Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Boundedness of some fibered K-trivial varieties

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Irreducible Calabi-Yau varieties of fixed dimension that fiber over a lower-dimensional base by abelian varieties or by primitive symplectic varieties of a fixed deformation class are birationally bounded, and Lagrangian-fibered primitive…

desk verdict Genuinely new boundedness results for fibered CY and symplectic varieties, carefully written; the main risk is Proposition 5.16's unstated Čech identification, which a referee should check. read the letter →

arxiv 2507.00973 v1 pith:M7PDFSWS submitted 2025-07-01 math.AG hep-thmath.DG

Philip Engel , Stefano Filipazzi , François Greer , Mirko Mauri , Roberto Svaldi This is my paper · ORCID
classification math.AGhep-thmath.DG MSC 14J3214J4214E3014D07
keywords birationalboundednessCalabi-YauvarietiesprimitivesymplecticLagrangianfibrationsTate-ShafarevichgroupcanonicalbundleformulaperiodmappingsZarhintrick
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that unboundedness is impossible for a large class of fibered K-trivial varieties. For each fixed dimension d ≥ 3, irreducible Calabi-Yau varieties that admit a fibration by abelian varieties, or by a fixed deformation class of primitive symplectic varieties, are birationally bounded: all such varieties can be generated by a single finite-type family, up to birational equivalence. The paper also shows that Lagrangian-fibered primitive symplectic varieties of fixed dimension lie in finitely many locally trivial deformation classes, and that fibered Calabi-Yau 3-folds form a bounded family. The proof supplies a reusable mechanism: bound the base of the fibration via the canonical bundle formula and effective b-semiampleness, recover the fiber polarization from period maps via the Zarhin trick and the Kuga-Satake construction, and then use a refined Tate-Shafarevich theory to remove the usual rational-section hypothesis. If the arguments hold, they bring the long-standing finiteness question for K-trivial varieties within reach of the existing moduli theory.

What carries the argument

The argument is carried by three interlocking mechanisms. First, the canonical bundle formula converts a K-trivial fibration f: X → Y into a generalized pair (Y, B_Y, M_Y) on the base, where B_Y records the singularities of the fibration and M_Y is the Hodge line-bundle class; effective b-semiampleness (Theorem C) makes a bounded multiple of M_Y free, so the bases fit into a bounded family. Second, period maps to moduli of abelian or primitive symplectic varieties, combined with the Zarhin trick (A ⊕ A*)^⊕4 and a Kuga-Satake correspondence, recover the polarization type of the general fiber from a bounded Z-local system up to finite ambiguity. Third, the paper refines the Tate-Shafarevich group X_G = $H^{1}$_G(Z^+, P) of translation-birational twists: P is the translation sheaf of the fibration's Albanese model over a Galois cover Z^+ of the base, and a new multiplicity group measures the multiple fibers of the original fibration in codimension 1. The pivotal technical result is that the big open set Z^+ can be chosen to depend only on the induced pair (Y, B, M, Φ), not on the fibration itself, and that the algebraic twists are precisely the torsion analytic twists (Theorem 5.36), so the no-section case is controlled by the same bounded data as the section case.

What would settle it

To test the main theorem, search for an infinite sequence of irreducible Calabi-Yau 3-folds fibered by abelian surfaces over a fixed base whose Hodge number $h^{{2,0}}$ tends to infinity, since birational boundedness would force bounded Hodge numbers and such a sequence would refute Theorem A(AV). On the twist side, an abelian fibration whose algebraic Tate-Shafarevich class is nonzero but whose analytic image is zero would contradict Theorem 5.36, the comparison that removes the rational-section hypothesis.

Watch

Extended reading notes

Core claim

The central claim, in the paper's own terms, is Theorem A and Theorem B. Theorem A states that irreducible Calabi-Yau varieties of fixed dimension d ≥ 3 admitting a fibration by abelian varieties (case AV) or by a fixed deformation class of primitive symplectic varieties (case PS) are birationally bounded. Theorem B states that Lagrangian-fibered primitive symplectic varieties of fixed dimension 2d lie within a finite number of locally trivial deformation equivalence classes. The paper derives Corollary 1.2, that fibered Calabi-Yau 3-folds are bounded, and Corollary 1.3, that under the hyperkähler SYZ conjecture there are finitely many locally trivial deformation classes of primitive symplectic variety in each fixed dimension with b2 ≥ 5. The proof is carried out by first bounding the bases of all such fibrations in codimension 1, then bounding the polarization type of the general fiber, and finally using a new Tate-Shafarevich theory for twists with multiple fibers to pass from fibrations with a rational section to all fibrations.

Load-bearing premise

The load-bearing premise is that the Tate-Shafarevich twist data can be made uniform in the fibration: a single big open set Z^+ works for every fibration with the same induced pair, and the algebraic twists coincide with the torsion analytic twists; if either claim fails, the boundedness theorems only cover fibrations with rational sections.

Editorial extensions

If this is right

  • Fibered Calabi-Yau 3-folds form a bounded family; in particular their Hodge numbers and Chern numbers are bounded, and they are parameterized birationally by a finite-type base.
  • If the hyperkähler SYZ conjecture holds, each fixed dimension has only finitely many locally trivial deformation classes of primitive symplectic variety with b2 ≥ 5.
  • Lagrangian-fibered primitive symplectic varieties of fixed dimension are contained in finitely many locally trivial deformation classes, even though they do not form a bounded family of varieties.
  • Effective b-semiampleness holds for fibrations by abelian varieties or by primitive symplectic varieties of bounded second Betti number, giving a uniform constant multiplying the Hodge line bundle into a free series on some birational modification of the base.
  • The Tate-Shafarevich group of an abelian-fibered irreducible Calabi-Yau variety is finite and varies constructibly in families; for primitive symplectic fibrations it is an extension of a finite constructible group by a quotient of C, with C identified with H^2(X,O_X)/H^2(Y,O_Y).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Tate-Shafarevich technology — constructible finiteness of the non-divisible part, extension by a continuous group for symplectic fibers — should transfer to other boundedness problems for abelian torsors over a fixed base, including a Shafarevich-type finiteness statement for symplectic fibrations with a fixed polarized fiber type.
  • The conditional corollary offers a sharp test of the SYZ conjecture: producing, in any fixed dimension, infinitely many primitive symplectic varieties with b2 ≥ 5 in pairwise distinct locally trivial deformation classes would simultaneously disprove the SYZ conjecture; the paper shows that the conjecture is the only missing input for that finiteness.
  • The layout of the proof suggests that birational boundedness for more general Calabi-Yau varieties could be approached by finding any fibration with controlled fibers and then verifying two technical pillars: a base bounded independently of the fibration, and an algebraic-analytic comparison for twists. Calabi-Yau 4-folds fibered by K3 surfaces are a natural next case.
  • Because the effective b-semiampleness input is modular, a full proof of the effective b-semiampleness conjecture for arbitrary lc-trivial fibrations would automatically extend the boundedness results to any fiber type for which the period map and twist groups are controlled.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a general framework for boundedness of fibered K-trivial varieties. Theorem A asserts birational boundedness of irreducible Calabi–Yau varieties of fixed dimension that admit fibrations by abelian varieties or by primitive symplectic varieties of a fixed analytic deformation class; Theorem B asserts that Lagrangian-fibered primitive symplectic varieties of fixed dimension lie in finitely many locally trivial deformation classes. The overall strategy is: bound the base via the canonical bundle formula and effective b-semiampleness (Theorem C); bound the polarization type via the Zarhin trick and a Kuga–Satake construction in families (Section 3); and reconstruct fibrations without rational sections from their Albanese fibrations via a refined Tate–Shafarevich theory that allows multiple fibers (Sections 5–7). The visible portion (roughly Sections 1 through 5.6) is carefully structured, with detailed proofs, explicit near-optimality examples (Examples 4.21–4.24), and clearly flagged conditional statements. The main unresolved point in the visible text is the proof of Proposition 5.16, which relies on an asserted but not written-out Čech complex computation; this computation is load-bearing for removing the rational-section hypothesis in Theorems A(AV) and B.

Significance. If the main theorems hold, they resolve open questions that have been inaccessible for some time: birational boundedness of abelian-fibered and K3-fibered Calabi–Yau 3-folds (Corollary 1.2) and, conditionally on the hyperkähler SYZ or generalized abundance conjecture, finiteness of deformation classes of hyperkähler varieties with b2 ≥ 5 (Corollary 1.3). The paper also contains substantial independent results: effective b-semiampleness for abelian and primitive symplectic fibrations (Theorem C), relative boundedness for abelian fibrations with rational sections (Theorem D), K-triviality of the relative Albanese fibration (Theorem E), and finiteness of π1 of the regular locus for Lagrangian-fibered irreducible symplectic varieties (Theorem F). The work is notable for its explicit accounting of external dependencies, its construction of the multiplicity group and translation sheaf from first principles, and its series of examples that test the limits of the hypotheses. The main caveat is that the central local computation in Proposition 5.16 is asserted rather than demonstrated, and the final sections of the proof were not available for verification in the version I reviewed.

major comments (2)
  1. [§5.2, Proposition 5.16] In the proof of the case z ∈ ∂_i ⊂ ∂^+, the text states twice that “a Čech complex computation verifies” that the relevant connecting homomorphism is identified with the nilpotent monodromy operator N_i, and that the bottom map H^1(U^+ ∩ ∂^+, μ(i)) → H^3(U^+, Γ)_tors is an isomorphism when codim_z(Z' \ Z^+) = 2. These identifications are load-bearing: they imply injectivity of the coboundary map H^1(U^+ ∩ ∂^+, μ(i)) → H^2_an(U^+, P^0) in the component sequence (26), which in turn is used to conclude t_{z,an}(f) = 0. Without this step, the big open set Z^+ cannot be shown to be independent of f, and the removal of the rational-section hypothesis in Theorems A(AV) and B loses its foundation. Please replace the two Čech computation sentences with a complete proof, or at minimum with a precise lemma statement that specifies the isomorphisms, the cocharacter construction, and the relevant sign and torsion conventions.
  2. [§5.2–§5.3, Remark 5.3 and Corollary 5.18] The independence of Z^+ from the individual fibration f, asserted in Remark 5.3 and used to prove Corollary 5.18 (and later the constructibility results of Section 7), rests entirely on the omitted Čech computation in Proposition 5.16. Since Corollary 5.18 is the bridge that upgrades the rational-section results of Section 4 to the full Theorems A(AV) and B, the gap is not a cosmetic one. The authors should provide a standalone proof of the two local identifications — the comparison of the component-group coboundary with the logarithmic monodromy N_i, and the comparison of the cocharacter map with the isomorphism H^1(U^+ ∩ ∂^+, μ(i)) ≃ H^3(U^+, Γ)_tors — with all intermediate isomorphisms written out. This is the kind of local statement where a sign error or a failure of the cocharacter to descend would change the conclusion, so the level of detail should match the rest of the paper.
minor comments (4)
  1. [§1.3, Outline] The outline refers to “Section 4.7 begins (Prop. 5.2)”, but Proposition 5.2 appears in Section 5.2; the cross-reference should be corrected.
  2. [§3.7, Proof of Theorem 3.42] The name “Hirzebuch–Mumford proportionality” should read “Hirzebruch–Mumford proportionality”.
  3. [§1.1, Previous work] The sentence “There are at least 30108 known distinct topological types” is unclear as printed; please insert the appropriate thousands separator or spacing.
  4. [§5.1, Construction 5.5] The sentence “By ensuring T_i is the bending locus of an appropriate convex PL function, we can assume that V → A_{g,Λ}[3] ∪_i R_{≥ 0}N_i is relatively projective” would benefit from a reference or a one-sentence justification, since relative projectivity of the Mumford construction is used implicitly later.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems reduce to external results and to a Tate–Shafarevich theory developed from definitions, with no prediction forced by construction.

full rationale

I walked the claimed derivation chain of Theorems A and B. The main inputs are: the canonical bundle formula (Theorem 2.16, cited to the standard literature); effective b-semiampleness (Theorem 3.42, proved internally from effective basepoint-freeness for lc pairs and the very-ampleness of the Hodge bundle on period domains); boundedness of rationally connected klt bases via [26, Thm. 1.5]; the Zarhin trick and Kuga-Satake construction as external tools; and a long Tate-Shafarevich analysis (Sections 5-7) developed from definitions, with the obstruction maps and exponential sequence set up explicitly. The one result with an overlapping author citation, [26, Thm. 1.5], is an external boundedness theorem for rationally connected klt pairs with torsion canonical class; it does not assume or encode the fibered Calabi-Yau boundedness being proved, so it is independent support rather than a circular input. The forward reference in Corollary 3.45 to Theorem B is conditional and is not used in the proof of Theorem B itself, so it does not create a circular loop. The compressed 'Cech complex computation' in Proposition 5.16 is a place where the proof is terse, and a verification gap there would affect the removal of the rational-section hypothesis; however, that is a correctness or rigor concern, not a circularity: the argument attempts to identify a coboundary with the nilpotent monodromy operator using independent monodromy and Mumford-construction data, and it does not rename the conclusion as an input. Similarly, the multiplicity group and Tate-Shafarevich groups are defined from the fibration and its Kulikov model, then used to classify twists; no parameter is fitted to the target boundedness statement and then called a prediction. I therefore find no step where a claimed output reduces by definition or by self-citation to an input, and no fitted-input-called-prediction pattern. The appropriate score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 3 invented entities

No free parameters: this is pure mathematics, and all constants in the paper (for instance I in Theorem C and the index bounds in Lemma 3.38) are structural bounds depending only on dimension, fiber type, or b2, with no data fitted. The axioms are standard theorems from the MMP and Hodge theory literature, plus two open conjectures that are explicitly labeled and used only in clearly conditional corollaries. The three listed invented entities are new mathematical constructions with rigorous definitions, exact sequences, and worked examples, rather than empirical postulates.

assumptions (8)
  • standard math Beauville-Bogomolov decomposition for compact Kähler spaces with klt singularities
    Theorem 2.2 provides the three building blocks (AV), (ICY), (SIS). Cited from [84, 55, 54, 92, 83, 101, 40, 19]; not proved in this paper.
  • standard math Canonical bundle formula for lc-trivial fibrations (Theorem 2.16)
    Central tool relating KX to f*(KY + BY + MY) on the base; established in [122, 123, 75, 114, 74, 10, 11, 73, 60, 12]. Used throughout Sections 3 to 8.
  • standard math MMP: existence of good minimal models and termination with scaling for klt/lc pairs
    Invoked in Proposition 2.14 to construct good models of lc-trivial fibrations and in Section 5 to run MMPs producing models with big open total spaces; theorems from [94, 136, 99].
  • standard math Effective basepoint-free theorem and Kodaira vanishing for lc varieties
    Used in the proof of Theorem 3.42 (Theorem C) to obtain effective very-ampleness of the Hodge bundle on the level-3 cover of the period space; cited from [70, Thm. 1.1] and [72, Thm. 1.1].
  • standard math Boundedness of rationally connected klt pairs with bounded coefficients ([26, Thm. 1.5])
    Used in Lemma 4.14 to conclude log boundedness in codimension 1 of the pairs (Y, BY + Gamma Y). Deep external theorem; one present author is a co-author of [26], but the result is independently established there.
  • standard math Surjectivity of the period map and local Torelli for primitive symplectic varieties
    Used in Corollary 3.45 (existence of a PS variety with a nef isotropic class) and in Theorem 4.10(PS) (birational PS varieties with the same periods); cited from [21, Thm. 1.1].
  • domain assumption Generalized abundance conjecture for symplectic varieties (Conjecture 2.6)
    Open conjecture, explicitly labeled. Used conditionally in Corollary 3.45 and Corollary 1.3 to derive effective b-semiampleness and finiteness of deformation classes with b2 >= 5. If false, those corollaries fail but Theorems A through D and F remain.
  • domain assumption Hyperkähler SYZ conjecture (Conjecture 2.7)
    Open conjecture, explicitly labeled. Used only for Corollary 1.3: finiteness of deformation classes with b2 >= 5 follows only if every such variety admits a Lagrangian fibration after a locally trivial deformation.
invented entities (3)
  • Multiplicity group XG (Definitions 5.20, 5.28) independent evidence
    purpose: Measures the multiple fibers of an abelian fibration in codimension 1 via the multiplicity class m(f); records the obstruction to f being a twist of its Albanese fibration and enables the removal of the rational-section hypothesis in Theorems A(AV) and B.
    New mathematical construction, not a physical entity. It is rigorously defined, sits in an exact sequence 0 -> X -> XG -> XG -> 0, and is computed explicitly in Examples 5.25 and 5.27, so internal logical evidence exists; the empirical notion of independent evidence does not directly apply to pure mathematics.
  • Translation sheaf P (Definition 5.9) independent evidence
    purpose: Sheaf of translation-birational automorphisms of the normalized base change of an abelian fibration over étale charts; its first cohomology defines the Tate-Shafarevich groups XG = H1_G(Z+, P).
    Technical object defined and used within the proof system; identified with sections of the smooth locus of the Kulikov model, giving it a concrete geometric meaning and enabling the explicit computations in Examples 5.25 and 5.31.
  • Secondary Kuga-Satake variety (Definition 3.36) independent evidence
    purpose: Restores an integral action of Pin(M) on the Kuga-Satake torus so the Kuga-Satake construction can be performed in families over the period domain, a load-bearing step for Theorem A(PS).
    New construction refining the classical Kuga-Satake variety; the paper proves an isogeny to the primary Kuga-Satake variety and shows the polarization descends (Lemma 3.37), giving internal logical evidence consistent with the paper's explicit handling of the orthogonal-action subtlety in Remark 3.39.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Boundedness of some fibered K-trivial varieties." pith.science (2026). https://pith.science/paper/M7PDFSWS

@misc{pith2026250700973,
  author       = {Pith},
  title        = {Pith review of: Boundedness of some fibered K-trivial varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7PDFSWS}},
  note         = {Machine review of arXiv:2507.00973}
}
abstract

We prove that irreducible Calabi-Yau varieties of a fixed dimension, admitting a fibration by abelian varieties or primitive symplectic varieties of a fixed analytic deformation class, are birationally bounded. We prove that there are only finitely many deformation classes of primitive symplectic varieties of a fixed dimension, admitting a Lagrangian fibration. We also show that fibered Calabi-Yau 3-folds are bounded. Conditional on the generalized abundance or hyperk\"ahler SYZ conjecture, our results prove that there are only finitely many deformation classes of hyperk\"ahler varieties, of a fixed dimension, with $b_2 \geq 5$.

Figures

Figures reproduced from arXiv: 2507.00973 by the authors.

Figure 1
Figure 1. Right: Abelian surface fibration f + : X+ → Y + over a curve, with 4 singular fibers (multiplicities in red), over points p1, p2, p3, p4. B = 5 6 [p1] + 1 6 [p2]+2 3 [p3]+1 2 [p4]. The G-cover Z + → Y + is a cyclic 6-to-1 cover, branched over {p1, p2, p3, p4}. Left: Normalized base change g + : W+ → Z +. Analytic local sections of g + : W+ → Z + in purple. Above: Kulikov model h + : V + → Z + of (W+) Alb. Global sec… view at source ↗
Figure 2
Figure 2. Left and Right: Mumford constructions over ∂1 and ∂4. Mon￾odromy matrices: M1 = diag(2, 2) and M4 = (3). Standard simplicial tilings: T1 and T4 of R 2/Z(2, 0) ⊕ Z(0, 2) and R/Z(3), respectively [PITH_FULL_IMAGE:figures/full_fig_p062_2.png] view at source ↗
Figure 3
Figure 3. Symplectic resolution XAlb → XAlb of the Albanese fibration of a Type II(5) Hwang–Oguiso fiber, resolving two A4 singularities (red) on the original multiplicity 5 component (blue). space of family over a (g − 1)-dimensional base of cuspidal curve bundles over abelian (g − 1)- folds. In appropriate analytic-local coordinates, the C5-action is the same as the model above (up to increasing the dimension of the base an… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Symplectic resolution XAlb → XAlb of the Albanese fibration of a Type E△(6) Hwang–Oguiso fiber, resolving A1, A2, A5 singularities (red) on the original multiplicity 6 component (blue). with α6 a nontrivial order 6 torsion point on E. Then f : X → Y is W/C6 → Z/C6. A m…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the boundedness of elliptic Calabi-Yau 4-folds

    math.AG 2026-07 accept novelty 6.0 of 10

    Elliptic Calabi–Yau 4-folds not crepant to a product quotient of a Calabi–Yau 3-fold times an elliptic curve form a bounded family.

  2. A Picard rank bound for base surfaces of elliptic Calabi-Yau 3-folds

    hep-th 2025-07 conditional novelty 6.0 of 10

    The Picard rank of any rational base surface of an elliptic Calabi-Yau 3-fold (with the relevant 1/6-lc condition) is at most 568.

Reference graph

Works this paper leans on

203 extracted references · 62 canonical work pages · cited by 2 Pith papers

  1. [1]

    Abasheva, Shafarevich–Tate groups of holomorphic Lagrangian fibrations II , arXiv preprint, arXiv:2407.09178 (2024)

    A. Abasheva, Shafarevich–Tate groups of holomorphic Lagrangian fibrations II , arXiv preprint, arXiv:2407.09178 (2024)

    arXiv 2024

  2. [2]

    Abasheva and V

    A. Abasheva and V. Rogov,Shafarevich–Tate groups of holomorphic Lagrangian fibrations, Math. Z.311 (2025), no. 1, Paper No. 4

    2025

  3. [3]

    Abramovich, M

    D. Abramovich, M. Olsson, and A. Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008) 1057–1091

    2008

  4. [4]

    Abramovich and A

    D. Abramovich and A. Vistoli,Compactifying the space of stable maps, J. Amer. Math. Soc.15 (2002), no. 1, 27–75

    2002

  5. [5]

    V.Alexeev, Complete moduli in the presence of semiabelian group action, Ann.ofMath.(2)(2002)611–708

    2002

  6. [6]

    ———, Root systems and hyperkähler varieties, arXiv preprint arXiv:2206.14070 (2022)

    work page Pith review arXiv 2022

  7. [7]

    Alexeev and P

    V. Alexeev and P. Engel,Compact moduli of K3 surfaces, Ann. of Math. (2)198 (2023), no. 2, 727–789

    2023

  8. [8]

    BOUNDEDNESS OF SOME FIBERED K-TRIVIAL V ARIETIES 121

    ———, On lattice-polarized K3 surfaces, arXiv preprint, arXiv:2505.22557 (2025) . BOUNDEDNESS OF SOME FIBERED K-TRIVIAL V ARIETIES 121

    arXiv 2025

Show all 203 references
  1. [9]

    Alexeev and I

    V. Alexeev and I. Nakamura,On Mumford’s construction of degenerating abelian varieties, Tohoku Math. J. (2) 51 (1999), no. 3, 399 – 420

    1999

  2. [10]

    Ambro,Shokurov’s boundary property, J

    F. Ambro,Shokurov’s boundary property, J. Differential Geom.67 (2004), no. 2, 229–255

    2004

  3. [11]

    Math.141 (2005), no

    ———, The moduli b-divisor of an lc-trivial fibration, Compos. Math.141 (2005), no. 2, 385–403

    2005

  4. [12]

    Ambro, P

    F. Ambro, P. Cascini, V. V. Shokurov, and C. Spicer, Positivity of the Moduli Part, arXiv preprint, arXiv:2111.00423 (2021)

    2021 arXiv

  5. [13]

    Artin,Algebraic approximation of structures over complete local rings, Publ

    M. Artin,Algebraic approximation of structures over complete local rings, Publ. Math. Inst. Hautes Études Sci. 36 (1969) 23–58

    1969

  6. [14]

    I, in Global Analysis (Papers in Honor of K

    ———, Algebraization of formal moduli. I, in Global Analysis (Papers in Honor of K. Kodaira), 21–71, Univ. Tokyo Press, Tokyo (1969)

    1969

  7. [15]

    Existence of modifications, Ann

    ———, Algebraization of formal moduli: II. Existence of modifications, Ann. of Math. (2)91 (1970), no. 1, 88–135

    1970

  8. [16]

    Ascher, D

    K. Ascher, D. Bejleri, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, and X. Wang,Moduli of boundary polarized Calabi–Yau pairs, arXiv preprint, arXiv:2307.06522 (2023)

    2023 arXiv

  9. [17]

    A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai, Smooth compactifications of locally symmetric varieties, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second edition (2010), ISBN 978-0-521-73955-9. With the collaboration of Peter Scholze

    2010

  10. [18]

    W. L. Baily, Jr. and A. Borel,Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2)84 (1966) 442–528

    1966

  11. [19]

    Bakker, H

    B. Bakker, H. Guenancia, and C. Lehn,Algebraic approximation and the decomposition theorem for Kähler Calabi–Yau varieties, Invent. Math.228 (2022), no. 3, 1255–1308

    2022

  12. [20]

    Bakker and C

    B. Bakker and C. Lehn,A global Torelli theorem for singular symplectic varieties, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 3, 949–994

    2021

  13. [21]

    Reine Angew

    ———, The global moduli theory of symplectic varieties, J. Reine Angew. Math.790 (2022) 223–265

    2022

  14. [22]

    Barlet and J

    D. Barlet and J. Varouchas,Fonctions holomorphes sur l’espace des cycles, Bull. Soc. Math. France117 (1989), no. 3, 327–341

    1989

  15. [23]

    Barth, K

    W. Barth, K. Hulek, C. Peters, and A. van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Springer Berlin Heidelberg (2015), ISBN 9783642577390

    2015

  16. [24]

    Beauville,Variétés Kähleriennes dont la première classe de Chern est nulle, J

    A. Beauville,Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom.18 (1983), no. 4, 755–782 (1984)

    1983

  17. [25]

    Birkar,Existence of log canonical flips and a special LMMP, Publ

    C. Birkar,Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci.115 (2012) 325–368

    2012

  18. [26]

    Birkar, G

    C. Birkar, G. Di Cerbo, and R. Svaldi,Boundedness of elliptic Calabi–Yau varieties with a rational section, J. Differential Geom.128 (2024), no. 2, 463–519

    2024

  19. [27]

    Birkar and D.-Q

    C. Birkar and D.-Q. Zhang,Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. Inst. Hautes Études Sci.123 (2016) 283–331

    2016

  20. [28]

    Birkenhake and H

    C. Birkenhake and H. Lange, Complex abelian varieties, Vol. 302 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition (2004), ISBN 3-540-20488-1

    2004

  21. [29]

    Bogomolov, L

    F. Bogomolov, L. Kamenova, and M. Verbitsky,Sections of Lagrangian fibrations on holomorphic symplec- tic manifolds, arXiv preprint, arXiv:2407.07877 (2024)

    2024 arXiv

  22. [30]

    Boissière, M

    S. Boissière, M. Nieper-Wißkirchen, and A. Sarti,Higher dimensional Enriques varieties and automor- phisms of generalized Kummer varieties, J. Math. Pures Appl. (9)95 (2011), no. 5, 553–563

    2011

  23. [31]

    Borel,Arithmetic properties of linear algebraic groups, in Proc

    A. Borel,Arithmetic properties of linear algebraic groups, in Proc. I.C.M. Stockholm, 10–22 (1962). 122 ENGEL, FILIPAZZI, GREER, MAURI, AND SV ALDI

    1962

  24. [32]

    Actualités Scientifiques et Industrielles, No

    ———, Introductionauxgroupesarithmétiques, Publicationsdel’InstitutdeMathématiquedel’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris (1969)

    1969

  25. [33]

    Differential Geom

    ———, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geom. 6 (1972), no. 4, 543–560

    1972

  26. [34]

    Braun,The local fundamental group of a Kawamata log terminal singularity is finite, Invent

    L. Braun,The local fundamental group of a Kawamata log terminal singularity is finite, Invent. Math.226 (2021), no. 3, 845–896

    2021

  27. [35]

    Braun and F

    L. Braun and F. Figueroa,Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs, arXiv preprint, arXiv:2401.01315 (2024)

    2024 arXiv

  28. [36]

    Campana, Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J

    F. Campana, Fundamental group and positivity of cotangent bundles of compact Kähler manifolds, J. Algebraic Geom. 4 (1995), no. 3, 487–502

    1995

  29. [37]

    Torino, Turin (2004)

    ———, Orbifoldes à première classe de Chern nulle, in The Fano Conference, 339–351, Univ. Torino, Turin (2004)

    2004

  30. [38]

    ———, Isotrivialité de certaines familles kählériennes de variétés non projectives, Math. Z. 252 (2006), no. 1, 147–156

    2006

  31. [39]

    ———, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes com- pactes, J. Inst. Math. Jussieu10 (2011), no. 4, 809–934

    2011

  32. [40]

    ———, The Bogomolov–Beauville–Yau decomposition for KLT projective varieties with trivial first Chern class—without tears, Bull. Soc. Math. France149 (2021), no. 1, 1–13

    2021

  33. [41]

    Campana, K

    F. Campana, K. Oguiso, and T. Peternell,Non-algebraic hyperkähler manifolds, J. Differential Geom.85 (2010), no. 3, 397–424

    2010

  34. [42]

    Carlson, S

    J. Carlson, S. Müller-Stach, and C. Peters, Period Mappings and Period Domains, Cambridge Studies in Advanced Mathematics, Cambridge University Press (2003), ISBN 9780521814669

    2003

  35. [43]

    Chandra, A

    A. Chandra, A. Constantin, C. S. Fraser-Taliente, T. R. Harvey, and A. Lukas,Enumerating Calabi– Yau Manifolds: Placing Bounds on the Number of Diffeomorphism Classes in the Kreuzer-Skarke List, Fortschritte der Physik72 (2024), no. 5, 2300264

    2024

  36. [44]

    Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K3 surfaces, and the Tate conjecture, Ann

    F. Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K3 surfaces, and the Tate conjecture, Ann. of Math. (2) (2016) 487–526

    2016

  37. [45]

    Conrad, M

    B. Conrad, M. Lieblich, and M. Olsson,Nagata compactification for algebraic spaces, J. Inst. Math. Jussieu 11 (2012), no. 4, 747–814

    2012

  38. [46]

    Debarre, Tores et variétés abéliennes complexes, Société mathématique de France (1999)

    O. Debarre, Tores et variétés abéliennes complexes, Société mathématique de France (1999)

    1999

  39. [47]

    ———, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York (2001), ISBN 0-387-95227-6

    2001

  40. [48]

    Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ

    P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. Inst. Hautes Études Sci. (1968), no. 35, 259–278

    1968

  41. [49]

    163 ofLecture Notes in Mathematics, Springer-Verlag (1970)

    ———, Equations Differentielles a Points Singuliers Reguliers, Vol. 163 ofLecture Notes in Mathematics, Springer-Verlag (1970)

    1970

  42. [50]

    ———, Théorie de Hodge: III, Publ. Math. Inst. Hautes Études Sci.44 (1974) 5–77

    1974

  43. [51]

    ———, Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 1-19 (1987)

    1987

  44. [52]

    Dolgachev and M

    I. Dolgachev and M. Gross,Elliptic three-folds I: Ogg-Shafarevich theory, arXiv preprint, alg-geom/9210009 (1992)

    1992 arXiv

  45. [53]

    Douady, Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, in Contemporary Problems in Theory Anal

    A. Douady, Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, in Contemporary Problems in Theory Anal. Functions (Internat. Conf., Erevan, 1965) (Russian), 141–143, Izdat. “Nauka”, Moscow (1966). BOUNDEDNESS OF SOME FIBERED K-TRIVI...

    1966

  46. [54]

    Druel, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent

    S. Druel, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math.211 (2018), no. 1, 245–296

    2018

  47. [55]

    Druel and H

    S. Druel and H. Guenancia,A decomposition theorem for smoothable varieties with trivial canonical class, J. Éc. polytech. Math.5 (2018) 117–147

    2018

  48. [56]

    A. H. Durfee,Intersection homology Betti numbers, Proceedings of the American Mathematical Society 123 (1995), no. 4, 989–993

    1995

  49. [57]

    Dutta, D

    Y. Dutta, D. Mattei, and E. Shinder, Twists of intermediate Jacobian fibrations , arXiv preprint, arXiv:2411.01953 (2024)

    2024 arXiv

  50. [58]

    Elkik,Rationalité des singularités canoniques, Invent

    R. Elkik,Rationalité des singularités canoniques, Invent. Math.64 (1981), no. 1, 1–6

    1981

  51. [59]

    Faltings,Endlichkeitssätze für abelsche Varietäten über Zahlkörpern., Invent

    G. Faltings,Endlichkeitssätze für abelsche Varietäten über Zahlkörpern., Invent. Math.73 (1983) 349–366

    1983

  52. [60]

    Filipazzi,On a generalized canonical bundle formula and generalized adjunction, Ann

    S. Filipazzi,On a generalized canonical bundle formula and generalized adjunction, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)21 (2020) 1187–1221

    2020

  53. [61]

    Geom.11 (2024), no

    ———, On the boundedness ofn-folds with κ(X) = n − 1, Algebr. Geom.11 (2024), no. 3, 318–345

    2024

  54. [62]

    Filipazzi, C

    S. Filipazzi, C. D. Hacon, and R. Svaldi,Boundedness of elliptic Calabi–Yau threefolds, J. Eur. Math. Soc. (2024) . In print

    2024

  55. [63]

    Filipazzi and J

    S. Filipazzi and J. Moraga,Strong (δ, n)-complements for semi-stable morphisms, Doc. Math. 25 (2020) 1953–1996

    2020

  56. [64]

    Filipazzi and R

    S. Filipazzi and R. Svaldi,Invariance of plurigenera and boundedness for generalized pairs, Mat. Contemp. 47 (2020) 114–150

    2020

  57. [65]

    Sigma11 (2023) Paper No

    ———, On the connectedness principle and dual complexes for generalized pairs, Forum Math. Sigma11 (2023) Paper No. e33, 39

    2023

  58. [66]

    Friedman,On threefolds with trivial canonical bundle, Complex geometry and Lie theory (Sundance, UT, 1989) 53 (1991) 103–134

    R. Friedman,On threefolds with trivial canonical bundle, Complex geometry and Lie theory (Sundance, UT, 1989) 53 (1991) 103–134

    1991

  59. [67]

    L. Fu, Z. Li, T. Takamatsu, and H. Zou,Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture, arXiv preprint, arXiv:2505.15295 (2025)

    2025 arXiv

  60. [68]

    Fu and G

    L. Fu and G. Menet,On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four, Math. Z. 299 (2021), no. 1-2, 203–231

    2021

  61. [69]

    Fujino,A canonical bundle formula for certain algebraic fiber spaces and its applications, Nagoya Math

    O. Fujino,A canonical bundle formula for certain algebraic fiber spaces and its applications, Nagoya Math. J. 172 (2003) 129–171

    2003

  62. [70]

    ———, Effective base point free theorem for log canonical pairs—Kollár type theorem, Tohoku Math. J. (2) 61 (2009), no. 4, 475–481

    2009

  63. [71]

    ———, On Kawamata’s theorem, in Classification of algebraic varieties, EMS Ser. Congr. Rep., 305–315, Eur. Math. Soc., Zürich (2011)

    2011

  64. [72]

    Japan Acad

    ———, Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs, Proc. Japan Acad. Ser. A Math. Sci.91 (2015), no. 8, 112–117

    2015

  65. [73]

    Fujino and Y

    O. Fujino and Y. Gongyo,On the moduli b-divisors of lc-trivial fibrations, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 4, 1721–1735

    2014

  66. [74]

    Fujino and S

    O. Fujino and S. Mori,A canonical bundle formula, J. Differential Geom.56 (2000), no. 1, 167–188

    2000

  67. [75]

    Fujita,Zariski decomposition and canonical rings of elliptic threefolds, J

    T. Fujita,Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan38 (1986), no. 1, 19–37

    1986

  68. [76]

    Fulger, J

    M. Fulger, J. Kollár, and B. Lehmann,Volume and Hilbert functions ofR-divisors, Mich. Math. J. 65 (2016), no. 2, 371–387

    2016

  69. [77]

    Gachet,Well-clipped cones behave themselves under all finite quotients, the cone conjecture under most, arXiv preprint, arXiv:2504.01753 (2025)

    C. Gachet,Well-clipped cones behave themselves under all finite quotients, the cone conjecture under most, arXiv preprint, arXiv:2504.01753 (2025) . 124 ENGEL, FILIPAZZI, GREER, MAURI, AND SV ALDI

    2025 arXiv

  70. [78]

    Graber, J

    T. Graber, J. Harris, and J. Starr,Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67

    2003

  71. [79]

    Grassi,On minimal models of elliptic threefolds, Math

    A. Grassi,On minimal models of elliptic threefolds, Math. Ann.290 (1991), no. 2, 287–301

    1991

  72. [80]

    Grassi and D

    A. Grassi and D. Wen,Higher dimensional elliptic fibrations and Zariski decompositions, Commun. Con- temp. Math. 24 (2022), no. 04, 2150024

    2022

  73. [81]

    Grauert,Über Modifikationen und exzeptionelle analytische Mengen, Math

    H. Grauert,Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann.146 (1962) 331–368

    1962

  74. [82]

    Grauert, T

    H. Grauert, T. Peternell, and R. Remmert, editors, Several complex variables. VII, Vol. 74 ofEncyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin (1994), ISBN 3-540-56259-1

    1994

  75. [83]

    D. Greb, H. Guenancia, and S. Kebekus,Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups, Geom. Topol.23 (2019), no. 4, 2051–2124

    2019

  76. [84]

    D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell,Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. (2011), no. 114, 87–169

    2011

  77. [85]

    D. Greb, S. Kebekus, and T. Peternell,Singular spaces with trivial canonical class, in Minimal models and extremal rays (Kyoto, 2011), Vol. 70 ofAdv. Stud. Pure Math., 67–113, Math. Soc. Japan, [Tokyo] (2016)

    2016

  78. [86]

    D. Greb, C. Lehn, and S. Rollenske,Lagrangian fibrations on hyperkähler manifolds—on a question of Beauville, Ann. Sci. Éc. Norm. Supér. (4)46 (2013), no. 3, 375–403 (2013)

    2013

  79. [87]

    P. A. Griffiths,Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping), Publications Mathématiques de l’IHÉS38 (1970) 125–180

    1970

  80. [88]

    Gross,A finiteness theorem for elliptic Calabi–Yau threefolds, Duke Math

    M. Gross,A finiteness theorem for elliptic Calabi–Yau threefolds, Duke Math. J.74 (1994), no. 2, 271–299

    1994

  81. [89]

    9, 3409–3468

    ———, Elliptic three-folds II: Multiple fibres, Transactions of the American Mathematical Society349 (1997), no. 9, 3409–3468

    1997

  82. [90]

    Grothendieck and J.-L

    A. Grothendieck and J.-L. Verdier,Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics (1972)

    1972

  83. [91]

    Guan,On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four, Math

    D. Guan,On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four, Math. Res. Lett.8 (2001), no. 5, 663–669

    2001

  84. [92]

    Guenancia, Semistability of the tangent sheaf of singular varieties, Algebr

    H. Guenancia, Semistability of the tangent sheaf of singular varieties, Algebr. Geom. 3 (2016), no. 5, 508–542

    2016

  85. [93]

    C. D. Hacon, J. McKernan, and C. Xu,Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 4, 865–901

    2018

  86. [94]

    C. D. Hacon and C. Xu,Existence of log canonical closures, Invent. Math.192 (2013), no. 1, 161–195

    2013

  87. [95]

    ———, Boundedness of log Calabi–Yau pairs of Fano type, Math. Res. Lett.22 (2015), no. 6, 1699–1716

    2015

  88. [96]

    L. H. Halle and J. Nicaise,Motivic zeta functions of degenerating Calabi–Yau varieties, Math. Ann.370 (2018) 1277–1320

    2018

  89. [97]

    Han and C

    J. Han and C. Jiang,Birational boundedness of rationally connected log Calabi–Yau pairs with fixed index, Algebr. Geom. Phys.1 (2024), no. 1, 59–79

    2024

  90. [98]

    Hartshorne,Stable reflexive sheaves, Math

    R. Hartshorne,Stable reflexive sheaves, Math. Ann.254 (1980), no. 2, 121–176

    1980

  91. [99]

    Hashizume and Z.-Y

    K. Hashizume and Z.-Y. Hu,On minimal model theory for log abundant lc pairs, J. Reine Angew. Math. 767 (2020) 109–159

    2020

  92. [100]

    Hironaka,Flattening theorem in complex-analytic geometry, Amer

    H. Hironaka,Flattening theorem in complex-analytic geometry, Amer. J. Math.97 (1975) 503–547

    1975

  93. [101]

    Höring and T

    A. Höring and T. Peternell,Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math.216 (2019), no. 2, 395–419

    2019

  94. [102]

    Hunt,A bound on the Euler number for certain Calabi-Yau3-folds, J

    B. Hunt,A bound on the Euler number for certain Calabi-Yau3-folds, J. Reine Angew. Math.411 (1990) 137–170

    1990

  95. [103]

    Huybrechts,Finiteness results for compact hyperkähler manifolds, J

    D. Huybrechts,Finiteness results for compact hyperkähler manifolds, J. Reine Angew. Math.558 (2003) 15–22. BOUNDEDNESS OF SOME FIBERED K-TRIVIAL V ARIETIES 125

    2003

  96. [104]

    158 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2016), ISBN 978-1-107-15304-2

    ———, Lectures on K3 surfaces, Vol. 158 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2016), ISBN 978-1-107-15304-2

    2016

  97. [105]

    Hwang and K

    J.-M. Hwang and K. Oguiso,Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration, Amer. J. Math.131 (2009), no. 4, 981–1007

    2009

  98. [106]

    ———, Multiple fibers of holomorphic Lagrangian fibrations, Commun. Contemp. Math.13 (2011), no. 02, 309–329

    2011

  99. [107]

    Ivashkovich,The Hartogs-type extension theorem for meromorphic maps intor compact Kähler mani- folds., Invent

    S. Ivashkovich,The Hartogs-type extension theorem for meromorphic maps intor compact Kähler mani- folds., Invent. Math.109 (1992), no. 1, 47–54

    1992

  100. [108]

    Javanpeykar and D

    A. Javanpeykar and D. Litt,Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields, Manuscripta Math.173 (2024), no. 1, 23–44

    2024

  101. [109]

    Jejjala, W

    V. Jejjala, W. Taylor, and A. Turner,Identifying equivalent Calabi–Yau topologies: A discrete challenge from math and physics for machine learning, arXiv preprint, arXiv:2202.07590v1 (2022)

    2022 arXiv

  102. [110]

    M. J. Jeon, Resolution of indeterminacy of rational maps to proper tame stacks , arXiv preprint, arXiv:2506.14969v1 (2025)

    2025

  103. [111]

    Kamenova, Survey of finiteness results for hyperkähler manifolds, in Phenomenological approach to algebraic geometry, Vol

    L. Kamenova, Survey of finiteness results for hyperkähler manifolds, in Phenomenological approach to algebraic geometry, Vol. 116 ofBanach Center Publ., 77–86, Polish Acad. Sci. Inst. Math., Warsaw (2018)

    2018

  104. [112]

    ———, Finiteness of stable Lagrangian fibrations, São Paulo J. Math. Sci.18 (2024), no. 2, 801–806

    2024

  105. [113]

    Kamenova and C

    L. Kamenova and C. Lehn, Non-hyperbolicity of holomorphic symplectic varieties , arXiv preprint, arXiv:2212.11411 (2022)

    2022 arXiv

  106. [114]

    Kawamata,Subadjunction of log canonical divisors

    Y. Kawamata,Subadjunction of log canonical divisors. II, Amer. J. Math.120 (1998), no. 5, 893–899

    1998

  107. [115]

    ———, Flops connect minimal models, Publ. Res. Inst. Math. Sci.44 (2008), no. 2, 419–423

    2008

  108. [116]

    Kebekus and C

    S. Kebekus and C. Schnell,Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc.34 (2021), no. 2, 315–368

    2021

  109. [117]

    Kerr and G

    M. Kerr and G. Pearlstein,Boundary components of Mumford–Tate domains, Duke Math. J.165 (2016), no. 4, 661 – 721

    2016

  110. [118]

    Kim, A Remark on Fujino’s work on the canonical bundle formula via period maps, arXiv preprint, arXiv:2405.05489 (2024)

    H. Kim, A Remark on Fujino’s work on the canonical bundle formula via period maps, arXiv preprint, arXiv:2405.05489 (2024)

    2024 arXiv

  111. [119]

    Kim, The Néron model of a higher-dimensional Lagrangian fibration , arXiv preprint, arXiv:2410.21193 (2024)

    Y.-J. Kim, The Néron model of a higher-dimensional Lagrangian fibration , arXiv preprint, arXiv:2410.21193 (2024)

    2024 arXiv

  112. [120]

    Kim and R

    Y.-J. Kim and R. Laza,A conjectural bound on the second Betti number for hyper-Kähler manifolds, Bull. Soc. Math. France148 (2020), no. 3, 467–480

    2020

  113. [121]

    Kirschner, Period Mappings with Applications to Symplectic Complex Spaces, Lecture Notes in Math- ematics, 2140, Springer International Publishing, Cham, 1st ed

    T. Kirschner, Period Mappings with Applications to Symplectic Complex Spaces, Lecture Notes in Math- ematics, 2140, Springer International Publishing, Cham, 1st ed. 2015. edition (2015), ISBN 3-319-17521-1

    2015

  114. [122]

    Kodaira,On the structure of compact complex analytic surfaces

    K. Kodaira,On the structure of compact complex analytic surfaces. II, Amer. J. Math.88 (1966) 682–721

    1966

  115. [123]

    III, Amer

    ———, On the structure of compact complex analytic surfaces. III, Amer. J. Math.90 (1968) 55–83

    1968

  116. [124]

    Kollár,Higher direct images of dualizing sheaves

    J. Kollár,Higher direct images of dualizing sheaves. I, Ann. of Math. (2)123 (1986), no. 1, 11–42

    1986

  117. [125]

    ———, Higher direct images of dualizing sheaves. II, Ann. of Math. (2)124 (1986), no. 1, 171–202

    1986

  118. [126]

    35 of Oxford Lecture Ser

    ———, Kodaira’s canonical bundle formula and adjunction, in Flips for 3-folds and 4-folds, Vol. 35 of Oxford Lecture Ser. Math. Appl., 134–162, Oxford Univ. Press, Oxford (2007)

    2007

  119. [127]

    200 ofCambridge Tracts in Mathematics, Cam- bridge University Press, Cambridge (2013)

    ———, Singularities of the minimal model program, Vol. 200 ofCambridge Tracts in Mathematics, Cam- bridge University Press, Cambridge (2013)

    2013

  120. [128]

    ———, Abelian fiber spaces and their Tate-Shafarevich twists with sections , arXiv preprint, arXiv:2504.21705 (2025)

    2025 arXiv

  121. [129]

    126 ENGEL, FILIPAZZI, GREER, MAURI, AND SV ALDI

    ———, Néron models, minimal models, and birational group actions, arXiv preprint, arXiv:2502.13800 (2025) . 126 ENGEL, FILIPAZZI, GREER, MAURI, AND SV ALDI

    2025 arXiv

  122. [130]

    Kollár and M

    J. Kollár and M. Larsen,Quotients of Calabi–Yau varieties, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Vol. 270 ofProgr. Math., 179–211, Birkhäuser Boston, Boston, MA (2009), ISBN 978-0-8176-4746-9

    2009

  123. [131]

    Kollár, R

    J. Kollár, R. Laza, G. Saccà, and C. Voisin,Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier (Grenoble)68 (2018), no. 7, 2837–2882

    2018

  124. [132]

    Kollár and S

    J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703

    1992

  125. [133]

    134 ofCambridge Tracts in Mathematics, Cam- bridge University Press, Cambridge (1998), ISBN 0-521-63277-3

    ———, Birational geometry of algebraic varieties, Vol. 134 ofCambridge Tracts in Mathematics, Cam- bridge University Press, Cambridge (1998), ISBN 0-521-63277-3. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

    1998

  126. [134]

    Kreuzer and H

    M. Kreuzer and H. Skarke,Calabi–Yau data, online website, http://hep.itp.tuwien.ac.at/~kreuzer/ CY/

  127. [135]

    Kurnosov,On an inequality for Betti numbers of hyper-Kähler manifolds of dimension six, Mat

    N. Kurnosov,On an inequality for Betti numbers of hyper-Kähler manifolds of dimension six, Mat. Zametki 99 (2016), no. 2, 309–313

    2016

  128. [136]

    Lai,Varieties fibered by good minimal models, Math

    C.-J. Lai,Varieties fibered by good minimal models, Math. Ann.350 (2011), no. 3, 533–547

    2011

  129. [137]

    Lazarsfeld, Positivity in algebraic geometry

    R. Lazarsfeld, Positivity in algebraic geometry. I, Vol. 48 ofErgebnisse der Mathematik und ihrer Grenzge- biete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer-Ver...

    2004

  130. [138]

    Lazić and T

    V. Lazić and T. Peternell, On generalised abundance, I, Publ. Res. Inst. Math. Sci. 56 (2020), no. 2, 353–389

    2020

  131. [139]

    Lehn,Normal crossing singularities and Hodge theory over Artin rings, Asian J

    C. Lehn,Normal crossing singularities and Hodge theory over Artin rings, Asian J. Math.19 (2015), no. 2, 235–250

    2015

  132. [140]

    ———, Deformations of Lagrangian subvarieties of holomorphic symplectic manifolds, Math. Res. Lett. 23 (2016), no. 2, 473–497

    2016

  133. [141]

    C. Lehn, G. Mongardi, and G. Pacienza,The Morrison-Kawamata cone conjecture for singular symplectic varieties, Selecta Math. (N.S.)30 (2024), no. 4, Paper No. 79, 36

    2024

  134. [142]

    Li,On the relative Morrison–Kawamata cone conjecture (II), arXiv preprint, arXiv:2309.04673 (2023)

    Z. Li,On the relative Morrison–Kawamata cone conjecture (II), arXiv preprint, arXiv:2309.04673 (2023)

    2023 arXiv

  135. [143]

    Li and H

    Z. Li and H. Zhao, On the relative Morrison-Kawamata cone conjecture , arXiv preprint, arXiv:2206.13701v5 (2022)

    2022

  136. [144]

    Y. Liu, Z. Liu, and C. Xu,Irreducible symplectic varieties with a large second Betti number, arXiv preprint, arXiv:2410.01566 (2024)

    2024 arXiv

  137. [145]

    Markman,Modular Galois covers associated to symplectic resolutions of singularities, J

    E. Markman,Modular Galois covers associated to symplectic resolutions of singularities, J. Reine Angew. Math. 644 (2010) 189–220

    2010

  138. [146]

    ———, Lagrangian fibrations of holomorphic-symplectic varieties of K 3 [n]-type, inAlgebraicandComplex Geometry: In Honour of Klaus Hulek’s 60th Birthday, 241–283, Springer (2014)

    2014

  139. [147]

    Martinelli, S

    D. Martinelli, S. Schreieder, and L. Tasin,On the number and boundedness of log minimal models of general type, Ann. Sci. Éc. Norm. Supér. (4)53 (2020), no. 5, 1183–1207

    2020

  140. [148]

    Matsusaka,Polarized varieties with a given Hilbert polynomial, Amer

    T. Matsusaka,Polarized varieties with a given Hilbert polynomial, Amer. J. Math.94 (1972), no. 4, 1027– 1077

    1972

  141. [149]

    J.104 (1986) 175–211

    ———, On polarized normal varieties, I, Nagoya Math. J.104 (1986) 175–211

    1986

  142. [150]

    1, 79–83

    D.Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology38(1999), no. 1, 79–83

    1999

  143. [151]

    ———, Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds, Math. Res. Lett. 7 (2000), no. 4, 389–391. BOUNDEDNESS OF SOME FIBERED K-TRIVIAL V ARIETIES 127

    2000

  144. [152]

    China Math.58 (2015), no

    ———, On base manifolds of Lagrangian fibrations, Sci. China Math.58 (2015), no. 3, 531–542

    2015

  145. [153]

    315 ofProgr

    ———, On deformations of Lagrangian fibrations, in K3 surfaces and their moduli, Vol. 315 ofProgr. Math., 237–243, Birkhäuser/Springer, [Cham] (2016)

    2016

  146. [154]

    Meyer,Mathematische Mittheilungen, Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich29 (1884) 209–222

    A. Meyer,Mathematische Mittheilungen, Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich29 (1884) 209–222

  147. [155]

    Moraga and T

    J. Moraga and T. Stark, The geometric cone conjecture in relative dimension two, arXiv preprint, arXiv:2409.13068 (2024)

    2024 arXiv

  148. [156]

    Morrison,The Clemens–Schmid exact sequence and applications, in Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Vol

    D. Morrison,The Clemens–Schmid exact sequence and applications, in Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Vol. 106 ofAnn. of Math. Stud., 101–119, Princeton Univ. Press, Princeton, NJ (1984)

    1984

  149. [157]

    Mumford, Abelian varieties, Vol

    D. Mumford, Abelian varieties, Vol. 5 ofTata Institute of Fundamental Research Studies in Mathematics, Tata Institute of Fundamental Research, Bombay; by Oxford University Press, London (1970)

    1970

  150. [158]

    Math.24 (1972), no

    ———, An analytic construction of degenerating abelian varieties over complete rings, Compos. Math.24 (1972), no. 3, 239–272

    1972

  151. [159]

    Math.42 (1977) 239–272

    ———, Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math.42 (1977) 239–272

    1977

  152. [160]

    Nakayama,The lower semicontinuity of the plurigenera of complex varieties, in Algebraic geometry, Sendai, 1985, Vol

    N. Nakayama,The lower semicontinuity of the plurigenera of complex varieties, in Algebraic geometry, Sendai, 1985, Vol. 10 ofAdv. Stud. Pure Math., 551–590, North-Holland, Amsterdam (1987)

    1987

  153. [161]

    14 ofMSJ Memoirs, Mathematical Society of Japan, Tokyo (2004), ISBN 4-931469-31-0

    ———, Zariski-decomposition and abundance, Vol. 14 ofMSJ Memoirs, Mathematical Society of Japan, Tokyo (2004), ISBN 4-931469-31-0

    2004

  154. [162]

    Namikawa, Toroidal compactification of Siegel spaces, Vol

    Y. Namikawa, Toroidal compactification of Siegel spaces, Vol. 812, Springer (1980)

    1980

  155. [163]

    Ann.319 (2001), no

    ———, Deformation theory of singular symplecticn-folds, Math. Ann.319 (2001), no. 3, 597–623

    2001

  156. [164]

    Reine Angew

    ———, On deformations ofQ-factorial symplectic varieties, J. Reine Angew. Math. 599 (2006) 97–110

    2006

  157. [165]

    Narasimhan, Introduction to the theory of analytic spaces, Vol

    R. Narasimhan, Introduction to the theory of analytic spaces, Vol. 25, Springer (1966)

    1966

  158. [166]

    Nicaise and C

    J. Nicaise and C. Xu,The essential skeleton of a degeneration of algebraic varieties, Amer. J. Math.138 (2016), no. 6, 1645–1667

    2016

  159. [167]

    V. V. Nikulin,Integral symmetric bilinear forms and some of their applications, Mathematics of the USSR- Izvestiya 14 (1980), no. 1, 103

    1980

  160. [168]

    M. V. Nori,Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4)16 (1983), no. 2, 305–344

    1983

  161. [169]

    K. Ohno, The Euler Characteristic Formula for Logarithmic Minimal Degenerations of Surfaces with Kodaira Dimension Zero and its application to Calabi-Yau Threefolds with a pencil, arXiv preprint, arXiv:0710.3641 (2007)

    2007 arXiv

  162. [170]

    Olsson,A boundedness theorem for Hom-stacks, Math

    M. Olsson,A boundedness theorem for Hom-stacks, Math. Res. Lett.14 (2007), no. 6, 1009–1021

    2007

  163. [171]

    K. G. O’Grady,Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math.1999 (1999), no. 512, 49–117

    1999

  164. [172]

    Algebraic Geom.12 (2003) 435–505

    ———, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom.12 (2003) 435–505

    2003

  165. [173]

    Perego and A

    A. Perego and A. Rapagnetta,Irreducible symplectic varieties from moduli spaces of sheaves on K3 and Abelian surfaces, arXiv:1802.01182.v1 (2018)

    2018 arXiv

  166. [174]

    Peters,Rigidity for variations of Hodge structure and Arakelov-type finiteness theorems, Compos

    C. Peters,Rigidity for variations of Hodge structure and Arakelov-type finiteness theorems, Compos. Math. 75 (1990), no. 1, 113–126

    1990

  167. [175]

    Pozzi, The Kuga–Satake Construction: A Modular Interpretation, Master’s thesis, Concordia University (2013)

    A. Pozzi, The Kuga–Satake Construction: A Modular Interpretation, Master’s thesis, Concordia University (2013)

    2013

  168. [176]

    Prendergast-Smith, The cone conjecture for abelian varieties, J

    A. Prendergast-Smith, The cone conjecture for abelian varieties, J. Math. Sci. Univ. Tokyo19 (2012), no. 2, 243–261

    2012

  169. [177]

    Procesi,The invariant theory of n× n matrices, Adv

    C. Procesi,The invariant theory of n× n matrices, Adv. Math.19 (1976) 306–381. 128 ENGEL, FILIPAZZI, GREER, MAURI, AND SV ALDI

    1976

  170. [178]

    An approach through invariants and representations

    ———, Lie groups, Universitext, Springer, New York (2007), ISBN 978-0-387-26040-2; 0-387-26040-4. An approach through invariants and representations

    2007

  171. [179]

    Yu. G. Prokhorov and V. V. Shokurov,Towards the second main theorem on complements, J. Algebraic Geom. 18 (2009), no. 1, 151–199

    2009

  172. [180]

    Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Vol

    M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Vol. 119 ofLecture Notes in Mathematics, Springer-Verlag, Berlin-New York (1970)

    1970

  173. [181]

    Reid,The moduli space of 3-folds with K= 0 may nevertheless be irreducible, Math

    M. Reid,The moduli space of 3-folds with K= 0 may nevertheless be irreducible, Math. Ann.278 (1987) 329–334

    1987

  174. [182]

    Rizov, Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic, J

    J. Rizov, Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic, J. Reine Angew. Math. 2010 (2010), no. 648, 13–67

    2010

  175. [183]

    Saccà,Compactifying Lagrangian fibrations, arXiv:2411.06505 (2024)

    G. Saccà,Compactifying Lagrangian fibrations, arXiv:2411.06505 (2024)

    2024 arXiv

  176. [184]

    Sawon,On the discriminant locus of a Lagrangian fibration, Math

    J. Sawon,On the discriminant locus of a Lagrangian fibration, Math. Ann.341 (2008), no. 1, 201–221

    2008

  177. [185]

    Algebraic Geom.25 (2016), no

    ———, A finiteness theorem for Lagrangian fibrations, J. Algebraic Geom.25 (2016), no. 3, 431–459

    2016

  178. [186]

    ———, A bound on the second Betti number of hyperkähler manifolds of complex dimension six, Eur. J. Math. 8 (2022), no. 3, 1196–1212

    2022

  179. [187]

    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) 211–320

    1973

  180. [188]

    6, 2165–2182

    S.SchreiederandA.Soldatenkov, The Kuga–Satake construction under degeneration, J.Inst.Math.Jussieu 19 (2020), no. 6, 2165–2182

    2020

  181. [189]

    Schwald,Low degree Hodge theory for klt varieties, arXiv preprint, arXiv:1612.01919 (2016)

    M. Schwald,Low degree Hodge theory for klt varieties, arXiv preprint, arXiv:1612.01919 (2016)

    2016 arXiv

  182. [190]

    Algébrique4 (2020) Art

    ———, Fujiki relations and fibrations of irreducible symplectic varieties, Épijournal Géom. Algébrique4 (2020) Art. 7, 19

    2020

  183. [191]

    E.Sernesi, Deformationsofalgebraicschemes, Vol.334of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (2006), ISBN 978-3-540-30608- 5; 3-540-30608-0

    2006

  184. [192]

    M.Shin, ComputationsofthecohomologicalBrauergroupofsomealgebraicstacks, Ph.D.thesis, University of California, Berkeley (2019)

    2019

  185. [193]

    Skorobogatov, Torsors and rational points, Vol

    A. Skorobogatov, Torsors and rational points, Vol. 144 ofCambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2001), ISBN 0-521-80237-7

    2001

  186. [194]

    Soldatenkov and M

    A. Soldatenkov and M. Verbitsky,The Moser isotopy for holomorphic symplectic and C-symplectic struc- tures, Annales de l’Institut Fourier (2025) 1–18

    2025

  187. [195]

    J. H. M. Steenbrink, Cohomologically insignificant degenerations, Compos. Math. 42 (1980/81), no. 3, 315–320

    1980

  188. [196]

    Voisin, Théorie de Hodge et géométrie algébrique complexe, Collection SMF, Société Mathématique de France (2002), ISBN 9782856291290

    C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Collection SMF, Société Mathématique de France (2002), ISBN 9782856291290

    2002

  189. [197]

    Wang,Structure of projective varieties with nef anticanonical divisor: the case of log terminal singular- ities, Math

    J. Wang,Structure of projective varieties with nef anticanonical divisor: the case of log terminal singular- ities, Math. Ann.384 (2022), no. 1-2, 47–100

    2022

  190. [198]

    Wells,Comparison of de Rham and Dolbeault cohomology for proper surjective mappings, Pac

    R. Wells,Comparison of de Rham and Dolbeault cohomology for proper surjective mappings, Pac. J. Math. 53 (1974), no. 1, 281–300

    1974

  191. [199]

    P. M. H. Wilson,Boundedness questions for Calabi–Yau threefolds, J. Algebraic Geom.30 (2021), no. 4, 631–684

    2021

  192. [200]

    ———, The topology of Calabi–Yau threefolds with Picard number three, arXiv preprint, arXiv:2202.05202 (2022)

    2022 arXiv

  193. [201]

    Yu. G. Zarhin,Hodge groups of K3 surfaces., J. Reine Angew. Math.341 (1983) 193–220. BOUNDEDNESS OF SOME FIBERED K-TRIVIAL V ARIETIES 129

    1983

  194. [202]

    Math.79 (1985), no

    ———, A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. Math.79 (1985), no. 2, 309–321

    1985

  195. [203]

    ———, Abelian varieties, quaternion trick and endomorphisms, in International Conference on Birational Geometry, Kaehler–Einstein Metrics and Degenerations, 857–864, Springer (2019). Department of Mathematics, Statistics, and Computer Science, University of Illinois in Chicago ...

    2019

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.