Pith. sign in

REVIEW 3 major objections 5 minor 163 references

A smooth proper moduli stack carrying a maximal-variation family of stable pairs has a big log canonical bundle, so such moduli spaces are naturally of log general type.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 23:51 UTC pith:EBJHSCCF

load-bearing objection Solid stack-level extension of Wei–Wu with real novelty, but Theorem A leans on the authors' own unpublished companion results; referee should check those before accepting. the 3 major comments →

arxiv 2607.15203 v1 pith:EBJHSCCF submitted 2026-07-16 math.AG

Moduli spaces of snc klt KSBA stable pairs are naturally of log general type

Sebastian Casalaina-Martin , Shend Zhjeqi This is my paper
classification math.AG MSC 14D2314J1014J1714C20
keywords KSBA stable pairslog general typeDeligne–Mumford stacksViehweg hyperbolicitybig line bundlesHodge modulesrelative simple normal crossingscoarse moduli 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 proves that moduli stacks of stable pairs are themselves of log general type. The precise statement is: if a smooth proper Deligne–Mumford stack X carries a family of KSBA stable pairs of maximal variation, and if away from a reduced divisor Δ the family is relatively simple normal crossings with klt fibers, then K_X+Δ is big. The proof extends the standard hyperbolicity strategy for smooth projective bases to stacks, constructing a Viehweg–Zuo sheaf from a Hodge module pushed forward from a cyclic cover of a self-product of the family. A corollary upgrades the conclusion to coarse moduli spaces: after resolving singularities, the coarse space together with the ramification and discriminant boundary is log canonical and of log general type. The gain over earlier results is visible even on the j-line, where less boundary weight is needed than previously known.

Core claim

The paper establishes that the log canonical bundle of a smooth proper Deligne–Mumford stack is big whenever the stack carries a family of KSBA stable pairs of maximal variation that is relatively simple normal crossings and has klt fibers away from a reduced divisor Δ. Because maximal variation means the classifying map to the moduli stack is generically finite, the theorem applies to integral components of moduli spaces of KSBA stable pairs. The corollary upgrades the usual statement on the coarse space: after resolving singularities, the pair consisting of the coarse space and the stack-theoretic ramification divisor plus the discriminant is log canonical and of log general type. In the e

What carries the argument

The central object is the Viehweg–Zuo sheaf: a coherent sheaf H with big determinant that embeds into a tensor power of the logarithmic cotangent bundle (Ω¹_X(log Δ))^{⊗s}. The construction forms self-products of the family, resolves them, uses a cyclic covering to create a Hodge module, pushes it forward with an ad hoc stack-level push forward, and extracts its graded pieces. The big part comes from the determinant of the direct image of the relative pluricanonical bundle, which is big for maximal variation; the weak positivity comes from the kernels of the deformation maps in the associated graded Higgs bundle. The inclusion of a big line bundle A(−Δ) into the first graded piece of the Hod

Load-bearing premise

The proof collapses unless the family is relatively simple normal crossings over the complement of Δ—meaning every stratum of the boundary divisor is smooth over the base—because the key vanishing lemma about the Hodge-theoretic sheaf is proved only under that stronger condition; assuming merely that the fibers are snc pairs is not enough.

What would settle it

A concrete check: on the moduli stack of marked elliptic curves, taking Δ empty, the universal family has snc fibers but is not relatively snc at the two automorphism points, and its canonical bundle is not big; this shows the relative-snc hypothesis is essential. For the theorem itself, a decisive test would be to construct a maximal-variation family satisfying all hypotheses and directly compute that K_X+Δ is big in a case where the boundary has stack automorphisms.

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

If this is right

  • Theorem A applies directly to moduli stacks rather than only to their coarse spaces, so log general type becomes a property of the moduli problem itself.
  • For any proper integral DM stack with a generically finite map to the moduli stack of marked KSBA stable pairs, the log resolution is of log general type whenever the relative-snc discriminant is not the whole base and the fibers are klt outside it.
  • The boundary divisor on the coarse space is the sum of the ramification divisor of the stack morphism and the discriminant, with coefficients that can be smaller than one; hence the pair (X, R+Δ) is log canonical and K_X+R+Δ is big.
  • Earlier results on the coarse space required rounding the boundary up, while the stack-level statement gives precise coefficients—for example, 13/6 of a point suffices on the elliptic-curve moduli line, not 3.
  • The theorem provides a foundation for proving hyperbolicity-type properties of moduli stacks of stable pairs, since it identifies the log canonical bundle as genuinely big under natural geometric hypotheses.

Where Pith is reading between the lines

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

  • Editorial inference: because the theorem is stable under generically finite base change, finite covers of moduli stacks of KSBA pairs should also be log general type, making the statement a birational and étale invariant of the moduli stack.
  • Editorial inference: the explicit boundary coefficients suggest that the minimal Q-divisor making K_X big could serve as a numerical measure of how far a moduli stack is from being canonically polarized; the paper does not develop this invariant.
  • Editorial inference: the ad hoc push forward used for Hodge modules is tailored to proper schematic morphisms; extending it to quasi-projective bases would likely yield quasi-projectivity or hyperbolicity theorems for moduli stacks, a direction the paper flags as future work.
  • Editorial inference: testing the corollary on concrete weighted pointed stable-curve spaces would give explicit numerical checks of the boundary weights; the paper works out only the elliptic-curve case.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proves a Deligne–Mumford stack version of Wei–Wu's Viehweg hyperbolicity theorem for families of stable pairs. Theorem A states that if X is a smooth proper integral DM stack over C with projective coarse moduli space, and f:(Y,D)→X is a maximal-variation family of KSBA stable pairs that is relative simple normal crossings over X−Δ with klt fibers there, then K_X+Δ is big. The proof follows the Popa–Schnell / Wei–Wu strategy: after birational preparations and a cyclic covering construction, it builds a Hodge module with strict support and a graded logarithmic Higgs subbundle F^• satisfying θ(F^k)⊆Ω^1_X(log Δ)⊗F^{k+1} and F^0 big. It then invokes the authors' companion results [CMZ26c, Thm. B] and [CMZ26a, Cor. B] to obtain a Viehweg–Zuo sheaf and conclude bigness of K_X+Δ. The paper also proves corollaries for KSBA moduli stacks and their coarse moduli spaces, comparing the resulting boundary divisors with those obtained by applying the Wei–Wu theorem on the coarse space. A substantial portion of the paper develops an 'ad hoc' pushforward of filtered D-modules and Hodge modules on DM stacks via étale descent, with appendices relating it to Beilinson–Drinfeld and Tubach's pushforwards.

Significance. If the companion results are correct, Theorem A is a meaningful extension of a central hyperbolicity statement from smooth projective varieties to smooth proper DM stacks, and Corollary C gives a sharper log-general-type statement on coarse moduli spaces by including the stack-theoretic ramification divisor with coefficients ≤1. The paper's main positive contribution is the construction of Hodge-theoretic input on stacks: the ad hoc pushforward, the cyclic-cover Hodge module, and the logarithmic Higgs subbundle with θ(F^k) contained in Ω^1(log Δ)⊗F^{k+1}. These are nontrivial technical developments and are presented in detail. However, the headline theorem is not self-contained: the final implication from the Higgs-bundle data to bigness of K_X+Δ is delegated to two unpublished companion preprints [CMZ26c, CMZ26a], and one intermediate step, Lemma 4.11, is delegated to a published lemma [WW23, Lem. 6.2] by a short étale-local reduction. The significance of the paper is therefore conditional on those external results, and the refereeing process should verify them explicitly.

major comments (3)
  1. [Section 6, proof of Theorem A] The last two sentences of the proof are load-bearing: after Theorem 5.1 produces F^•⊆E^• with θ(F^k)⊆Ω^1_X(log Δ)⊗F^{k+1} and F^0 big, the argument invokes [CMZ26c, Thm. B] to obtain a Viehweg–Zuo sheaf and then [CMZ26a, Cor. B] to conclude K_X+Δ is big. These two theorems are not proved in the manuscript and are cited only as companion preprints from the same authors. The reader cannot currently check whether all hypotheses of those theorems are satisfied by the specific F^• and E^• constructed here, particularly the saturation assumptions, the stack-theoretic bigness convention, and the logarithmic Higgs-bundle hypotheses. Please include the precise statements of [CMZ26c, Thm. B] and [CMZ26a, Cor. B], verify that the output of Theorem 5.1 meets them, and either provide proofs or state clearly that Theorem A is conditional on those companion results. This is the central issue for the so
  2. [Lemma 4.11 and Remark 4.12] Lemma 4.11 is the unique place where the strong relative-snc hypothesis is used, and it is proved by the sentence: 'As the triviality of ˇG^0_m|Us can be checked after a surjective étale base change, this reduces to the case of varieties, and so follows from [WW23, Lem. 6.2].' This reduction is not fully demonstrated. One must show that, after an étale base change to a smooth variety, the object ˇG^0_• restricts to the object for which [WW23, Lem. 6.2] is stated, and that the relative-snc and discriminant hypotheses are preserved. Since Theorem 5.3(c) and hence Theorem A depend on Lemma 4.11, the reduction should be written out rather than asserted. This is not a fatal objection, but it is a correctness-risk point that needs to be fixed in revision.
  3. [Sections 3.7, 3.9, Proposition 4.4] The ad hoc pushforward is developed carefully, but its use in Proposition 4.4 is delicate: the proof identifies gr^F_• H^0(h^ah_+ O_Z) with R^0 h_*(ω_{Z/X}⊗C_{Z→X,•}) by combining Lemma 3.10 with Saito's strictness theorem. The manuscript then defines M as the torsion-free quotient of H^0(h^ah_+ Q^H_Z[dim Z]) and asserts that gr^F_• M is a quotient of the same pushforward. It should be explained why taking the torsion-free quotient commutes with the associated graded in the stack setting, and why the descent data for the Hodge module and for the coherent sheaf pushforward are compatible under this operation. If this fails, the construction of G^• in (4.15) and the inclusion G^•⊆gr^F_•(M) in Theorem 4.3(d) would not follow. Please clarify this point.
minor comments (5)
  1. [Introduction, Theorem A statement] There is a typographical issue: 'Theorem A(Viehweg hyperbolicity...' should have a space after 'A'. Also 'smoothoutside' appears later in the introduction.
  2. [Lemma 1.7] The proof asserts 'clearly eΔ≤Δ^+_{X'}' without discussing the coefficient assumptions on Δ. This is immediate when Δ is reduced and X' is a log resolution with appropriate discrepancies, but if Δ is an arbitrary effective Q-divisor with coefficients >1, a short justification is needed.
  3. [Section 3.7.1 and Appendix D/E] The comparison with Tubach's pushforward is sketched in prose and in a diagram. If this comparison is only motivational, please say so explicitly; if it is used, a precise statement with hypotheses would help the reader.
  4. [Section 4.3.4] The notation G^• and ˇG^0_• is easy to confuse, especially in Proposition 4.9 and Lemma 4.11. Adding a sentence that ˇG^0_• denotes the left-hand term of (4.15) before the surjection would improve readability.
  5. [References] The companion papers [CMZ26c], [CMZ26b], [CMZ26a] are cited as 2026 arXiv preprints. Please provide version numbers or acceptance status where available, since Theorem A depends on two of them.

Circularity Check

2 steps flagged

Theorem A's final bigness step is delegated to the authors' own unpublished companion results [CMZ26c, Thm. B] and [CMZ26a, Cor. B].

specific steps
  1. self citation load bearing [Section 6, proof of Theorem A (final sentences)]
    "This data allows us to use [CMZ26c, Thm. B] to obtain a Viehweg–Zuo sheaf, i.e., a coherent sheaf H on X with big determinant, and for some s≥1, an inclusion H ,→ (Ω^1_X(log∆))⊗s. Finally, taking the saturation of H, then from [CMZ26a, Cor. B], such an inclusion implies K_X+∆ is big, completing the proof."

    These sentences are the final step of Theorem A. The paper's Sections 2–5 construct a Higgs submodule F• with θ(F•)⊆Ω^1_X(log∆)⊗F• and F^0 big; the remaining inference to a Viehweg–Zuo sheaf and then to bigness of K_X+∆ is supplied by two companion preprints by the same authors. No proof of those stack-theoretic results is included here, and they belong to the same 'fourth in a series' project. Thus the cited results are load-bearing self-citations, not independent external support; the derivation chain terminates in the authors' own unpublished claims.

  2. self citation load bearing [Introduction, paragraph beginning 'The proof of Theorem A is therefore reduced...']
    "It follows in the case of varieties from a theorem of Campana–Păun [CP19, Thm. 7.6, Thm. 1.2], and from [CMZ26a, Thm. A] in the case of stacks, that detQ is pseudo-effective. Taking determinants in the short exact sequence above, one obtains that a positive multiple of K_X+∆ is 'big plus pseudo-effective', and therefore is big (e.g., [CMZ26a, Cor. B])."

    This paragraph reduces the proof of Theorem A to constructing a Viehweg–Zuo sheaf and then cites [CMZ26a, Thm. A / Cor. B] for the stack case of the Campana–Păun pseudo-effectivity step and for the bigness criterion. The theorem's conclusion is therefore obtained by invoking the authors' prior work rather than by an argument contained in this paper. The self-citation is doing the decisive logical work: if [CMZ26a] had a gap or did not cover the constructed F•, Theorem A would be unsupported at exactly this point.

full rationale

There is no definitional circularity, no fitted parameter renamed as a prediction, and no ansatz smuggled in via citation: the internal geometric construction in Sections 2–5 is new and independent, and Lemma 4.11 is delegated to the external published result [WW23, Lem. 6.2], which is independent evidence rather than self-citation. However, the proof of Theorem A is not self-contained. The conclusion that K_X+∆ is big is obtained by applying the authors' own unpublished companion theorems [CMZ26c, Thm. B] and [CMZ26a, Cor. B]; these are load-bearing self-citations from the same series, and the paper does not reproduce or independently verify the stack-theoretic Popa–Schnell / Campana–Păun arguments that they supply. This is a real gap in self-containedness and a correctness risk, though not a reduction of the result to its inputs by construction. If the companion papers are sound, the mathematics is not circular; but as presented, the central claim's final implication rests on the authors' own unverified-in-this-paper results.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

The paper relies on standard Hodge-module and D-module theory, on positivity results for KSBA moduli (KP17), on the previous variety-level theorems of Wei-Wu and Popa-Schnell, and on the authors' own companion preprints for the final stack-level steps. There are no fitted numerical parameters. The only new mathematical construction is the ad hoc pushforward, which is defined explicitly and checked against existing pushforwards.

axioms (6)
  • standard math Saito's theory of polarizable Hodge modules and strictness of pushforwards (used throughout Sections 3–5, e.g., Proposition 4.8, equation (3.42)).
    The construction relies on the existence of Hodge modules, strictness, and the Riemann–Hilbert correspondence as black boxes.
  • domain assumption Kollár–Patakfalvi [KP17, Thm. 7.1]: positivity of det f_*ω^m_{Y/X}(mD) over KSBA moduli.
    Used in Proposition 2.1 to obtain the non-zero section (2.7).
  • domain assumption Wei–Wu [WW23, Thm. A and Lem. 6.2] for the variety case.
    The stack statement is meant to generalize; Lemma 4.11 reduces to [WW23, Lem. 6.2].
  • domain assumption Popa–Schnell [PS17] results on existence of Viehweg–Zuo sheaves in the variety case.
    Many propositions reduce étale-locally to [PS17, Props 2.8–2.15].
  • ad hoc to paper The companion preprints [CMZ26c, Thm. B] and [CMZ26a, Cor. B] (also CMZ26b) are correct.
    Section 6 uses [CMZ26c, Thm. B] to finish the VZ sheaf construction and [CMZ26a, Cor. B] to conclude K_X+Δ big; these preprints are not independently verified here.
  • domain assumption The moduli stack SP(a,n,ν) has projective coarse moduli space [KP17, Thm. 1.1], and the universal family is schematic/projective; finite flat covers by smooth projective varieties exist [KV04, Kre09].
    These structural facts about DM stacks and KSBA moduli are used throughout Section 1.3 and Section 2.
invented entities (1)
  • Ad hoc pushforward H^i(f^ah_+) of Hodge modules on DM stacks independent evidence
    purpose: Define pushforwards along schematic proper morphisms of DM stacks without a full six-functor formalism; used to construct the Hodge module M in Theorem 4.3.
    It is shown compatible with Tubach's pushforward (§3.7.1) and reduces to Saito's pushforward on étale charts; the main theorem provides a nontrivial application.

pith-pipeline@v1.3.0-alltime-deepseek · 62087 in / 18179 out tokens · 143865 ms · 2026-08-01T23:51:32.840715+00:00 · methodology

0 comments
read the original abstract

We generalize work of Wei-Wu on Viehweg hyperbolicity for families of stable pairs over smooth projective varieties to the case of smooth Deligne-Mumford stacks. The results of Wei-Wu build on results of Popa-Schnell and Viehweg-Zuo, utilize a result of Campana-P\u{a}un, and extend results of Kebekus-Kov\'acs and Patakfalvi. We apply our results to smooth proper moduli stacks parameterizing generically klt relatively simple normal crossings KSBA stable pairs, showing that such moduli stacks naturally have big log canonical bundles. We also consider implications for their coarse moduli spaces.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

163 extracted references · 7 canonical work pages · 1 internal anchor

  1. [1]

    Weibel , title =

    Charles A. Weibel , title =

  2. [2]

    European Journal of Mathematics , year =

    Sergey Rybakov , title =. European Journal of Mathematics , year =

  3. [3]

    Serre, Jean-Pierre , title =. S. 1954 , pages =

    1954

  4. [4]

    Positivity in the context of

    Sebastian Casalaina-Martin and Shend Zhjeqi , year=. Positivity in the context of. 2605.22989 , archivePrefix=

    Pith/arXiv arXiv

  5. [5]

    Movable curve classes and slope stability on

    Sebastian Casalaina-Martin and Shend Zhjeqi , year=. Movable curve classes and slope stability on. 2605.26101 , archivePrefix=

    Pith/arXiv arXiv

  6. [6]

    Foliations, slope stability, and positivity of log canonical bundles on

    Sebastian Casalaina-Martin and Shend Zhjeqi , year=. Foliations, slope stability, and positivity of log canonical bundles on. 2605.26443 , archivePrefix=

    Pith/arXiv arXiv

  7. [7]

    Popa, Mihnea and Schnell, Christian , TITLE =. Invent. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00222-016-0698-9 , URL =

    doi:10.1007/s00222-016-0698-9 2017

  8. [8]

    Michigan Math

    Cacciola, Salvatore and Lopez, Angelo Felice and Viviani, Filippo , TITLE =. Michigan Math. J. , FJOURNAL =. 2016 , NUMBER =. doi:10.1307/mmj/1472066146 , URL =

    arXiv 2016

  9. [9]

    and Nitsure, Nitin and Vistoli, Angelo , TITLE =

    Fantechi, Barbara and G\"ottsche, Lothar and Illusie, Luc and Kleiman, Steven L. and Nitsure, Nitin and Vistoli, Angelo , TITLE =. 2005 , PAGES =. doi:10.1090/surv/123 , URL =

    doi:10.1090/surv/123 2005

  10. [10]

    1998 , PAGES =

    Harris, Joe and Morrison, Ian , TITLE =. 1998 , PAGES =

    1998

  11. [11]

    Cornalba, Maurizio and Harris, Joe , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1988 , NUMBER =

    1988

  12. [12]

    Vakil, Ravi , TITLE =. Invent. Math. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-005-0481-9 , URL =

    doi:10.1007/s00222-005-0481-9 2006

  13. [13]

    , TITLE =

    Gieseker, D. , TITLE =. Amer. J. Math. , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/2373939 , URL =

    work page doi:10.2307/2373939 1979

  14. [14]

    Miyaoka, Yoichi , TITLE =. Invent. Math. , FJOURNAL =. 1977 , PAGES =. doi:10.1007/BF01389789 , URL =

    doi:10.1007/bf01389789 1977

  15. [15]

    Algebraic geometry,

    Miyaoka, Yoichi , TITLE =. Algebraic geometry,. 1987 , ISBN =. doi:10.2969/aspm/01010449 , URL =

    arXiv 1987

  16. [16]

    Greb, Daniel and Kebekus, Stefan and Peternell, Thomas and Taji, Behrouz , TITLE =. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , FJOURNAL =. 2019 , NUMBER =. doi:10.24033/asens.2414 , URL =

    doi:10.24033/asens.2414 2019

  17. [17]

    Wei, Chuanhao and Wu, Lei , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnab280 , URL =

    doi:10.1093/imrn/rnab280 2023

  18. [18]

    2008 , PAGES =

    Hotta, Ryoshi and Takeuchi, Kiyoshi and Tanisaki, Toshiyuki , TITLE =. 2008 , PAGES =. doi:10.1007/978-0-8176-4523-6 , URL =

    doi:10.1007/978-0-8176-4523-6 2008

  19. [19]

    An overview of

    Christian Schnell , year=. An overview of

  20. [20]

    D-Modules and mixed

    Blum, Harold and Murayama, Takumi and Mustata, Mircea , year=. D-Modules and mixed

  21. [21]

    Mustata, Mircea , year=

  22. [22]

    Surikaisekikenkyusho Kokyuroku , FJOURNAL =

    Saito, Masa-Hiko , TITLE =. Surikaisekikenkyusho Kokyuroku , FJOURNAL =. 1992 , PAGES =

    1992

  23. [23]

    Several complex variables and complex geometry,

    Saito, Morihiko , TITLE =. Several complex variables and complex geometry,. 1991 , ISBN =. doi:10.1090/pspum/052.2/1128566 , URL =

    doi:10.1090/pspum/052.2/1128566 1991

  24. [24]

    Saito, Morihiko , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 1988 , NUMBER =. doi:10.2977/prims/1195173930 , URL =

    arXiv 1988

  25. [25]

    Beilinson and V

    A. Beilinson and V. Drinfeld , TITLE =

  26. [26]

    , TITLE =

    G\'omez, Tom\'as L. , TITLE =. Affine flag manifolds and principal bundles , SERIES =. 2010 , ISBN =. doi:10.1007/978-3-0346-0288-4\_2 , URL =

    doi:10.1007/978-3-0346-0288-4 2010

  27. [27]

    Abramovich, Dan and Graber, Tom and Vistoli, Angelo , TITLE =. Amer. J. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1353/ajm.0.0017 , URL =

    doi:10.1353/ajm.0.0017 2008

  28. [28]

    Kresch, Andrew and Vistoli, Angelo , TITLE =. Bull. London Math. Soc. , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0024609303002728 , URL =

    doi:10.1112/s0024609303002728 2004

  29. [29]

    Vistoli, Angelo , TITLE =. Invent. Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF01388892 , URL =

    doi:10.1007/bf01388892 1989

  30. [30]

    Algebraic geometry---

    Kresch, Andrew , TITLE =. Algebraic geometry---. 2009 , ISBN =. doi:10.1090/pspum/080.1/2483938 , URL =

    doi:10.1090/pspum/080.1/2483938 2009

  31. [31]

    The stacks project , howpublished =

    The. The stacks project , howpublished =

  32. [32]

    Jacob Lurie , title =

  33. [33]

    2017 , month = sep, date =

    Lurie, Jacob , title =. 2017 , month = sep, date =

    2017

  34. [34]

    2000 , PAGES =

    Laumon, G\'erard and Moret-Bailly, Laurent , TITLE =. 2000 , PAGES =

    2000

  35. [35]

    2004 , PAGES =

    Lazarsfeld, Robert , TITLE =. 2004 , PAGES =. doi:10.1007/978-3-642-18808-4 , URL =

    doi:10.1007/978-3-642-18808-4 2004

  36. [36]

    Cascini, Paolo and Hacon, Christopher and Mustata, Mircea and Schwede, Karl , TITLE =. Amer. J. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1353/ajm.2014.0047 , URL =

    arXiv 2014

  37. [37]

    1998 , PAGES =

    Koll\'ar, J\'anos and Mori, Shigefumi , TITLE =. 1998 , PAGES =. doi:10.1017/CBO9780511662560 , URL =

    doi:10.1017/cbo9780511662560 1998

  38. [38]

    2013 , PAGES =

    Koll\'ar, J\'anos , TITLE =. 2013 , PAGES =. doi:10.1017/CBO9781139547895 , URL =

    doi:10.1017/cbo9781139547895 2013

  39. [39]

    2023 , PAGES =

    Koll\'ar, J\'anos , TITLE =. 2023 , PAGES =

    2023

  40. [40]

    Koll\'ar, J\'anos , TITLE =. J. Differential Geom. , FJOURNAL =. 1990 , NUMBER =

    1990

  41. [41]

    Patakfalvi, Zsolt , TITLE =. Adv. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.aim.2011.12.013 , URL =

    work page doi:10.1016/j.aim.2011.12.013 2012

  42. [42]

    , TITLE =

    Kebekus, Stefan and Kov\'acs, S\'andor J. , TITLE =. Adv. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.aim.2008.01.005 , URL =

    work page doi:10.1016/j.aim.2008.01.005 2008

  43. [43]

    , TITLE =

    Kov\'acs, S\'andor J. , TITLE =. Handbook of moduli. 2013 , ISBN =

    2013

  44. [44]

    Bhatt, Bhargav and Ho, Wei and Patakfalvi, Zsolt and Schnell, Christian , TITLE =. Compos. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1112/S0010437X13007288 , URL =

    doi:10.1112/s0010437x13007288 2013

  45. [45]

    and Patakfalvi, Zsolt , TITLE =

    Kov\'acs, S\'andor J. and Patakfalvi, Zsolt , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/jams/871 , URL =

    doi:10.1090/jams/871 2017

  46. [46]

    Compositio Math

    Esnault, H\'el\`ene and Viehweg, Eckart , TITLE =. Compositio Math. , FJOURNAL =. 1990 , NUMBER =

    1990

  47. [47]

    1992 , PAGES =

    Esnault, H\'el\`ene and Viehweg, Eckart , TITLE =. 1992 , PAGES =. doi:10.1007/978-3-0348-8600-0 , URL =

    doi:10.1007/978-3-0348-8600-0 1992

  48. [48]

    1995 , PAGES =

    Viehweg, Eckart , TITLE =. 1995 , PAGES =. doi:10.1007/978-3-642-79745-3 , URL =

    doi:10.1007/978-3-642-79745-3 1995

  49. [49]

    Algebraic varieties and analytic varieties (

    Viehweg, Eckart , TITLE =. Algebraic varieties and analytic varieties (. 1983 , ISBN =. doi:10.2969/aspm/00110329 , URL =

    arXiv 1983

  50. [50]

    Classification of algebraic and analytic manifolds (

    Viehweg, Eckart , TITLE =. Classification of algebraic and analytic manifolds (. 1983 , ISBN =

    1983

  51. [51]

    Viehweg, Eckart , TITLE =. Invent. Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF01393700 , URL =

    doi:10.1007/bf01393700 1989

  52. [52]

    , TITLE =

    Viehweg, E. , TITLE =. Invent. Math. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01231501 , URL =

    doi:10.1007/bf01231501 1990

  53. [53]

    Viehweg, Eckart , TITLE =. Invent. Math. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01231514 , URL =

    doi:10.1007/bf01231514 1990

  54. [54]

    , TITLE =

    Kashiwara, M. , TITLE =. Algebraic geometry (. 1983 , ISBN =. doi:10.1007/BFb0099962 , URL =

    doi:10.1007/bfb0099962 1983

  55. [55]

    Popa, Mihnea and Wu, Lei , TITLE =. Math. Res. Lett. , FJOURNAL =. 2016 , NUMBER =. doi:10.4310/MRL.2016.v23.n4.a8 , URL =

    doi:10.4310/mrl.2016.v23.n4.a8 2016

  56. [56]

    Laszlo, Yves and Olsson, Martin , TITLE =. Math. Z. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s00209-008-0348-z , URL =

    doi:10.1007/s00209-008-0348-z 2009

  57. [57]

    Foliations with positive slopes and birational stability of orbifold cotangent bundles , JOURNAL =

    Campana, Fr\'ed\'eric and P. Foliations with positive slopes and birational stability of orbifold cotangent bundles , JOURNAL =. 2019 , PAGES =. doi:10.1007/s10240-019-00105-w , URL =

    doi:10.1007/s10240-019-00105-w 2019

  58. [58]

    \'Epijournal G\'eom

    Schnell, Christian , TITLE =. \'Epijournal G\'eom. Alg\'ebrique , FJOURNAL =. 2017 , PAGES =. doi:10.46298/epiga.2017.volume1.3281 , URL =

    doi:10.46298/epiga.2017.volume1.3281 2017

  59. [59]

    Hyperbolicity properties of algebraic varieties , SERIES =

    Claudon, Beno\^it and Kebekus, Stefan and Taji, Behrouz , TITLE =. Hyperbolicity properties of algebraic varieties , SERIES =. [2021] 2021 , ISBN =

    2021

  60. [60]

    Greb, Daniel and Kebekus, Stefan and Peternell, Thomas , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2016 , NUMBER =. doi:10.1093/imrn/rnv126 , URL =

    doi:10.1093/imrn/rnv126 2016

  61. [61]

    Greb, Daniel and Kebekus, Stefan and Peternell, Thomas , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2014 , PAGES =. doi:10.1515/crelle-2012-0097 , URL =

    doi:10.1515/crelle-2012-0097 2014

  62. [62]

    Campana, Fr\'ed\'eric and Peternell, Thomas , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 2011 , NUMBER =. doi:10.24033/bsmf.2599 , URL =

    doi:10.24033/bsmf.2599 2011

  63. [63]

    The pseudo-effective cone of a compact

    Boucksom, S\'ebastien and Demailly, Jean-Pierre and P. The pseudo-effective cone of a compact. J. Algebraic Geom. , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S1056-3911-2012-00574-8 , URL =

    doi:10.1090/s1056-3911-2012-00574-8 2013

  64. [64]

    Algebraic varieties and automorphism groups , SERIES =

    Fujino, Osamu , TITLE =. Algebraic varieties and automorphism groups , SERIES =. 2017 , ISBN =. doi:10.2969/aspm/07510073 , URL =

    arXiv 2017

  65. [65]

    Beilinson, A. A. and Bernstein, J. and Deligne, P. , TITLE =. Analysis and topology on singular spaces,. 1982 , MRCLASS =

    1982

  66. [66]

    [2020] 2020 , PAGES =

    G\"ortz, Ulrich and Wedhorn, Torsten , TITLE =. [2020] 2020 , PAGES =. doi:10.1007/978-3-658-30733-2 , URL =

    doi:10.1007/978-3-658-30733-2 2020

  67. [67]

    1989 , PAGES =

    Matsumura, Hideyuki , TITLE =. 1989 , PAGES =

    1989

  68. [68]

    Hartshorne, Robin , TITLE =. Math. Ann. , FJOURNAL =. 1980 , NUMBER =. doi:10.1007/BF01467074 , URL =

    doi:10.1007/bf01467074 1980

  69. [69]

    1977 , PAGES =

    Hartshorne, Robin , TITLE =. 1977 , PAGES =

    1977

  70. [70]

    and Villamayor, O

    Encinas, S. and Villamayor, O. , TITLE =. Acta Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/BF02392749 , URL =

    doi:10.1007/bf02392749 1998

  71. [71]

    , TITLE =

    Bierstone, Edward and Milman, Pierre D. , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 2008 , NUMBER =. doi:10.2977/prims/1210167338 , URL =

    arXiv 2008

  72. [72]

    Israel J

    Temkin, Michael , TITLE =. Israel J. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s11856-018-1656-6 , URL =

    doi:10.1007/s11856-018-1656-6 2018

  73. [73]

    Duke Math

    Temkin, Michael , TITLE =. Duke Math. J. , FJOURNAL =. 2012 , NUMBER =. doi:10.1215/00127094-1699539 , URL =

    doi:10.1215/00127094-1699539 2012

  74. [74]

    2023 , eprint=

    Birational geometry of Deligne-Mumford stacks , author=. 2023 , eprint=

    2023

  75. [75]

    Knudsen, Finn Faye and Mumford, David , TITLE =. Math. Scand. , FJOURNAL =. 1976 , NUMBER =. doi:10.7146/math.scand.a-11642 , URL =

    doi:10.7146/math.scand.a-11642 1976

  76. [76]

    Grothendieck

    Fabio Nironi , year=. Grothendieck. 0811.1955 , archivePrefix=

    Pith/arXiv arXiv 1955

  77. [77]

    , TITLE =

    Kleiman, Steven L. , TITLE =. Compositio Math. , FJOURNAL =. 1980 , NUMBER =

    1980

  78. [78]

    and Karu, K

    Abramovich, D. and Karu, K. , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002229900024 , URL =

    doi:10.1007/s002229900024 2000

  79. [79]

    Alper, Jarod , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2013 , NUMBER =. doi:10.5802/aif.2833 , URL =

    doi:10.5802/aif.2833 2013

  80. [80]

    2010 , edition =

    Daniel Huybrechts and Manfred Lehn , title =. 2010 , edition =

    2010

Showing first 80 references.