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Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces

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Pith's one-line read A Nakano-positive singular Hermitian metric on a weakly pseudoconvex complex space forces all positive-degree cohomology of its Grauert–Riemenschneider L2 canonical sheaf to vanish; without Kähler, H^1 vanishes.

desk verdict Serious L2 theory on singular spaces; main theorems plausible, but they lean on same-author preprints and one stated theorem misses the exhaustion condition it needs. read the letter →

arxiv 2606.16275 v2 pith:UDLUQIYI submitted 2026-06-15 math.CV

Yuta Watanabe This is my paper · ORCID
classification math.CV MSC 32S2014F1832L1032L2032C1532J25
keywords L2-estimatessingularHermitianmetricsNakanopositivityGriffithscomplexspacesvanishingtheoremsL2-Dolbeaultresolutionweaklypseudoconvex
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

On weakly pseudoconvex complex spaces, the paper establishes a vanishing theorem: a singular Hermitian metric on a holomorphic vector bundle that is Nakano positive in the sense of $L^2$-estimates forces all positive-degree cohomology of its Grauert–Riemenschneider canonical $L^2$-subsheaf to vanish; without a Kähler metric, first cohomology still vanishes. To get there, the paper introduces definitions of singular Griffiths and Nakano positivity appropriate to complex spaces — Nakano positivity is defined through solvability of the $\partial$-equation with the standard Bochner–Kodaira–Nakano estimate on the regular locus — and proves they behave well under resolution of singularities. Two further tools are built: a global $L^2$-existence theorem on the regular locus of weakly pseudoconvex spaces, and an $L^2$-Dolbeault fine resolution of the canonical $L^2$-subsheaf, which identifies its sheaf cohomology with $L^2$-Dolbeault cohomology. A sympathetic reader would care because this is the first time the standard $L^2$ vanishing machine is made to work for arbitrary singular vector-bundle metrics on singular spaces, not just line-bundle-type singularities or tame metrics.

What carries the argument

The machinery has four components. (1) Singular Griffiths/Nakano positivity on complex spaces: Griffiths positivity is defined via log-norm plurisubharmonicity; Nakano positivity ($\theta$-Nakano positivity 'in the sense of $L^2$-estimates') is defined as a property of the $\partial$-problem — for every Stein coordinate patch, every $\partial$-closed form with finite $B^{-1}$-norm admits a $\partial$-potential with the standard estimate, with $B = [(\theta + i\partial\partial\psi)\otimes id_E, \Lambda_\omega]$. (2) The Grauert–Riemenschneider canonical $L^2$-subsheaf $\omega^{GR}_X(E,h)$, whose sections are $L^2$-integrable $(n,0)$-forms on the regular locus; it equals $\pi_*(K_{\tilde{X}} \otimes E(\pi^*h))$ for any resolution. (3) A global $L^2$-existence theorem (Theorem 5.3) on the regular locus of weakly ps

What would settle it

Construct a weakly pseudoconvex complex space $X$ with a singular positive line bundle for which some $(X_c)_{\mathrm{reg}} \setminus A$ has non-Stein components, or otherwise prove the exhaustion premise fails; this would undercut Theorem 5.3. Alternatively, search for a weakly pseudoconvex Kähler complex space with a singular positive line bundle and a $\theta$-Nakano positive singular Hermitian metric, plus a $\partial$-closed $(n,q)$-form $f$ with finite $\int \langle B^{-1}f, f\rangle$ norm, that has no $\partial$-potential with the stated $L^2$ bound — that would directly falsify the global $L^2$-existence theorem and hence the vanishing theorem.

Watch

Extended reading notes

Core claim

On a weakly pseudoconvex complex space of pure dimension that admits a singular positive line bundle, a singular Hermitian metric that is Nakano positive in the $L^2$-estimate sense makes $H^q(X, \omega^{GR}_X(E,h))$ vanish for all $q>0$; this is the content of Theorem 7.6. The paper's central insight is to formulate Nakano positivity on complex spaces not through curvature, but through the solvability of the $\partial$-equation with the standard estimate on the regular locus. With that formulation, positivity pulls back to any resolution of singularities, the Grauert–Riemenschneider canonical $L^2$-subsheaf $\omega^{GR}_X(E,h)$ — defined by $L^2$-integrability of sections on the regular locus — is coherent, and the global $L^2$ $\partial$-

Load-bearing premise

The paper assumes, as an imported structural premise, that a weakly pseudoconvex complex space admitting a singular positive line bundle can be exhausted by open subsets $(X_{c_j})_{\mathrm{reg}} \setminus A_j$ that are Stein manifolds; this is the fact on which the global $L^2$-existence theorem and hence the vanishing theorem rest.

Editorial extensions

If this is right

  • For any holomorphic vector bundle with a Nakano-positive singular Hermitian metric on a weakly pseudoconvex Kähler complex space, the higher cohomology groups H^q(X, ω^GR_X(E,h)) vanish for all q>0, giving singular-space analogues of the classical Nakano–Nadel vanishing theorem.
  • Without a Kähler metric, first cohomology H^1 still vanishes; in the compact case, the Kähler assumption can be dropped entirely and vanishing forces X to be Moishezon.
  • A Griffiths-positive (and a.e. Griffiths semi-positive) metric yields the same vanishing after twisting by determinant: H^q(X, ω^GR_X(E⊗detE, h⊗deth)) = 0 for q>0.
  • The L2-Dolbeault complex resolves ω^GR_X(E,h) by fine sheaves, so the structure sheaf's cohomology is computed by the L2 ∂-complex — a new cohomological isomorphism for singular spaces with general singular metrics (beyond line-bundle-type singularities and tame metrics).
  • Higher direct images R^qπ_*(K_ X̃ ⊗ E(π^*h)) vanish for q>0 under the local Griffiths–Nakano boundedness condition, a relative vanishing theorem for higher direct image sheaves on complex spaces.

Reading between the lines

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

  • The L2-based definition of Nakano positivity could serve as a template for other analytic positivity notions on singular spaces, since it is insensitive to codimension-one behavior and bypasses the failure of naive curvature pull-backs.
  • The reliance on the exhaustion premise (Theorem 5.1) suggests a close tie between the vanishing theorem and the existence of 'enough' Stein neighborhoods on weakly pseudoconvex spaces; if that exhaustion fails for some class of spaces, the vanishing might still hold but would require a different route.
  • One could test the strength of the theorem by checking whether it recovers, on smooth spaces, the known Nakano–Nadel and relative vanishing results, and by seeking examples of singular metrics of line-bundle type where the L2-subsheaf coincides with the classical multiplier ideal sheaf — the theorem then reduces to a familiar statement.
  • The compact/Moishezon conclusion suggests a converse direction worth exploring: if all higher cohomology of ω^GR_X(E,h) vanishes for some positive singular metric, does X necessarily admit a singular positive line bundle or have a projectivizable desingularization?
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a theory of singular Griffiths and Nakano positivity for Hermitian metrics on holomorphic vector bundles over reduced complex spaces of pure dimension. It introduces definitions via local plurisubharmonicity and L2-estimates, proves that these notions behave well under resolution of singularities, and shows that Griffiths positivity implies Nakano positivity after twisting by the determinant (Theorem 4.13). It then constructs L2-Dolbeault fine resolutions of the Grauert-Riemenschneider canonical L2-sheaf (Theorem 1.8) and proves a global L2-existence theorem on weakly pseudoconvex Kähler complex spaces (Theorem 1.11). The main application is a Nakano-Nadel vanishing theorem (Theorem 1.12/7.6): under a weak pseudoconvexity and positivity assumption, H^q(X, ω^GR_X(E,h)) = 0 for q > 0, with H^1 vanishing in the non-Kähler case. The approach reduces to known results on manifolds via desingularization, but key structural inputs are imported from several same-author preprints without proofs.

Significance. If the main theorems are correct, the paper would establish that the Hörmander–Andreotti–Vesentini L2 machinery works for singular vector-bundle metrics on singular spaces, substantially generalizing previous line-bundle and smooth-metric results (Ruppenthal, Shentu–Zhao, Inayama, Watanabe). The paper is honest about its limitations (Remarks 3.2, 7.7) and contains several genuinely new technical results, such as the pullback equivalence of Nakano positivity (Theorem 4.12) and the L2-resolution theorem. The strengths include careful treatment of currents and plurisubharmonic extension on non-normal spaces, and correct identification of where positivity degenerates along the exceptional divisor. However, the central claims are not independently checkable from the manuscript because of the heavy reliance on unpublished same-author preprints, and one imported theorem is stated insufficiently for its use. The significance would be high if the gaps are closed.

major comments (3)
  1. [§5, Theorem 5.1 and proof of Theorem 5.3] Theorem 5.1 is the load-bearing structural input for the global L2-existence theorem. As stated, it only asserts an increasing sequence {c_j} with c_1 > inf_X Ψ, not that c_j → +∞. The proof of Theorem 5.3 applies the local L2 estimates on S_j = ((X_{c_j})_reg \ A_j) \ H_j and then uses [Wat25a, Lemma 3.18] to obtain a weak limit u on all of X. This requires the domains S_j to exhaust X, i.e. c_j → +∞ (or at least ∪_j X_{c_j} = X). Without exhaustiveness the weak-limit step cannot produce a global solution. In addition, the analytic subsets A_j are not specified to be closed in X nor to have any codimension, and 'increasing' alone does not imply exhaustion. The author must either state and prove the full version of [Wat26b, Theorem 4.4] with c_j → +∞, or provide an alternative exhaustion argument. This is not a cosmetic issue: Theorem 1.12 depends on it.
  2. [§6, Theorem 6.5; §2, Lemma 2.2; §7, Theorem 7.3] Several key steps are imported from same-author preprints that are not proved here and are not otherwise publicly verified: [Wat25c, Theorem 1.1] (Nakano-Nadel vanishing on weakly pseudoconvex manifolds) is used in the proof of Theorem 6.5; [Wat26b, Theorem 4.4] is used for Theorem 5.1; [Wat25b, Lemma 2.2] is used for the Negativity Lemma; [Wat24b, Theorem 3.2] is used in Theorem 7.3. While referencing preprints is acceptable, for a central theorem of this scope the referee cannot verify correctness without seeing these proofs. The burden is on the author to either include the necessary statements with complete proofs in an appendix, or to ensure these results are accepted and publicly available with full details. At minimum, the precise statements (including the exhaustive sequence issue) must be quoted correctly.
  3. [§7.2, proof of Theorem 7.6] The argument 'the integral ... become convergent if χ grows fast enough' is too terse. For a given f ∈ Γ(X, L^{n,q}_{E,h}), one must show there exists a smooth convex increasing χ such that ∫_X ⟨B^{-1}_{θ,χ∘Ψ,ω} f,f⟩_{h,ω} e^{-χ∘Ψ} dV_ω < +∞. Since f is only locally L2 with respect to h, its growth as Ψ → +∞ is uncontrolled; the existence of such χ is plausible (as in the standard trick of choosing χ depending on f) but requires a proof. This step is essential for applying Theorem 1.11. Also, in the same proof the notation 'L^{n,q}_{L,h}' should be 'L^{n,q}_{E,h}'.
minor comments (5)
  1. [Throughout] There are numerous typographical errors, most notably 'resolusion' for 'resolution' (e.g., pages 2, 4, 9, 27, 28), 'resulution' on page 4, and inconsistent spelling of 'Kähler' in a few places. The paper would benefit from a careful proofreading pass.
  2. [§2.4, Definition 2.4 and L1_loc] The definition of L1_loc(U) uses integrals over K_reg with respect to a volume measure that is not explicitly defined on a singular space. It would be helpful to state that the measure is induced by a local embedding into C^N, as is done later in §2.4.
  3. [§4, Theorem 4.14(b)] In the display after 'there exists u ∈ L^2_{n,q−1}(S_reg \ H, E; ...)' the equality '= L^2_{n,q}(S,E⊗L; ...)' appears to have a typo: the left-hand side should be L^2_{n,q}(S_reg \ H, E; ...) rather than L^2_{n,q−1}. The dimension q−1 vs q is inconsistent.
  4. [§5, Definition of weak pseudoconvexity] The definition of exhaustion is incomplete: a function Ψ is an exhaustion if all sublevel sets X_c = {x ∈ X | Ψ(x) < c} are relatively compact, but one usually also requires that the sublevel sets exhaust X as c → +∞, i.e., that Ψ is proper and bounded below. The current wording does not imply that the sequence c_j in Theorem 5.1 is unbounded.
  5. [References] [Wat25c] is listed as 'preprint' without an arXiv number or publication status; please update if it has been accepted. Similarly, [Wat26b] is an arXiv preprint from 2026; its status should be made clear so that readers can assess the availability of the cited results.

Circularity Check

1 steps flagged · score 4.0 of 10

Central global L2-existence rests on an imported same-author Stein-exhaustion theorem, but no internal step reduces to its own input by construction.

  1. self citation load bearing [§5, Theorem 5.1 and Proof of Theorem 5.3; applied in §7.2, Theorem 7.6]
    "Theorem 5.1 ([Wat26b, Theorem 4.4]). If a weakly pseudoconvex complex space X admits a singular positive line bundle, then there exist an increasing sequence of real numbers {c_j}_{j∈N} with c_1 > inf_X Ψ and analytic subsets A_j of each X_{c_j} such that each open subset (X_{c_j})_reg \ A_j = (X_{c_j} \ A_j) ∩ X_reg is Stein manifold."

    The proof of the global L2-existence theorem (Theorem 5.3 = Theorem 1.11), which is the engine for the Nakano-Nadel vanishing Theorem 7.6, begins 'By Theorem 5.1...' and constructs solutions only on S_j = ((X_{c_j})_reg \ A_j) \ H_j, relying on those being Stein. The global solution on all of X is then obtained by 'applying [Wat25a, Lemma 3.18]' as a weak limit. Thus the global ∂-existence theorem, and therefore H^q(X, ω^GR_X(E,h)) = 0, has no derivation inside this paper unless the same-author preprint [Wat26b, Theorem 4.4] supplies the asserted Stein exhaustion. That theorem is not proved, machine-checked, or independently reproduced here; it is a load-bearing structural premise. Moreover, the quoted statement only says {c_j} is increasing, not that c_j → +∞, so the weak-limit step's cov

full rationale

The paper's original content—Definitions 3.1/3.3, the pullback theorems 4.1 and 4.12, the coherence results, and the L2-Dolbeault resolution Theorem 1.8—is not constructed from the final vanishing theorem. Theorem 4.12 is proved by direct restriction/pushforward on Stein charts and does not assume the target. Theorem 1.8 proves exactness at singular points using the local L2-existence theorem on a Stein neighborhood, not the global vanishing. The global Nakano-Nadel theorem is then obtained by applying the global L2-existence theorem to each ∂-closed form; this is the standard Hörmander–Andreotti–Vesentini route and is not definitionally identical to the Nakano-positivity condition in Definition 3.3, which only quantifies over Stein coordinates. No equation in the paper is shown to equal another by construction, and no fitted parameter is renamed as a prediction. The one genuinely load-bearing dependency not established in the text is the Stein-exhaustion theorem imported from [Wat26b], a same-author preprint that is not machine-checked or independently reproduced. Because this is a different statement from the vanishing claim and the central derivation has independent content, the circularity score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No empirical free parameters appear; constants such as C, ε, δ are existential choices in proofs, not fitted values. No new physical or geometric entities are postulated; the new sheaves are definitions, not independent axioms.

assumptions (6)
  • domain assumption Canonical desingularization π: (X̃, Exc) → X exists for non-compact reduced complex spaces, with Exc simple normal crossing.
    Theorem 2.1 (Bierstone-Milman/Hironaka) is used throughout; it is a standard theorem but not proved here.
  • domain assumption Theorem 5.1: a weakly pseudoconvex complex space admitting a singular positive line bundle has exhaustion sublevel sets whose regular parts minus analytic subsets are Stein manifolds.
    Imported verbatim from same-author [Wat26b, Thm 4.4]; proof not included; drives global L2 existence in Theorem 5.3.
  • standard math Strong openness property for multiplier L2-subsheaves of vector bundles.
    Used in Theorem 4.16 to preserve E(π*h) when adding small weights; cited from GZ15/LXYZ24.
  • domain assumption Negativity Lemma / compensation lemma: there exists a quasi-plurisubharmonic ψ on π^{-1}(V) whose Levi form compensates for degeneracy of π*ω along Exc.
    Lemma 2.2 from [Wat25b, Lemma 3.2]; essential for Theorem 4.16 and compact-case vanishing.
  • domain assumption Nakano-Nadel vanishing on weakly pseudoconvex manifolds ([Wat25a, Cor 6.2], [Wat25c, Thm 1.1]).
    Key black box in proof of Theorem 6.5; Wat25c is an unpublished same-author preprint with no arXiv ID.
  • standard math Coherence of the L2-subsheaf E(h) for Nakano semipositive h on manifolds ([Ina22]).
    Used to prove coherence of ω^GR via direct image theorem.

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Pith. "Pith review of Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces." pith.science (2026). https://pith.science/paper/UDLUQIYI

@misc{pith2026260616275,
  author       = {Pith},
  title        = {Pith review of: Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UDLUQIYI}},
  note         = {Machine review of arXiv:2606.16275}
}
abstract

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we introduce appropriate notions of singular Griffiths/Nakano positivity on complex spaces and establish various properties of these notions. By applying these results, we provide $L^2$-Dolbeault fine resolutions and cohomological isomorphisms, and $L^2$-existence theorems. As an application, we obtain Nakano-Nadel vanishing theorems on weakly pseudoconvex complex spaces.

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Cited by 1 Pith paper

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

  1. Steenbrink vanishing theorem for big line bundles

    math.CV 2026-08 conditional novelty 6.0 of 10

    Big line bundles on compact complex spaces satisfy a Steenbrink-type cohomology vanishing theorem after passing to a log resolution and twisting by multiplier ideal sheaves.

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Works this paper leans on

43 extracted references · 3 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Andersson and H

    M. Andersson and H. Samuelsson, A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas, Invent. Math. 190 (2012), 261--297

    2012

  2. [2]

    Andreotti and E

    A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Publ. Math. Inst. Hautes \' E tudes Sci. 25 (1965), 81-130

    1965

  3. [3]

    Berndtsson and M

    B. Berndtsson and M. P a un, Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), no. 2, 341--378

    2008

  4. [4]

    Bierstone and P.D

    E. Bierstone and P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207-302

    1997

  5. [5]

    M. A. A. de Cataldo, Singular Hermitian metrics on vector bundles, J. Reine Angew. Math. 502, (1998), 93-122

    1998

  6. [6]

    Demailly, Estimations L^2 pour l'op\' e rateur d'un fibr\' e vectoriel holomorphe semi-positif au dessus d'une vari\' e t\' e K\" a hl\' e rienne compl\` e te , Ann

    J.-P. Demailly, Estimations L^2 pour l'op\' e rateur d'un fibr\' e vectoriel holomorphe semi-positif au dessus d'une vari\' e t\' e K\" a hl\' e rienne compl\` e te , Ann. Sci. Ec. Norm. Sup. 15, (1982), 457-511

    1982

  7. [7]

    Demailly, Mesures de Monge-Amp\` e re et caract\' e risation g\' e om\' e trique des vari\' e t\' e s alg\' e briques affines , M\' e m

    J.-P. Demailly, Mesures de Monge-Amp\` e re et caract\' e risation g\' e om\' e trique des vari\' e t\' e s alg\' e briques affines , M\' e m. Soc. Math. France (N.S.) (1985), no. 19, p. 124

    1985

  8. [8]

    Demailly, Singular Hermitian metrics on positive line bundles, Proc

    J.-P. Demailly, Singular Hermitian metrics on positive line bundles, Proc. Conf. Complex algebraic varieties (Bayreuth, April 2-6, 1990), edited by K. Hulek, T. Peternell, M. Schneider, F. Schreyer, Lecture Notes in Math., Vol. 1507, Springer-Verlag, Berlin, 1992

    1990

Show all 43 references
  1. [9]

    Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, in School on Vanishing Theorems and Effective Results in Algebraic Geometry, ICTP Lecture Notes, vol

    J.-P. Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, in School on Vanishing Theorems and Effective Results in Algebraic Geometry, ICTP Lecture Notes, vol. 6, The Abdus Salam International Centre for Theoretical Physics, 2001, 1-148

    2001

  2. [10]

    Demailly, Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics, vol

    J.-P. Demailly, Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics, vol. 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012

    2012

  3. [11]

    Demailly, Complex analytic and differential geometry, https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf

    J.-P. Demailly, Complex analytic and differential geometry, https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf

  4. [12]

    F. Deng, J. Ning, Z. Wang and X. Zhou, Positivity of holomorphic vector bundles in terms of L^p -estimates for , Math. Ann. 385 (2023), 575--607

    2023

  5. [13]

    Diederich, Some aspects of the Levi problem: recent developments, in Geometric Complex Analysis, Proc

    K. Diederich, Some aspects of the Levi problem: recent developments, in Geometric Complex Analysis, Proc. 3rd IRIMSJ Conf., World Scientific, 1996, 163-181

    1996

  6. [14]

    Forn ss and R

    J.E. Forn ss and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), 47-72

    1980

  7. [15]

    Grauert, T

    H. Grauert, T. Peternell and R. Remmert, editors. Several Complex Variables VII, Sheaf-Theoretical Methods in Complex Analysis, volume 74 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1994

    1994

  8. [16]

    Grauert and R

    H. Grauert and R. Remmert, Plurisubharmonische Funktionen in komplexen R\" a umen , Math. Z. 65 (1956), 175-194

    1956

  9. [17]

    Guan and X

    Q. Guan and X. Zhou, A Proof of Demailly's Strong Openness Conjecture, Annals of Mathematics, 182 (2015), pp. 605-616

    2015

  10. [18]

    Hacon, M

    C. Hacon, M. Popa and C. Schnell, Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and P a un , Local and global methods in algebraic geometry, 143--195, Contemp. Math., 712, Amer. Math. Soc., Providence, RI, 2018

    2018

  11. [19]

    Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann

    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109-203; II, 205-326

    1964

  12. [20]

    H\"ormander, L^2 estimates and existence theorems for the operator, Acta Math

    L. H\"ormander, L^2 estimates and existence theorems for the operator, Acta Math. 113 (1965), 89-152

    1965

  13. [21]

    H\"ormander, An introduction to complex analysis in several variables, Third edition, NorthHolland Mathematical Library, 7

    L. H\"ormander, An introduction to complex analysis in several variables, Third edition, NorthHolland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, (1990)

    1990

  14. [22]

    Inayama, Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type, Algebr

    T. Inayama, Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type, Algebr. Geom. 9 (1) (2022) 69-92

    2022

  15. [23]

    Inayama, S

    T. Inayama, S. Matsumura and Y. Watanabe, Singular Nakano positivity of direct image sheaves of adjoint bundles, accepted for publication in Anal. PDE

  16. [24]

    Jonsson and M

    M. Jonsson and M. Musta t a , Valuations and asymptotic invariants for sequences of ideals, Annales de L'Institut Fourier, 62(6):2145-2209, 2012

    2012

  17. [25]

    Kazama, On pseudoconvexity of complex abelian Lie groups, J

    H. Kazama, On pseudoconvexity of complex abelian Lie groups, J. Math. Soc. Japan, 25 (1973), 329-333

    1973

  18. [26]

    Z. Liu, B. Xiao, H. Yang and X. Zhou, Multiplier Submodule Sheaves and a problem of Lempert, arXiv:2111.13452v2

    arXiv

  19. [27]

    Moishezon, On n -dimensional compact varieties with n algebraically independant meromorphic functions

    B. Moishezon, On n -dimensional compact varieties with n algebraically independant meromorphic functions. I, II, III, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 133-174, 345-386, 621-656

    1966

  20. [28]

    A. M. Nadel, Multiplier ideal sheaves and K\" a hler-Einstein metrics of positive scalar curvature , Ann. Math. (2) 132 (1990), no. 3, p. 549-596

    1990

  21. [29]

    Ohsawa and K

    T. Ohsawa and K. Takegoshi, On the Extension of L^2 Holomorphic Functions, Math. Z. 195, (1987), 197-204

    1987

  22. [30]

    P a un and S

    M. P a un and S. Takayama, Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebraic Geom. 27, (2018), 211-272

    2018

  23. [31]

    Raufi, Singular hermitian metrics on holomorphic vector bundles, Ark

    H. Raufi, Singular hermitian metrics on holomorphic vector bundles, Ark. Mat. 53, (2015), no. 2, 359-382

    2015

  24. [32]

    Ruppenthal, L^2 -theory for the -operator on compact complex spaces, Duke Math

    J. Ruppenthal, L^2 -theory for the -operator on compact complex spaces, Duke Math. J. 163 (2014), no. 15, 2887-2934

    2014

  25. [33]

    Shentu and C

    J. Shentu and C. Zhao, MacPherson's Conjecture via H\" o rmander's estimate , Int. Math. Res. Not. IMRN 2023 (2023), 2170-2187

    2023

  26. [34]

    Shentu and C

    J. Shentu and C. Zhao, L^2 -extension of adjoint bundles and Koll\' a r's conjecture , Math. Ann. 386 (2023), 1429-1462

    2023

  27. [35]

    Shentu and C

    J. Shentu and C. Zhao, L^2 -Dolbeault resolusion of the lowest Hodge piece of a Hodge module, Adv. Math. 442 no.2, 109547

  28. [36]

    Takegoshi, Relative vanishing theorems in analytic spaces, Duke Math

    K. Takegoshi, Relative vanishing theorems in analytic spaces, Duke Math. J. 52 (1985), 273-279

    1985

  29. [37]

    Watanabe, -trace and Griffiths positivity for singular Hermitian metrics, arXiv:2402.06658

    Y. Watanabe, -trace and Griffiths positivity for singular Hermitian metrics, arXiv:2402.06658

    arXiv

  30. [38]

    Watanabe, Bigness of adjoint linear subsystem and approximation theorems with ideal sheaves on weakly pseudoconvex manifolds, arXiv:2412.02007v2

    Y. Watanabe, Bigness of adjoint linear subsystem and approximation theorems with ideal sheaves on weakly pseudoconvex manifolds, arXiv:2412.02007v2

  31. [39]

    Watanabe, Nakano-Nadel type, Bogomolov-Sommese type vanishing and singular dual Nakano semi-positivity, Ann

    Y. Watanabe, Nakano-Nadel type, Bogomolov-Sommese type vanishing and singular dual Nakano semi-positivity, Ann. Fac. Sci. Toulouse Math. (6) 34(2), 339-394 (2025)

    2025

  32. [40]

    Watanabe, Global embeddings of weakly pseudoconvex complex spaces and refined Runge-type approximation theorems, arXiv:2512.03572v3

    Y. Watanabe, Global embeddings of weakly pseudoconvex complex spaces and refined Runge-type approximation theorems, arXiv:2512.03572v3

  33. [41]

    Watanabe, Nakano-Nadel vanishing theorem on weakly pseudoconvex manifolds, preprint

    Y. Watanabe, Nakano-Nadel vanishing theorem on weakly pseudoconvex manifolds, preprint

  34. [42]

    Watanabe, L^2 -Dolbeault isomorphisms and vanishing theorems for logarithmic sheaves twisted by multiplier ideal sheaves, Math

    Y. Watanabe, L^2 -Dolbeault isomorphisms and vanishing theorems for logarithmic sheaves twisted by multiplier ideal sheaves, Math. Z. 312 (2026), Paper No. 40, 28pp

    2026

  35. [43]

    Watanabe, L^2 -Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics, arXiv:2602.03332

    Y. Watanabe, L^2 -Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics, arXiv:2602.03332

    arXiv

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Reviewed August 2, 2026 · model on record in the stance chip above.