Closed-form solutions of tautological systems at maximal-unipotent-monodromy points for Calabi-Yau hypersurfaces and complete intersections in flag varieties

Lian, Song and Yau (J. Eur. Math. Soc. 15 (2013), 1457-1483) attached to every partial flag variety $X=\mathrm{SL}_n/P$ a tautological system: a regular holonomic D-module on the space of sections of an ample equivariant line bundle, generated by differential operators from the vanishing ideal of the embedded variety, by Lie-algebra vector fields, and by Euler operators, whose solutions include all period integrals of the universal family of Calabi-Yau hypersurfaces (and, by Definition 5.6 of that paper, complete intersections) in $X$. The structural problems the source left open have since been solved: the holonomic rank is known in full generality - for flag varieties with the anticanonical bundle (Bloch-Huang-Lian-Srinivas-Yau 2014; Huang-Lian-Zhu 2016), for complete intersections in homogeneous spaces (Huang-Lian-Yau-Yu 2021; Lee-Lian-Zhang 2023), and for arbitrary reductive groups, homogeneous spaces and equivariant line bundles, with a functorial mixed-Hodge-module construction and non-vanishing criteria (Görlach-Reichelt-Sevenheck-Steiner-Walther 2026); the full solution sheaf is geometrically realized by chain integrals (Huang-Lian-Yau-Zhu 2016). The explicit analytic side remains open: beyond the power-series computations for projective space and $G(2,4)$ in the source paper, no closed formula is known for the general solutions near a point of maximal unipotent monodromy - the flag-variety analog of the Gamma-function series of the toric GKZ theory. The problem asks for exactly that: a uniform closed-form Frobenius basis of the local solution sheaf at such boundary points, together with the monodromy analysis identifying them.

Background

Let $X$ be a smooth Fano variety, i.e. a smooth projective variety whose anticanonical line bundle $K_X^{-1}$ - the dual of the determinant line bundle of holomorphic $1$-forms on $X$ - is ample. A section $f\in H^0(X,K_X^{-1})$ cuts out a hypersurface $Y_f=\{f=0\}$; when $Y_f$ is smooth it is a Calabi-Yau variety (its canonical bundle is trivial), and its complex structures are probed by period integrals $\int_\gamma\mathrm{Res}(\omega/f)$, where $\omega$ is a canonically constructed meromorphic top form on $X$ with a single pole along $Y_f$ and $\gamma$ runs over middle-dimensional cycles. As $f$ varies, these integrals form the period sheaf on the space of sections minus the discriminant locus of singular hypersurfaces. One seeks the Picard-Fuchs system: the system of linear PDEs annihilating all periods. Its solution sheaf contains the period sheaf and can be strictly larger.

For $X$ toric (a variety containing a dense algebraic torus as open subset, e.g. projective space), this system is the Gel'fand-Kapranov-Zelevinsky (GKZ) $A$-hypergeometric system, a combinatorially defined holonomic D-module (a PDE system with finite-dimensional solution space) built from the lattice of integer relations among the exponents of monomial sections. The toric theory is complete: the holonomic rank (the dimension of the solution space) is the normalized volume of the exponent polytope (lattice volume normalized so that a fundamental cell has volume $1$), and near a distinguished boundary point of the parameter space where the monodromy (the linear action on solutions along loops) is a single unipotent Jordan block - a point of maximal unipotent monodromy (MUM), the large complex structure limit of mirror symmetry - a full basis of solutions is given by explicit Gamma-function power series (Hosono-Lian-Yau 1996; Li-Lian-Yau 2012).

Lian-Song-Yau (2013) extended the Picard-Fuchs construction to homogeneous spaces. For a partial flag variety $X=F(d_1,\dots,d_r;n)=\mathrm{SL}_n/P_S$ - the variety of flags of subspaces $0\subset E_1\subset\cdots\subset E_r\subset\mathbb C^n$ with $\dim E_i=d_i$, a Grassmannian being the one-step case - the monomial-exponent mechanism fails, because the binomial relations among monomials are too few to cut out the embedded variety. They introduced instead the tautological system $\tau(X,\varphi,G,\beta)$: the D-module on the space of sections generated by (i) operators $p(\partial_\zeta)$ for polynomials $p$ in the vanishing ideal of $X$ under the embedding $\varphi$, (ii) the vector fields $Z_x$ induced by the Lie algebra of $G$, and (iii) Euler operators. They proved the system regular holonomic (regular: all solutions grow at most polynomially near singularities) and that all period integrals of the universal family of Calabi-Yau hypersurfaces, and of complete intersections, solve it (Theorem 5.3 and Definition 5.6). For the anticanonical embedding of a flag variety the quadratic generators are explicit binomials $\partial_{a_u}\partial_{a_v}-\partial_{a_w}\partial_{a_t}$ indexed by exponent quadruples with $u+v=w+t$; Remark 5.4 of that paper observes that for an arbitrary ample line bundle $L$ this breaks down (e.g. for $G(d,n)$ with $L=\mathcal O(1)$ the ideal is generated by the non-binomial Plücker relations, the classical quadratic equations defining the Grassmannian in its Plücker embedding). Section 6 computed explicit power series only for $X=\mathbb P^{n-1}$ (where the system is the extended GKZ system and the toric Gamma formula applies) and for $X=G(2,4)$.

The paper closes (Section 7) with the rank problem - compute the dimension of the solution space - and the coincidence problem - decide when the solution sheaf equals the period sheaf and by how much it exceeds it. Both have since been settled. Bloch-Huang-Lian-Srinivas-Yau (2014) proved rank formulas for Calabi-Yau hypersurfaces in generalized flag varieties (the solution space as a Lie algebra homology group and as the middle cohomology $H^{\dim X}(X-Y_f)$ of the complement) and exhibited rank-one points as large-complex-structure-limit candidates; Huang-Lian-Zhu (2016) proved the rank conjecture in full generality and extended it to arbitrary projective manifolds with a group action. Combined with the residue exact sequence $0\to H^d(X)_{\mathrm{prim}}\to H^d(X-Y_f)\to H^{d-1}(Y_f)_{\mathrm{van}}\to 0$ and the period-sheaf rank formula stated in the source paper, this answers the coincidence question: the solution sheaf exceeds the period sheaf exactly by the primitive middle cohomology of the ambient flag variety, and Huang-Lian-Yau-Zhu (2016) realized the extra solutions geometrically as chain integrals over relative cycles. For complete intersections, Huang-Lian-Yau-Yu (2021) constructed the tautological system for zero loci of vector-bundle sections and conjectured the rank formula, which Lee-Lian-Zhang (2023) proved for ample complete intersections in projective homogeneous spaces of semisimple groups, together with the existence of rank-one points. Finally, Görlach-Reichelt-Sevenheck-Steiner-Walther (2026) solved the holonomic rank problem in full generality - any connected reductive group acting transitively, any very ample equivariant line bundle - via a functorial construction showing these systems underlie mixed Hodge modules (the axiomatic Hodge-theoretic structure that D-modules of geometric origin carry), and gave necessary and sufficient non-vanishing criteria (these systems can in fact be zero, a possibility overlooked in earlier studies). An $\ell$-adic version in characteristic $p$ was constructed by Fu-Huang-Lian-Yau-Zhang-Zhu (2021).

What remains open is the analytic, closed-form side of the toric theory: an explicit Gamma-type basis of solutions near a MUM point for general flag varieties.

Problem Statement

Let $X=F(d_1,\dots,d_r;n)=\mathrm{SL}_n/P_S$ be a partial flag variety with its anticanonical embedding, and let $\tau(X,\varphi,\mathrm{SL}_n,1)$ be the tautological system on $H^0(X,K_X^{-1})$ (Theorem 5.3 of Lian-Song-Yau); analogously, let $\tau(X,\mathrm{SL}_n,L_1,\dots,L_s,1,\dots,1)$ be the tautological system for Calabi-Yau complete intersections cut out by sections of ample equivariant line bundles with $L_1\otimes\cdots\otimes L_s\simeq K_X^{-1}$ (Definition 5.6 of the same paper). The holonomic rank of these systems is known, and the solution sheaf is geometrically realized by chain integrals. The open problem is the flag-variety analog of the toric GKZ Gamma series: construct, uniformly for all partial flag varieties and their Calabi-Yau complete intersections, an explicit basis of the local solution sheaf of the tautological system near a point of maximal unipotent monodromy (a large-complex-structure-limit point), by closed formulas - explicit hypergeometric-type (Gamma-function) power series together with their logarithmic Frobenius powers. This comprises two jointly supporting parts of one objective: (a) identifying and certifying suitable MUM points in the parameter space, including the monodromy analysis establishing maximal unipotence there; and (b) the closed-form series themselves, generalizing the known formulas for $\mathbb P^{n-1}$ (extended GKZ Gamma series) and the published $G(2,4)$ computation. The target, as in the toric theory, is the general closed form.

References

1. Bong H. Lian, Ruifang Song, Shing-Tung Yau, Periodic integrals and tautological systems, J. Eur. Math. Soc. 15 (2013), no. 4, 1457-1483. DOI 10.4171/jems/396, https://doi.org/10.4171/jems/396 (arXiv:1105.2984)
2. Spencer Bloch, An Huang, Bong H. Lian, Vasudevan Srinivas, Shing-Tung Yau, On the holonomic rank problem, J. Differential Geom. 97 (2014), no. 1, 11-35. DOI 10.4310/jdg/1404912101, https://doi.org/10.4310/jdg/1404912101 (arXiv:1302.4481)
3. An Huang, Bong H. Lian, Xinwen Zhu, Period integrals and the Riemann-Hilbert correspondence, J. Differential Geom. 104 (2016), no. 2, 325-369. DOI 10.4310/jdg/1476367060, https://doi.org/10.4310/jdg/1476367060 (arXiv:1303.2560)
4. An Huang, Bong H. Lian, Shing-Tung Yau, Xinwen Zhu, Chain integral solutions to tautological systems, Math. Res. Lett. 23 (2016), no. 6, 1721-1736. DOI 10.4310/mrl.2016.v23.n6.a7, https://doi.org/10.4310/mrl.2016.v23.n6.a7 (arXiv:1508.00406)
5. An Huang, Bong H. Lian, Shing-Tung Yau, Chenglong Yu, Period integrals of vector bundle sections and tautological systems, Math. Res. Lett. 28 (2021), no. 2, 415-434. DOI 10.4310/mrl.2021.v28.n2.a4, https://doi.org/10.4310/mrl.2021.v28.n2.a4 (arXiv:1801.01194; arXiv version titled Period integrals of local complete intersections and tautological systems)
6. Tsung-Ju Lee, Bong H. Lian, Dingxin Zhang, On a conjecture of Huang-Lian-Yau-Yu, Comm. Anal. Geom. 31 (2023), no. 6, 1341-1352. DOI 10.4310/cag.2023.v31.n6.a1, https://doi.org/10.4310/cag.2023.v31.n6.a1 (arXiv:2005.00676)
7. Paul Görlach, Thomas Reichelt, Christian Sevenheck, Avi Steiner, Uli Walther, Tautological systems, homogeneous spaces and the holonomic rank problem, J. Éc. polytech. Math. 13 (2026), 519-591. DOI 10.5802/jep.332, https://doi.org/10.5802/jep.332 (arXiv:2211.05356)
8. Lei Fu, An Huang, Bong Lian, Shing-Tung Yau, Dingxin Zhang, Xinwen Zhu, $\ell$-adic Tautological Systems, arXiv:2104.12956 (2021), https://arxiv.org/abs/2104.12956
9. Bong H. Lian, Shing-Tung Yau, Period Integrals of CY and General Type Complete Intersections, Invent. Math. 191 (2013), no. 1, 35-89. DOI 10.1007/s00222-012-0391-6, https://doi.org/10.1007/s00222-012-0391-6 (arXiv:1105.4872)
10. Tsung-Ju Lee, Hui-Wen Lin, Tautological systems under the conifold transition on $G(2,4)$, arXiv:1602.00774 (2016), https://arxiv.org/abs/1602.00774
11. Luis Narváez Macarro, Christian Sevenheck, Tautological systems and free divisors, Adv. Math. 352 (2019), 372-405. DOI 10.1016/j.aim.2019.06.007, https://doi.org/10.1016/j.aim.2019.06.007 (arXiv:1805.02101)
12. Paul Görlach, Christian Sevenheck, Duality theory of tautological systems, arXiv:2510.01980 (2025), https://arxiv.org/abs/2510.01980
13. Paul Görlach, Thomas Reichelt, Christian Sevenheck, Uli Walther, Tautological systems and local cohomology, arXiv:2607.14602 (2026), https://arxiv.org/abs/2607.14602
14. An Huang, Bong H. Lian, Shing-Tung Yau, Chenglong Yu, Period integrals and tautological systems, Surveys in Differential Geometry 22 (2017), no. 1, 275-289. DOI 10.4310/sdg.2017.v22.n1.a10, https://doi.org/10.4310/sdg.2017.v22.n1.a10
15. Shinobu Hosono, Bong H. Lian, Shing-Tung Yau, GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Comm. Math. Phys. 182 (1996), no. 3, 535-577. DOI 10.1007/BF02506417, https://doi.org/10.1007/BF02506417
16. Si Li, Bong H. Lian, Shing-Tung Yau, Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces, Amer. J. Math. 134 (2012), no. 5, 1345-1384. DOI 10.1353/ajm.2012.0039, https://doi.org/10.1353/ajm.2012.0039
17. Jingyue Chen, An Huang, Bong H. Lian, Holonomic systems for period mappings, Nucl. Phys. B 898 (2015), 693-706. DOI 10.1016/j.nuclphysb.2015.07.009, https://doi.org/10.1016/j.nuclphysb.2015.07.009 (arXiv:1512.05111)