-homotopy argument, and its possible failure — for instance through flat factors of the target — is itself part of what must be analyzed) and depend smoothly on $t$, so that $E(t) = E(f_t)$ is a well-defined smooth function on the base; establish the precise hypotheses on the target and the homotopy class under which this smooth dependence holds, and exhibit explicit examples of any failure.

(2) Nonsmooth Kuranishi spaces. Extend the energy functional and the plurisubharmonicity statement to the obstructed case in which the Kuranishi space $B$ is a singular analytic set: define $E$ on all of $B$, not merely on its smooth locus, and prove that $E$ is plurisubharmonic on $B$ in the Fornaess–Narasimhan sense for reduced complex spaces — that is, $E$ is locally the restriction of a plurisubharmonic function on an open neighbourhood of $B$ in the ambient space of infinitesimal deformations — so that the plurisubharmonicity conclusion of Theorem 5.2 holds without the unobstructedness assumption.

(3) Asymptotics of the energy. Determine the asymptotic behavior of the energy function $E(t)$ near the central fiber of the Kuranishi family, that is, determine the higher-order terms of the expansion of $E(t)$ in the coordinates of the Kuranishi base beyond the second variation computed in Theorem 5.2, as the complex structure tends to $X_0$.

The common objective of the three items is a rigorous, non-formal theory of the harmonic-map energy functional on the Kuranishi (Teichmüller-type) spaces of compact Kähler–Einstein manifolds of general type, with plurisubharmonicity of the energy as its central conclusion — the higher-dimensional counterpart of Toledo's theorem for Riemann surfaces. A complete solution must make the formal variation formulas and the plurisubharmonicity of Theorem 5.2 rigorous consequences rather than assumptions; the three items are jointly the completion of that single objective, since without (1) the energy function is not even well defined on the base, without (2) the plurisubharmonicity statement is limited to a special class of manifolds, and without (3) the function's usefulness for the geometry of the moduli space cannot be assessed.

References

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