Lifting harmonic maps from the symmetric-space quotient back to Griffiths period domains, and harmonic-map finiteness and rigidity for general period mappings

Jost and Yau (1993) proved Shafarevich-type finiteness, rigidity, and existence theorems for holomorphic maps into moduli spaces of curves and into Siegel quotients using harmonic map theory, and proved one structural fact for general period domains: if $\phi$ is a Griffiths period mapping into a quotient $D/\Gamma$ of a period domain $D=H\backslash G$ by its monodromy group, then the composition of the lifted period map with the projection $\pi:D\to K\backslash G$ to the Riemannian symmetric space of a maximal compact subgroup $K\supset H$ is harmonic. The paper explicitly deferred the further development of this correspondence: whether it yields finiteness and rigidity theorems for period maps into general period domains, and how a projected harmonic map can be lifted back to a holomorphic horizontal map on the period domain, where the lift's indeterminacy depends on the monodromy action.

Background

A polarized variation of Hodge structure over a complex manifold $S$ records how the Hodge decomposition (the splitting of the complex cohomology into pieces of type $(p,q)$) of the cohomology of a family of algebraic varieties varies over $S$; its analytic content is encoded in the period map. After trivializing the underlying local system (a locally constant sheaf of complex vector spaces), the set of possible Hodge structures of a given weight and polarization is parametrized by a period domain $D=H\backslash G$, a homogeneous complex manifold in which $G$ is a real reductive Lie group (the automorphism group of the polarization, for example $\mathrm{Sp}(2g,\mathbb{R})$ for weight-one structures) and $H\subset G$ is the compact subgroup stabilizing a reference Hodge structure. The lifted period map $\tilde\phi:\tilde S\to D$ on the universal cover is holomorphic and horizontal, meaning that its differential takes values in a distinguished subbundle of the tangent bundle of $D$ (Griffiths transversality), and it is equivariant for the monodromy group $\Gamma$, an arithmetic subgroup of $G$ (a subgroup commensurable with the integer points $G(\mathbb{Z})$), in the sense that $\tilde\phi(\gamma\cdot s)=\gamma\cdot\tilde\phi(s)$; the period map itself lands in the quotient $D/\Gamma$. For weight one, $D$ is the Siegel upper half space, a Hermitian symmetric space (a symmetric space that is simultaneously a complex manifold, with the symmetries acting holomorphically) whose arithmetic quotients carry nonpositive curvature. For general weights $D$ is not a symmetric space, and the natural $G$-invariant Hodge metric on it is indefinite, so the standard curvature-driven tools of geometric analysis are unavailable on $D$ itself.

A harmonic map between Riemannian manifolds is a stationary point of the Dirichlet energy $\int|du|^2$; when the target has suitable nonpositive curvature, harmonic representatives in a homotopy class are unique and often forced to be holomorphic, which is the mechanism behind harmonic-map rigidity. Jost and Yau (1993) applied this machinery to arithmetic questions over function fields. For targets that are moduli spaces $M_{g,n}$ of genus-$g$ curves with $n$ punctures, they combined the negative curvature and strict convexity properties of the Weil-Petersson metric (the natural $L^2$ metric on the Teichmüller space of marked Riemann surfaces, the cover of $M_{g,n}$ on which curves are marked up to isotopy) with Schwarz-lemma estimates (distance-decreasing estimates for holomorphic maps) to prove boundedness, rigidity (a locally liftable holomorphic map into $M_{g,n}$ is determined by its homotopy class, up to finite covers, unless constant), and finiteness of curve fibrations and of sections. For families of principally polarized abelian varieties (abelian varieties, the higher-dimensional analogues of elliptic curves, equipped with a nondegenerate alternating form defining the polarization), where the period domain is the symmetric Siegel space, they proved that the family is rigid exactly when there is no nonzero monodromy-invariant parallel vector field along the lifted period map, and that only finitely many homotopy classes of locally liftable maps arise from actual families. For general period domains they established only a single structural fact: if $K\supset H$ is a maximal compact subgroup of $G$ and $\pi:H\backslash G\to K\backslash G$ is the projection to the associated Riemannian symmetric space (the quotient of $G$ by a maximal compact subgroup, the model nonpositively curved target), then for any Griffiths period mapping $\phi$ the composition $\pi\circ\tilde\phi$ is harmonic. The further use of this observation, namely lifting the projected harmonic map back to $D$ and deriving finiteness and rigidity statements for general period domains, was explicitly deferred to future work, together with the remark that analogous results for families of varieties would require suitable geometric properties of Weil-Petersson-type metrics on the corresponding moduli spaces.

Later work developed pieces of the deferred program by other routes. Zhiqin Lu (1999) proved that for Calabi-Yau classifying spaces (the period domains parametrizing Hodge structures of Calabi-Yau type, a Calabi-Yau manifold being a projective manifold with trivial canonical bundle) the induced map to the associated symmetric space is pluriharmonic (harmonic along every complex line) and constructed a Hodge metric whose Ricci curvature is negative away from zero. Kang Zuo (2000) established negativity of kernels of Kodaira-Spencer maps (the maps encoding the infinitesimal variation of Hodge structure) on Hodge bundles (the holomorphic subbundles of the flat cohomology bundle whose fibers are the Hodge summands), and Lu and Sun (2004) developed the Weil-Petersson geometry of moduli spaces of polarized Calabi-Yau manifolds; through these and related Higgs-bundle methods (a Higgs bundle being a holomorphic vector bundle with a holomorphic one-form-valued endomorphism whose square vanishes, the algebraic counterpart of a variation of Hodge structure), Yi Zhang (2004) and Liu, Todorov, Yau and Zuo (2005) obtained rigidity and Shafarevich-type finiteness for families of polarized Calabi-Yau varieties in important classes. On the analytic side, Daskalopoulos, Katzarkov and Wentworth (2000) gave existence criteria for equivariant harmonic maps into Teichmüller space, and Daskalopoulos and Mese (2020, 2021) proved the holomorphic rigidity of Teichmüller space and extended equivariant pluriharmonic map theory to quasi-projective domains (complements of divisors in projective varieties) with symmetric-space, building (a combinatorial analogue of a symmetric space glued from Euclidean apartments), and completed-Teichmüller targets. Jost, Yang and Zuo (2015) returned to the harmonic map and period map interface with a harmonic-map version of the nilpotent orbit theorem of Schmid, which describes the asymptotic behavior of a period map near a boundary divisor. None of these works, however, produces a lift theory from the symmetric-space quotient back to a general period domain, or harmonic-map proofs of finiteness and rigidity for period maps into general period domains; the field's finiteness results for families were obtained instead through the negativity of deformation Higgs bundles and arithmetic geometry.

Problem Statement

Let $N$ be a compact Kähler manifold (a compact complex manifold carrying a compatible Riemannian metric), $A\subset N$ a divisor with simple normal crossings (a complex codimension-one subvariety that locally looks like a union of coordinate hyperplanes), and $S=N\setminus A$. Let $D=H\backslash G$ be a Griffiths period domain for polarized Hodge structures of a given weight on a given local system, where $G$ is the real reductive Lie group preserving the polarization and $H\subset G$ is the compact stabilizer of a reference Hodge structure; in general $D$ is not a Hermitian symmetric space. Let $\Gamma\subset G(\mathbb{Z})$ be the arithmetic monodromy group, and let $\phi:S\to D/\Gamma$ be a Griffiths period mapping, that is, a holomorphic locally liftable map whose lift $\tilde\phi:\tilde S\to D$ to the universal cover is horizontal (its differential takes values in the Griffiths horizontal subbundle) and $\Gamma$-equivariant. Let $K\supset H$ be a maximal compact subgroup of $G$ and $\pi:D\to K\backslash G$ the projection to the associated Riemannian symmetric space. Jost and Yau (1993) proved that the composition $\pi\circ\tilde\phi:\tilde S\to K\backslash G$ is harmonic, and explicitly deferred the further development of this observation. The problem is to carry out that deferred program:

(1) The lift problem. Characterize when a $\Gamma$-equivariant harmonic map $u:\tilde S\to K\backslash G$ that arises, up to homotopy, from a period mapping admits a lift $\tilde u:\tilde S\to D$ that is holomorphic and horizontal and hence recovers the period map. Resolve the indeterminacy of such lifts, which depends on the action of $\Gamma$, and give verifiable conditions under which an equivariant harmonic map into the symmetric-space quotient (satisfying the regularity available for maps on the quasi-projective base $S=N\setminus A$, possibly allowing infinite energy near $A$) is forced to be the projection of a holomorphic horizontal period map.

(2) Finiteness and rigidity for general period domains. Using the correspondence established in (1), prove for general, non-Hermitian-symmetric period domains $D$ the analogues of the results that Jost and Yau obtained for moduli of curves and for the Siegel case: up to homotopy and passage to finite covers, only finitely many locally liftable maps $S\to D/\Gamma$ arise from actual variations of Hodge structure over $S$ of the given topological type; and a period map is determined by its homotopy class whenever it admits no flat deformations (in the Siegel case the exact obstruction is a nonzero $\Gamma$-invariant parallel vector field along the lifted period map).

Determine, under precise hypotheses on the period domain, the monodromy group, the base, and the class of admissible maps, whether the lift characterization and the finiteness and rigidity statements hold, or whether the harmonic-map correspondence cannot yield such statements in the stated generality.

References

1. J. Jost and S.-T. Yau, "Harmonic Mappings and Algebraic Varieties Over Function Fields", American Journal of Mathematics 115 (1993), no. 6, pp. 1197 ff. DOI: 10.2307/2374964, https://doi.org/10.2307/2374964.
2. Z. Lu, "On the geometry of classifying spaces and horizontal slices", American Journal of Mathematics 121 (1999), no. 1, pp. 177-198. DOI: 10.1353/ajm.1999.0007, https://doi.org/10.1353/ajm.1999.0007; preprint arXiv:math/0505579, https://arxiv.org/abs/math/0505579.
3. K. Corlette, "Harmonic Maps, Rigidity, and Hodge Theory", in: Proceedings of the International Congress of Mathematicians, August 3-11, 1994, Zürich, Birkhäuser (1995), pp. 465-471. DOI: 10.1007/978-3-0348-9078-6_39, https://doi.org/10.1007/978-3-0348-9078-6_39.
4. K. Zuo, "On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications", Asian Journal of Mathematics 4 (2000), no. 1, pp. 279-302. DOI: 10.4310/ajm/2000.v4.n1.a17, https://doi.org/10.4310/ajm/2000.v4.n1.a17.
5. G. Daskalopoulos, L. Katzarkov, and R. Wentworth, "Harmonic Maps to Teichmüller Space", Mathematical Research Letters 7 (2000), no. 1, pp. 133-146. DOI: 10.4310/mrl.2000.v7.n1.a12, https://doi.org/10.4310/mrl.2000.v7.n1.a12.
6. Z. Lu and X. Sun, "Weil-Petersson Geometry on Moduli Space of Polarized Calabi-Yau Manifolds", Journal of the Institute of Mathematics of Jussieu 3 (2004), no. 2, pp. 185-229. DOI: 10.1017/s1474748004000076, https://doi.org/10.1017/s1474748004000076.
7. Y. Zhang, "Rigidity for Families of Polarized Calabi-Yau Varieties", Journal of Differential Geometry 68 (2004), no. 2. DOI: 10.4310/jdg/1115669511, https://doi.org/10.4310/jdg/1115669511.
8. K. Liu, A. Todorov, S.-T. Yau, and K. Zuo, "Shafarevich's Conjecture for CY Manifolds I", Pure and Applied Mathematics Quarterly 1 (2005), no. 1, pp. 28-67. DOI: 10.4310/pamq.2005.v1.n1.a3, https://doi.org/10.4310/pamq.2005.v1.n1.a3.
9. J. Jost, Y.-H. Yang, and K. Zuo, "Harmonic maps and singularities of period mappings", Proceedings of the American Mathematical Society 143 (2015), no. 8, pp. 3351-3356. DOI: 10.1090/proc/12566, https://doi.org/10.1090/proc/12566.
10. G. Daskalopoulos and C. Mese, "Rigidity of Teichmüller space", Inventiones mathematicae 224 (2020), no. 3, pp. 791-916. DOI: 10.1007/s00222-020-01020-2, https://doi.org/10.1007/s00222-020-01020-2; preprint arXiv:1502.03367, https://arxiv.org/abs/1502.03367.
11. G. Daskalopoulos and C. Mese, "Infinite energy maps and rigidity" (2021). arXiv:2112.13961, https://arxiv.org/abs/2112.13961.
12. K. Zuo, "On the Negativity of Moduli Spaces for Polarized Manifolds", Vietnam Journal of Mathematics 49 (2021), pp. 527-546. DOI: 10.1007/s10013-021-00507-6, https://doi.org/10.1007/s10013-021-00507-6.
13. A. Javanpeykar and D. Litt, "Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields", manuscripta mathematica 173 (2023), no. 1-2, pp. 23-44. DOI: 10.1007/s00229-023-01463-w, https://doi.org/10.1007/s00229-023-01463-w; preprint arXiv:1907.13536, https://arxiv.org/abs/1907.13536.
14. L. Kamenova, "Finiteness of stable Lagrangian fibrations", Sao Paulo Journal of Mathematical Sciences (2023). DOI: 10.1007/s40863-023-00395-3, https://doi.org/10.1007/s40863-023-00395-3; preprint arXiv:2308.05844, https://arxiv.org/abs/2308.05844.
15. G. Pearlstein and C. Peters, "Deformations and rigidity for mixed period maps", Algebraic Geometry 11 (2024), no. 5, pp. 620-675. DOI: 10.14231/ag-2024-019, https://doi.org/10.14231/ag-2024-019.
16. P. Engel, A. Lin, and S. Tayou, "Shafarevich's Conjecture for Families of Hypersurfaces over Function Fields", International Mathematics Research Notices 2025 (2025), no. 15. DOI: 10.1093/imrn/rnaf225, https://doi.org/10.1093/imrn/rnaf225; preprint arXiv:2410.06387, https://arxiv.org/abs/2410.06387.
17. M. Green, P. Griffiths, and C. Robles, "Period maps at infinity" (2025). arXiv:2509.08508, https://arxiv.org/abs/2509.08508.
18. K. Zuo, "Hyperbolic Geometry of Moduli Spaces of Algebraic Varieties via Hodge Theory and Beyond", in: Current Developments in Hodge Theory, Simons Symposia, Springer (2025), pp. 15-61. DOI: 10.1007/978-3-031-99683-2_2, https://doi.org/10.1007/978-3-031-99683-2_2.
19. P. Engel and S. Tayou, "On the non-abelian Hodge locus I", Selecta Mathematica (New Series) 32 (2026), no. 4. DOI: 10.1007/s00029-026-01185-3, https://doi.org/10.1007/s00029-026-01185-3; preprint arXiv:2305.00943, https://arxiv.org/abs/2305.00943.
20. X. Zhong, "Rigidity of the period map up to finite covers" (2026). arXiv:2608.29351, https://arxiv.org/abs/2608.29351.
21. K. Liu and Y. Shen, "Sections of Hodge bundles I: Global theory and applications to period maps" (2026). arXiv:2602.13947, https://arxiv.org/abs/2602.13947.
22. J. A. Carlson, S. Müller-Stach, and C. Peters, "Period Mappings and Period Domains", Cambridge University Press, 2nd edition (2017), ISBN 9781107189867; chapter 14, "Harmonic Maps and Hodge Theory", pp. 383-402. DOI: 10.1017/9781316995846, https://doi.org/10.1017/9781316995846.