Sections of Hodge bundles I: Global theory and applications to period maps

Kefeng Liu; Yang Shen

REVIEW 3 major objections 5 minor 1 cited by

For polarized families, lifted period maps land in a complex Euclidean subspace of the period domain, yielding global affine coordinates on Calabi–Yau type Teichmüller spaces.

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2026-08-02 23:35 UTC pith:3VEFNTUH

load-bearing objection The global Euclidean-image theorem is not just underproved; it looks false, and the K3 period map gives a concrete way to see it. the 3 major comments →

arxiv 2602.13947 v2 pith:3VEFNTUH submitted 2026-02-15 math.AG

Sections of Hodge bundles I: Global theory and applications to period maps

Kefeng Liu , Yang Shen This is my paper

classification math.AG MSC 14D0732G2032Q25

keywords period mapsHodge bundlesBeltrami differentialsunipotent orbitsHodge filtrationCalabi–Yau type manifoldsaffine structuresTeichmüller spaces

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper aims to pin down the global image of period maps for families of polarized manifolds. The authors argue that the lifted period map on the universal cover always lies in N⁻, a copy of complex Euclidean space sitting inside the period domain as a unipotent orbit. The engine is a comparison of two ways of building sections of Hodge bundles—one read off from the matrix representation of the period map, the other produced by Beltrami differentials and harmonic projection. Proving that these sections agree lets every block of the period map extend holomorphically across the universal cover. A corollary is that the Teichmüller space of Calabi–Yau type manifolds carries a global complex affine structure, meaning a global holomorphic immersion into C^N.

Core claim

The central claim is that, for any analytic family of polarized manifolds, the lifted period map Φ:tilde S→D has image contained in N⁻∩D, the unipotent orbit viewed as a complex Euclidean subspace of the period domain; equivalently, every subspace of the Hodge filtration at every point of the universal cover is isomorphic to the corresponding subspace at the base point. The proof identifies the sections Ω^(p) obtained from the upper-triangular blocks of Φ with the sections tilde Ω^(p) built from an integrable Beltrami differential and the harmonic-theory operator T=∂*G∂. Since the latter are globally defined, the former extend across the locus where the matrix representation might degenerate

What carries the argument

The load-bearing object is the unipotent subgroup N⁻ = exp(n⁻), realized as a complex Euclidean space inside the flag domain; in matrix form its elements are block upper-triangular matrices with identity blocks on the diagonal. The comparison lemma is the mechanism: the Hodge sections Ω^(p) defined by the period map's blocks coincide with the sections tilde Ω^(p) defined by the closed formula e^{iφ(t)}(I+T i_{φ(t)})^{-1}σ₀, where φ(t) is the integrable Beltrami differential encoding the deformed complex structure and T=∂*G∂ is the Green-operator expression from harmonic theory. Agreement of the two constructions makes the Φ-blocks holomorphically extendable, which confines the period map to

Load-bearing premise

Everything rests on the claim that the Hodge sections read off from the period map's matrix blocks agree with the sections constructed from Beltrami differentials and harmonic projection; if that equality fails at any point, the image of the lifted period map need not stay in the Euclidean subspace.

What would settle it

Take a one-parameter polarized family with a point where some block determinant det(Φ^{(p,q)}) vanishes, and compute the Beltrami/harmonic section tilde Ω^(p) there using the closed formula. If tilde Ω^(p) is not holomorphic at that point, or if the Φ-blocks fail to have finite limits, Lemma 4.2 and hence the Euclidean-image theorem collapse. Alternatively, find any point on the universal cover for which Φ(t) lies outside N⁻—for instance a polarized family that realizes a deformation to the complex conjugate manifold while preserving the polarization—and the central claim is directly refuted.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Every lifted period map of a polarized family admits global holomorphic coordinates given by its upper-triangular blocks, so period domains can be studied as open subsets of complex Euclidean space rather than only as general flag domains.
Hodge sections defined through the period map extend across the whole universal cover, so limits of the Φ-blocks exist at points that previously looked like singular locus of the matrix representation.
For Calabi–Yau type manifolds, the (0,1)-block Φ^(0,1) is a global holomorphic immersion from Teichmüller space into C^N, providing global affine coordinates on that moduli space.
The canonical bundle of the Teichmüller space of Calabi–Yau type manifolds is trivial, because the affine immersion pulls back a nowhere-vanishing top form from C^N.
The authors note that these features are consistent with conjectured contractibility and unique-geodesic properties of marked moduli spaces in settings with eight or more supercharges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same section-comparison argument might be pushed to punctured neighborhoods of boundary points, turning the Euclidean coordinates into a tool for describing maximal unipotent degenerations and mirror-symmetric limits.
One could test the affine-structure claim numerically in low-dimensional cases such as elliptic curves or K3 surfaces: if the (0,1)-block immersion degenerates at a finite point of Teichmüller space, the mechanism behind the theorem would be in question.
The paper's own example of deforming a projective manifold to its complex conjugate shows that polarization does real work here; a non-polarized analogue would need a different ambient object than the unipotent orbit N⁻.
The global affine structure, if it holds for all Calabi–Yau type manifolds, may constrain globally defined metrics on Teichmüller space by forcing coordinate expressions for their curvature to take a restricted form.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The global Euclidean-image theorem is not just underproved; it looks false, and the K3 period map gives a concrete way to see it.

read the letter

Short version: the paper has a real idea — comparing two constructions of Hodge-bundle sections, one from the period matrix on the big cell and one from Beltrami differentials — and Section 3's closed formula, Theorem 3.15, is a reasonable piece of deformation theory. But the global conclusion, Theorem 4.3/0.2, is not supported, and I think it is false.

The hinge is Lemma 4.2. The proof compares types but never establishes the compatibility of trivializations: the identification H_t ≅ H_{t0} used to define eΩ^(p) (via the diffeomorphisms d_t and harmonic projection) must agree with the flat marking used to define the blocks Φ^(p,q). The text says the polarization implies it, but that is not proved. Worse, the type comparison as printed is garbled: already at p=1, the H^{n-1,1} component of the right-hand side contains terms A^(0) i_φ η^(0), so comparing to η^(1) does not force A^(0)=0, A^(1)=I. This is not a small gap; it is exactly the point where the two constructions are joined.

The extension in Theorem 4.3 then claims that since eΩ^(p) extends, the Φ-blocks converge. If the sections are not actually equal on S∨, that conclusion does not follow.

I also think the theorem is false. For the Teichmüller space of marked polarized K3 surfaces, the period map is an isomorphism onto the type IV period domain. Fix a base period vector η. The set of period lines σ satisfying Q(σ, \bar η)=0 is a nonempty divisor in that domain. At those points the component along η vanishes, so the flag is not in the N^- big cell. The lifted period map from the universal cover of the moduli space with level structure is an isomorphism onto this period domain, so it hits that divisor. Hence S\S∨ is nonempty, contradicting Theorem 0.2. This is a concrete, checkable counterexample; if I am wrong, the missing hypothesis should be identified.

The affine-structure application inherits the problem: Ψ(t)=Φ^(0,1)(t) will have poles on that divisor. A global affine structure on K3 Teichmüller space may exist via the Harish-Chandra embedding, but this argument does not produce it.

For peer review: I would not send this to a referee. The main theorem is false as stated, and the central identification is a wrong claim, not a fixable gap. The Section 3 machinery may be worth salvaging, but it does not save the paper.

Referee Report

3 major / 5 minor

Summary. The paper develops two constructions of global sections of Hodge bundles for analytic families of polarized manifolds: one from the matrix representation of the lifted period map in the unipotent orbit N^-, and one from deformation theory via Beltrami differentials and harmonic projection. The central claim (Theorem 0.2 / Theorem 4.3) is that for any such family, the lifted period map on the universal cover has image contained in N^- ∩ D, i.e. the Hodge filtration at every point is transverse to a fixed reference filtration in the sense of the big cell. This is presented as a partial solution to a conjecture of Griffiths. As an application, the paper constructs a global complex affine structure on the Teichmüller space of Calabi–Yau type manifolds (Theorem 0.3 / Theorem 5.7). The main bridge between the two section constructions is Lemma 4.2, which asserts that the period-map sections Ω^(p) and the deformation-theoretic sections eΩ^(p) coincide on the open dense set eS∨.

Significance. If the main theorem is correct, it is a striking and potentially very influential result: it would give a global Euclidean realization of period maps on universal covers, with direct consequences for asymptotic analysis, compactifications, and affine structures on moduli spaces. The paper also contains an interesting explicit closed formula for sections of Hodge bundles using Green's operators and quasi-isometry estimates (Theorem 3.15), building on the authors' earlier published work; these techniques have independent value. However, the proof of the key identification in Lemma 4.2 is not established as written, and the characterization of eS∨ in Lemma 1.7 is stated incorrectly. The central claim therefore rests on a load-bearing gap. The paper is not ready for acceptance in its present form, but the deficiencies are specific and potentially repairable.

major comments (3)

[Section 4, Lemma 4.2] Lemma 4.2 is the hinge of Theorem 4.3. The proof compares types in the fixed basis η, but it never establishes that the isomorphism H^n(X_t) → H^n(X_{t0}) used to define the deformation-theoretic sections eΩ^(p) (via the diffeomorphisms d_t of Lemma 3.14 and the harmonic projection Hpr) agrees with the flat Gauss–Manin trivialization used to define the blocks Φ^(p,q)(t) of the period map. If the two identifications differ by a t-dependent automorphism not in the parabolic subgroup B, the coefficients of eΩ^(p) need not be the Φ-blocks, and the equality Ω^(p)=eΩ^(p) fails. The displayed type comparison is also written as a single vector identity; to conclude A^(0)=...=A^(p-1)=O and A^(p)=I one must compare each H^{n-i,i}(X_{t0}) component separately, which is not done. This gap is load-bearing: Theorem 4.3 and Theorem 0.2 depend on it.
[Section 1, Lemma 1.7 and Proposition 1.8] Lemma 1.7 states that Φ(z)∈N^- iff F^k_z is isomorphic to F^k_{z0} for all k. This condition is trivially true for every z, since any two subspaces of the same dimension are isomorphic. The proof uses a determinant criterion, which is a transversality condition (non-vanishing of certain minors), so the statement does not match the proof. Proposition 1.8 then uses Lemma 1.7 to conclude that eS minus eS∨ is an analytic subvariety of codimension at least 1. This conclusion is also not fully justified: the argument only rules out eS minus eS∨ = eS, not the case where it has nonempty interior. One must additionally use that a holomorphic map from a connected complex manifold to a flag variety that maps an open set into a proper analytic subvariety maps the whole manifold into it, contradicting the base point in eS∨. These points need correction, though a standard repair may be possible.
[Section 4, Theorem 4.3] The extension argument in Theorem 4.3 jumps from the existence of the extended section eΩ^(p) to the convergence of the Φ-blocks. This is valid only if the equality in Lemma 4.2 holds and if the coefficients of eΩ^(p) in the fixed flat frame are the Φ-blocks on eS∨. Since Lemma 4.2 is not established, the conclusion that the limits exist and that the period map takes values in N^- is unsupported. In addition, even if the coefficients converge along one curve, the independence of the limit from the chosen curve and the fact that the limiting matrix is in N^- (rather than its closure) should be stated explicitly. As written, the proof of the main theorem is incomplete at this step.

minor comments (5)

[Throughout] There are numerous typos: 'Gauss-Mannin' should be 'Gauss-Manin'; 'whihc' should be 'which'; in equations (48), (51) and in the proof of Lemma 4.2, η^(β) should be η^(p+k); Lemma 3.14 has a sentence saying φ(t)∈A^{1,0} instead of A^{0,1}; the notation eS∨ is used inconsistently.
[Section 3.3, Proposition 3.11] In inequality (41), the estimate ∥φ⌟Φ∥^2 ≤ ∥φ∥∥Φ∥^2 is dimensionally incorrect as written; the correct bound is ∥φ⌟Φ∥^2 ≤ ∥φ∥^2∥Φ∥^2, which still yields the contradiction since ∥φ∥<1. This should be corrected for clarity.
[Section 3.4, Theorem 3.15] In the proof, the notation d_{to}^*(F^pH^n(X_t,C)) is unclear; presumably d_t^* is intended. Also, the independence of the section from the choice of d_t is stated but not proved in detail; a reference or a few sentences would help.
[Section 5.1, Definition 5.1 and Lemma 5.3] The 'T-class assumption' is a strong technical hypothesis. The paper states that Lemma 5.3 is obvious, but the independence of T from the level m requires that the universal covers of the finite covering maps coincide; this is standard but deserves a bit more justification, especially since the analytic families fm are only assumed to exist for m≥m0.
[Section 5.2, Theorem 5.7] The proof of the affine structure uses Ψ(t)=Φ^(0,1)(t). The block Φ^(0,1) is defined after tensoring with the Tate Hodge structure; the paper should be careful with the indexing of the Hodge filtration so that Φ^(0,1) is indeed a vector of size N = dim H^1(X,Θ_X). Also, the 'global affine structure' definition in Definition 5.6 only requires an immersion, which is satisfied, but the authors should state that the local Torelli assumption gives dim T = N, not merely an injective differential.

Circularity Check

0 steps flagged

No significant circularity: the central coincidence in Lemma 4.2 compares independently defined objects, and the cited prior results are re-derived in the paper.

full rationale

The paper's main claim is a comparison between two independent constructions: the period-map sections Omega^(p)(t) defined via the N^- matrix representation of Phi(t) on \tilde S^vee (equation 17), and the deformation-theoretic sections tilde Omega^(p)(t) defined via Beltrami differentials and harmonic projection (equation 47). Lemma 4.2 attempts to prove equality of these two objects by type comparison; even if that proof is terse or incomplete, it is not an assumption of the conclusion. The later extension argument in Theorem 4.3 uses the global existence of tilde Omega^(p) to deduce that the Phi-blocks have finite limits; this is a logical consequence, not a restatement of the definition of N^- or \tilde S^vee. The affine-structure result in Theorem 5.7 is derived from Theorem 5.4 together with the Calabi-Yau type contraction-isomorphism condition (55), which is an explicitly stated geometric hypothesis, not a fitted parameter renamed as a prediction. The paper's citations to the authors' previous work ([9], [10], [12]) concern the Cartan formula and quasi-isometry estimates; both are re-derived in Sections 3.2-3.3 and are also published independently, so they do not function as load-bearing self-citations. Any concern that Lemma 4.2's proof is insufficiently detailed is a correctness or rigor issue, not a circularity issue. No step in the claimed derivation chain reduces an output to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No fitted parameters or invented entities appear. The central claim rests on standard mathematical background and on the stated domain assumptions about polarized families and smooth fine moduli spaces.

axioms (4)

domain assumption The family f:X→S is an analytic family of polarized manifolds (a line bundle L with ample fibers exists).
Used in Section 3 to construct the symplectic form equality via Moser and the norm<1 Beltrami differential; also needed for positivity in the period domain.
ad hoc to paper The moduli space Z_m is smooth and admits a universal analytic family for all m≥m_0 (the T-class assumption).
Section 5.1, Definition 5.1. This is needed to define the Teichmüller space as a universal cover and to run the period map machinery; many moduli spaces are not fine.
domain assumption Calabi–Yau type manifolds satisfy the infinitesimal Torelli theorem via condition (ii) in Definition 5.5.
Used in Section 5.2 to prove non-degeneracy of the Jacobian of Φ^(0,1).
standard math The deformation-theoretic closed formula (Theorem 3.13) is valid, depending on the quasi-isometry estimate for T=∂*G∂ and invertibility of I+T iφ.
Propositions 3.5 and 3.11/Corollary 3.12, based on Green's operator theory on compact Kähler manifolds, citing Liu–Rao–Yang [10].

pith-pipeline@v1.3.0-alltime-deepseek · 12 in / 50514 out tokens · 1041949 ms · 2026-08-02T23:35:53.565919+00:00 · methodology

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read the original abstract

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichm\"uller space of Calabi--Yau type manifolds.

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

19 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[1]

Carlson, S

J. Carlson, S. Muller-Stach, and C. Peters,Period Mappings and Period Domains, 2nd edition, Cambridge University Press, (2017)

2017
[2]

Clemens, Geometry of formal Kuranishi theory,Adv

H. Clemens, Geometry of formal Kuranishi theory,Adv. Math.,198(2005), pp. 311–365

2005
[3]

Griffiths, Periods of integrals on algebraic manifolds I,Amer

P. Griffiths, Periods of integrals on algebraic manifolds I,Amer. J. Math.,90(1968), pp. 568–626

1968
[4]

Griffiths, Periods of integrals on algebraic manifolds II,Amer

P. Griffiths, Periods of integrals on algebraic manifolds II,Amer. J. Math.,90(1968), pp. 805– 865

1968
[5]

Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems,Bull

P. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems,Bull. Amer. Math. Soc.,76, no.2 (1970), pp. 228–296

1970
[6]

Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann

G. Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann. of Math.,98, No. 3 (1973), pp. 411–426

1973
[7]

Kodaira and D

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, I and II,Annals of Mathematics,67(1958a), pp. 328-466
[8]

Kodaira and D

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, III,Annals of Mathematics,71(1960), pp. 43–76

1960
[9]

Liu and S

K. Liu and S. Rao, Remarks on the Cartan formula and its applications,Asian J. Math.,16 (2012), pp. 15–169

2012
[10]

K. Liu, S. Rao and X. Yang, Quasi-isometry and deformations of Calabi-Yau manifolds.Invent. Math.,199, no. 2 (2015), pp. 423–453

2015
[11]

Liu and Y

K. Liu and Y. Shen, Sections of Hodge bundles II: deformation of (p, p)-classes and applications to K¨ ahler geometry,preprint, (2026)

2026
[12]

Liu and S

K. Liu and S. Zhu, Solving equations with Hodge theory,arXiv:1803.01272, (2018)

Pith/arXiv arXiv 2018
[13]

Moser, On the volume elements on a manifold,Transactions of the American Mathematical Society,120(2) (1965), pp

J. Moser, On the volume elements on a manifold,Transactions of the American Mathematical Society,120(2) (1965), pp. 286–294

1965
[14]

Newlander and L

A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Annals of Mathematics, Vol.65, No. 3 (1957), pp. 391–404

1957
[15]

Popp,Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Math- emathics,620, Springer-Verlag, Berlin-New York, (1977)

H. Popp,Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Math- emathics,620, Springer-Verlag, Berlin-New York, (1977)

1977
[16]

Raman and C

S. Raman and C. Vafa, Swampland and the Geometry of Marked Moduli Spaces, arXiv:2405.11611 [hep-th]

Pith/arXiv arXiv
[17]

Schmid, Variation of Hodge structure: the singularities of the period mapping,Invent

W. Schmid, Variation of Hodge structure: the singularities of the period mapping,Invent. Math., 22(1973), pp. 211-319. 37 Kefeng Liu and Yang Shen

1973
[18]

Sommese, On the rationality of the period mapping,Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,5, Issue 4 (1978), pp

A. Sommese, On the rationality of the period mapping,Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,5, Issue 4 (1978), pp. 683–717

1978
[19]

Szendr¨ oi, Some finiteness results for Calabi–Yau threefolds,J

B. Szendr¨ oi, Some finiteness results for Calabi–Yau threefolds,J. London Math. Soc,(2)60 (1999), pp. 689–699. Mathematical Sciences Research Center, Chongqing University of Technology, Chongqing 400054, China; Department of Mathematics,University of California at Los Angeles, Los Angeles, CA 90095-1555, USA Email address:liu@math.ucla.edu Mathematical S...

1999

[1] [1]

Carlson, S

J. Carlson, S. Muller-Stach, and C. Peters,Period Mappings and Period Domains, 2nd edition, Cambridge University Press, (2017)

2017

[2] [2]

Clemens, Geometry of formal Kuranishi theory,Adv

H. Clemens, Geometry of formal Kuranishi theory,Adv. Math.,198(2005), pp. 311–365

2005

[3] [3]

Griffiths, Periods of integrals on algebraic manifolds I,Amer

P. Griffiths, Periods of integrals on algebraic manifolds I,Amer. J. Math.,90(1968), pp. 568–626

1968

[4] [4]

Griffiths, Periods of integrals on algebraic manifolds II,Amer

P. Griffiths, Periods of integrals on algebraic manifolds II,Amer. J. Math.,90(1968), pp. 805– 865

1968

[5] [5]

Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems,Bull

P. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems,Bull. Amer. Math. Soc.,76, no.2 (1970), pp. 228–296

1970

[6] [6]

Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann

G. Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann. of Math.,98, No. 3 (1973), pp. 411–426

1973

[7] [7]

Kodaira and D

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, I and II,Annals of Mathematics,67(1958a), pp. 328-466

[8] [8]

Kodaira and D

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, III,Annals of Mathematics,71(1960), pp. 43–76

1960

[9] [9]

Liu and S

K. Liu and S. Rao, Remarks on the Cartan formula and its applications,Asian J. Math.,16 (2012), pp. 15–169

2012

[10] [10]

K. Liu, S. Rao and X. Yang, Quasi-isometry and deformations of Calabi-Yau manifolds.Invent. Math.,199, no. 2 (2015), pp. 423–453

2015

[11] [11]

Liu and Y

K. Liu and Y. Shen, Sections of Hodge bundles II: deformation of (p, p)-classes and applications to K¨ ahler geometry,preprint, (2026)

2026

[12] [12]

Liu and S

K. Liu and S. Zhu, Solving equations with Hodge theory,arXiv:1803.01272, (2018)

Pith/arXiv arXiv 2018

[13] [13]

Moser, On the volume elements on a manifold,Transactions of the American Mathematical Society,120(2) (1965), pp

J. Moser, On the volume elements on a manifold,Transactions of the American Mathematical Society,120(2) (1965), pp. 286–294

1965

[14] [14]

Newlander and L

A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Annals of Mathematics, Vol.65, No. 3 (1957), pp. 391–404

1957

[15] [15]

Popp,Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Math- emathics,620, Springer-Verlag, Berlin-New York, (1977)

H. Popp,Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Math- emathics,620, Springer-Verlag, Berlin-New York, (1977)

1977

[16] [16]

Raman and C

S. Raman and C. Vafa, Swampland and the Geometry of Marked Moduli Spaces, arXiv:2405.11611 [hep-th]

Pith/arXiv arXiv

[17] [17]

Schmid, Variation of Hodge structure: the singularities of the period mapping,Invent

W. Schmid, Variation of Hodge structure: the singularities of the period mapping,Invent. Math., 22(1973), pp. 211-319. 37 Kefeng Liu and Yang Shen

1973

[18] [18]

Sommese, On the rationality of the period mapping,Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,5, Issue 4 (1978), pp

A. Sommese, On the rationality of the period mapping,Annali della Scuola Normale Superiore di Pisa-Classe di Scienze,5, Issue 4 (1978), pp. 683–717

1978

[19] [19]

Szendr¨ oi, Some finiteness results for Calabi–Yau threefolds,J

B. Szendr¨ oi, Some finiteness results for Calabi–Yau threefolds,J. London Math. Soc,(2)60 (1999), pp. 689–699. Mathematical Sciences Research Center, Chongqing University of Technology, Chongqing 400054, China; Department of Mathematics,University of California at Los Angeles, Los Angeles, CA 90095-1555, USA Email address:liu@math.ucla.edu Mathematical S...

1999