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Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes

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Pith's one-line read Slope stability and Hermitian-Einstein metrics are equivalent for big classes on normal spaces.

desk verdict A credible framework paper for big-class stability and HE metrics, but the main Kobayashi-Hitchin correspondence is not proved as stated because the HE-to-stability direction needs resolution independence the paper does not supply. read the letter →

arxiv 2501.04910 v6 pith:6C5DPFXW submitted 2025-01-09 math.AG math.DG

Satoshi Jinnouchi This is my paper · ORCID
classification math.AGmath.DG MSC 14J6032Q2614E30
keywords Kobayashi-HitchincorrespondencebigcohomologyclassslopestabilityHermitian-Einsteinmetricnon-pluripolarproductnormalvarietybirationalZariskidecompositionBogomolov-Giesekerinequality
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 extends the Kobayashi-Hitchin correspondence—the equivalence between slope stability and the existence of Hermitian-Einstein metrics—from Kaehler classes to big cohomology classes on compact normal complex spaces. The main theorem states that when a big class admits a birational Zariski decomposition with semiample positive part, a reflexive sheaf is slope stable with respect to the class if and only if it carries a Hermitian-Einstein metric adapted to a suitable closed positive current in the class. The proof works by transferring the statement along a birational map to a model where the positive part becomes a genuine Kaehler class, where the classical correspondence is known, and then pulling the metric and stability back. A corollary gives the correspondence for projective varieties of general type, and the same techniques yield a Bogomolov-Gieseker inequality for slope-stable reflexive sheaves with respect to nef and big classes.

What carries the argument

The load-bearing object is the positive product angle-bracket $alpha^{{n-1}}$ of a big class, defined via non-pluripolar products of closed positive currents with minimal singularities. It replaces the ordinary wedge power when the class is not nef, and it is used to define both the slope mu_alpha(E) and the Einstein constant $\lambda$. The second key mechanism is bimeromorphic invariance: under an orthogonality assumption (Assumption 3.1), both slope stability and the existence of T-Hermitian-Einstein metrics are unchanged under $\beta$-negative birational maps, which allows the problem to be pushed to a resolution or a canonical model where $\alpha$ becomes a pullback of a Kaehler class. The birational Zariski decomposition with semiample positive part is what guarantees such a model exists without needing the orthogonality assumption in the main theorem. Finally, the openness of slope stability in the class (Propositions 7.3 and 7.5) supplies the limit argument that yields the Bogomolov-Gieseker inequality for nef and big classes.

What would settle it

Produce a compact normal space X with a big class $\alpha$ admitting a birational Zariski decomposition with semiample positive part and a resolution pi: Y to X with an exceptional divisor D such that angle-bracket (pi^*$\alpha$)^{n-1} cdot [D] is nonzero; this would invalidate the resolution-independence step and with it Theorem 6.1. Equivalently, find a reflexive sheaf E on such an X that is slope stable with respect to one resolution but not another, or that is slope stable but provably admits no T-Hermitian-Einstein metric.

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Extended reading notes

Core claim

The central claim is Theorem 6.1: let X be a compact normal space, $\alpha$ a big Bott-Chern class, and E a reflexive sheaf. If $\alpha$ admits a birational Zariski decomposition whose positive part is big and semiample, then E is angle-bracket $alpha^{{n-1}}$-slope stable exactly when it admits a T-Hermitian-Einstein metric for some closed positive (1,1)-current T in $\alpha$. The slope is computed against the positive product angle-bracket $alpha^{{n-1}}$, defined through non-pluripolar products of currents with minimal singularities, and the metric is required to satisfy the Einstein condition $\sqrt$(-1) Lambda_T F_h = $\lambda$ Id on the ample locus away from the singularities, with finite $L^{2}$ norm. The proof shows resolution-independence of both notions under the stated hypothesis and uses the birational model where the positive part of the class is a pullback of a Kaehler class; on that model the result reduces to the known correspondence for normal varieties, and bimeromorphic invariance transfers it back. The same circle of ideas establishes the Bogomolov-Gieseker inequality for nef and big classes and the projective-flatness characterization in the equality case for minimal varieties of general type.

Load-bearing premise

The main theorem assumes that for a class with a birational Zariski decomposition whose positive part is semiample, the positive product angle-bracket (pi^*$\alpha$)^{n-1} pairs trivially with every exceptional divisor of every resolution of the space; if that orthogonality fails, the definitions of slope stability and Hermitian-Einstein metrics could depend on the resolution chosen and the correspondence would not be well-posed.

Editorial extensions

If this is right

  • For any normal projective variety of general type with log terminal singularities, a reflexive sheaf is c_1(K_X)^{n-1}-slope stable if and only if it admits a T-Hermitian-Einstein metric.
  • The tangent sheaf, cotangent sheaf, and their tensor and exterior products of such a variety are slope polystable and admit T-Hermitian-Einstein metrics.
  • A slope-stable reflexive sheaf with respect to a nef and big class satisfies the Bogomolov-Gieseker inequality; in the equality case on minimal varieties of general type it is projectively flat on the ample locus.
  • On K3 surfaces, any nef and big class is semiample, so the equality case of the Bogomolov-Gieseker inequality forces projective flatness on the ample locus.
  • The correspondence and the inequality remain valid for polystable sheaves by the same arguments.

Reading between the lines

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

  • If the orthogonality condition in Assumption 3.1 holds for all big classes—a conjecture the paper connects to the differentiability of volumes of big classes—then the Kobayashi-Hitchin correspondence would extend to every big class on compact normal spaces, not only those with a birational Zariski decomposition with semiample positive part.
  • The limit argument in Section 7 suggests a stability threshold: for a fixed reflexive sheaf, the locus of big classes for which it is slope stable is open in the cone of big classes, so moduli spaces of sheaves with respect to big classes could be studied through their behavior near the boundary with nef classes.
  • The same birational-transfer strategy could apply to other geometric PDEs on singular spaces, such as Hermitian-Yang-Mills equations with additional twisted or Higgs fields, whenever a bimeromorphic model with a Kaehler class is available.
  • The projective-flatness conclusion in the equality case may be testable numerically: computing the discriminant Delta(E) cdot alpha^{n-2} on explicit examples of minimal varieties would verify the bound and its sharpness.
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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

4 major / 5 minor

Summary. The paper extends the notions of slope stability and Hermitian-Einstein (HE) metrics from Kähler classes to arbitrary big cohomology classes on compact normal complex spaces, using the non-pluripolar product ⟨α^{n-1}⟩. The central result, Theorem 1.5/6.1, is a Kobayashi–Hitchin correspondence for reflexive sheaves on compact normal spaces when the big class admits a birational Zariski decomposition with semiample positive part. The proof strategy is to establish bimeromorphic invariance of stability and of the existence of HE metrics under an orthogonality assumption (Assumption 3.1), then descend to a Kähler class on a model via [Chen25]. A further result, Theorem 1.6/Proposition 7.7, proves the Bogomolov–Gieseker inequality for nef and big classes, with an equality characterization on minimal projective varieties of general type.

Significance. If correct, the main theorem would be a meaningful extension of the Kobayashi–Hitchin correspondence to singular spaces and big classes, with applications to canonical models, tangent sheaf polystability, and the Bogomolov–Gieseker inequality. The paper has a clear overall strategy and makes use of substantial external results (Chen25, BEGZ10, Nyström, BCHM10). The Bogomolov–Gieseker section is mostly self-contained and contains a useful openness argument. However, the proof of the main correspondence has a load-bearing gap concerning resolution independence under the stated hypotheses, and the formal statement of Theorem 6.1 is weaker than the abstract and Theorem 1.5. These issues are local and fixable, but they currently prevent the main claim from being established as written.

major comments (4)
  1. [§6, Theorem 6.1 and Definition 5.4] The statement of Theorem 6.1 proves only one direction: it says 'E is ⟨α^{n-1}⟩-stable if E admits a T-HE metric'. The abstract and Theorem 1.5 assert the full equivalence. Moreover, the proof of the converse direction begins with 'assume that µ^*E admits a T := (π^*ω + [D])-HE metric' on the specific resolution µ of the birational Zariski decomposition, whereas Definition 5.4 defines 'E admits a T-HE metric' existentially over some resolution. Passing from the given resolution to µ is exactly the resolution-independence statement of Lemma 5.5, whose proof invokes Assumption 3.1. Since Theorem 6.1 claims that Assumption 3.1 is not needed, this step is not justified. The paper must either prove the required orthogonality for the resolutions involved or replace Definition 5.4 with a resolution-choice that is compatible with the theorem.
  2. [§4.2, Lemma 4.7 and §6 proof] Lemma 4.7, which establishes independence of the chosen resolution in Definition 4.6, is proved under Assumption 3.1. The proof of Theorem 6.1 repeatedly applies Lemma 4.7 and Theorem 4.9 to conclude stability of µ^*E from stability of E, and vice versa, without Assumption 3.1. In the special case of a birational Zariski decomposition with semiample positive part, the needed orthogonality ⟨(µ^*α)^{n-1}⟩·[D] = 0 for π-exceptional divisors D may follow from the projection formula because ⟨µ^*α⟩ = π^*ω and D is π-exceptional, but this argument is not written. As it stands, the theorem's claim that Assumption 3.1 is unnecessary is not supported by the proof.
  3. [§5.3 and §6 proof] The transition from the ω-HE metric on π[∗]µ[∗]E, obtained from Chen25, to the T-HE metric on µ^*E with T = π^*ω + [D] uses Theorem 5.6, a statement whose proof relies on Assumption 3.1 (via equation (4.3)). The special case D being π-exceptional and π^*ω being semiample should make the check feasible by the projection formula, but the verification of the Einstein constant and the integrability condition ∫ |F|_T^2 T^n < ∞ is not carried out in the proof of Theorem 6.1. This is another load-bearing point that needs an explicit argument.
  4. [§4.2, Definition 4.6] Definition 4.6 defines ⟨α^{n-1}⟩-stability on a normal space by requiring stability of the reflexive pullback for every resolution. Lemma 4.7 then proves that this is equivalent to checking one resolution, under Assumption 3.1. Theorem 6.1 states that for classes with a birational Zariski decomposition with semiample positive part, Assumption 3.1 is unnecessary, but the paper does not prove that Definition 4.6 is well-posed for such classes without the assumption. The reader is left unable to determine whether the main theorem is about Definition 4.6 at all, or about a weaker notion where only the chosen resolution is tested.
minor comments (5)
  1. [§1.2, Abstract] There are several typographical inconsistencies: 'Kobayashi-Hichin' should be 'Kobayashi-Hitchin', and 'Kähler' is misspelled in a few places (e.g., 'compact Kähler manifolds' appears as 'compact K¨ aher manifolds' in §1.2).
  2. [§5.2, Definition 5.4] Definition 5.4 reads 'We fix a resolution π:Y→X' but the definition is existential in the resolution. The wording should be changed to avoid the apparent contradiction.
  3. [§5.2, Lemma 5.5 proof] The proof writes 'p2_*(p1^*h)' as the transferred metric; a pushforward of a Hermitian metric is not a standard operation and needs a precise definition, especially regarding regularity on the ample locus.
  4. [§7.1, Lemma 7.2 proof] The displayed formula in (7.5) contains several typos: 'T r(bp · F_{ν^*h0} · bp + \bar∂ bp ∧ ∂_{h0} bp)' should presumably be 'tr(bp · F_{ν^*h0} · bp + \bar∂ bp ∧ ∂_{h0} bp)', and the following inequality uses '∥Fh0∥L∞rk(F)' which should be 'rk(F)∥F_{h0}∥_{L∞}'.
  5. [Throughout] The notation π[∗]E is used in Example 4.5 before it is formally defined in Lemma 4.7; a short definition in §4.1 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Kobayashi–Hitchin result is reduced to independent external theorems (Chen25, BEGZ10, CT15, Nystr19, BCHM10), with no fitted parameters and no load-bearing self-citations.

full rationale

The derivation chain is not circular. Slope stability and T-Hermitian-Einstein metrics are defined by standard formulas involving the positive product, Chern classes, and the non-pluripolar Monge-Ampère normalization; the stability/HE equivalence is not built into these definitions. The bimeromorphic invariance results (Lemma 4.7, Theorem 4.9, Lemma 5.5) are proved under Assumption 3.1, which is itself an external orthogonality statement justified by Nyström on projective manifolds and by Collins–Tosatti for Zariski decompositions; it is not a restatement of the target theorem. The proof of Theorem 6.1 proceeds by pulling back to the resolution realizing the birational Zariski decomposition, invoking Theorem 3.4 to identify the positive product, Theorem 4.9 to pass to the semiample model, and Chen25 for the smooth-model Kobayashi–Hitchin correspondence in both directions. No parameter is fitted to any subset of the data and then renamed a prediction. The skeptical concern about Definition 5.4 versus the fixed resolution µ in the HE-to-stability direction is a real potential gap in the written proof, because the transfer between resolutions relies on Lemma 5.5/Assumption 3.1, which the theorem claims to avoid; however, this is a completeness or correctness issue, not circularity. The conclusion is not assumed in the hypotheses, and the missing orthogonality is not asserted by definition but would need an independent argument, for example via the projection formula after further pullback. Under the hard rule requiring an exhibited reduction of a result to its own input, no such reduction is present. There are no self-citations by the author to prior work carrying argumentative weight. For these reasons the paper receives a circularity score of 0.

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

No free parameters appear in the paper. The central claim is conditional on a nonstandard assumption, Assumption 3.1, and on external theorems from Chen25, BEGZ10 and BCHM10. The ledger would be nearly empty if those external results are accepted as standard, but Assumption 3.1 is a genuine added postulate in the compact Kaehler setting.

assumptions (6)
  • ad hoc to paper Assumption 3.1: for every bimeromorphic morphism pi:Y to X between compact Kaehler manifolds, big class alpha on X and pi-exceptional divisor D, the positive product satisfies `⟨(pi^*alpha)^{n-1}⟩ · [D] = 0`.
    Unproved in the general compact Kaehler setting; known for projective varieties by Nyström. It underpins the resolution-independence of slope stability in Lemma 4.7 and of T-Hermitian-Einstein metrics in Lemma 5.5.
  • standard math Chen25's Kobayashi-Hitchin correspondence for admissible Hermitian-Yang-Mills connections on compact normal Kaehler spaces.
    External theorem used as a black box in Theorem 6.1 to produce the Hermitian-Einstein metric and slope stability on the semiample model Y.
  • standard math Boucksom-Eyssidieux-Guedj-Zeriahi [BEGZ10] theory of non-pluripolar products and Monge-Ampere equations in big classes.
    Defines the positive products `⟨alpha^{n-1}⟩` used throughout the paper, including minimal singularity representatives and the volume characterization of bigness.
  • domain assumption Existence of canonical models and termination of the minimal model program for varieties of log general type [BCHM10].
    Used in Corollary 6.2 and Example 6.3 to provide the birational Zariski decomposition with semiample positive part for K_X.
  • standard math Collins-Tosatti [CT15] equality between non-Kaehler locus and null locus for nef and big classes.
    Used in Proposition 2.17 and in several arguments to identify exceptional loci with non-Kaehler loci.
  • domain assumption For Theorem 6.1, the class alpha admits a birational Zariski decomposition whose positive part is big and semiample.
    This is the theorem's hypothesis. The proof does not cover general big classes without such a decomposition.

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Pith. "Pith review of Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes." pith.science (2026). https://pith.science/paper/6C5DPFXW

@misc{pith2026250104910,
  author       = {Pith},
  title        = {Pith review of: Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6C5DPFXW}},
  note         = {Machine review of arXiv:2501.04910}
}
read the original abstract

In this paper, we introduce the notions of slope stability and the Hermitian Einstein metric for big cohomology classes. The main result is the Kobayashi Hitchin correspondence on compact normal spaces with big classes admitting the birational Zariski decomposition with semiample positive part. We also prove the Bogomolov Gieseker inequality for slope stable sheaves with respect to big and nef classes. Through this paper, the bimeromorphic invariance of slope stability and the existence of Hermitian Einstein metrics plays an essential role.

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Forward citations

Cited by 1 Pith paper

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

  1. The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

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    For projective klt varieties with big canonical or anticanonical divisor, the Miyaoka-Yau Chern class inequality holds when intersections are taken with the non-pluripolar product.

Reference graph

Works this paper leans on

40 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Abramovich, K

    D. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531--572; MR1896232

    work page 2002

  2. [2]

    Band and Y

    S. Band and Y. T. Siu, Stable sheaves and Einstein-Hermitian metrics, in Geometry and analysis on complex manifolds , 39--50, World Sci. Publ., River Edge, NJ, ; MR1463962

    work page

  3. [3]

    Bedford and B

    E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1--40; MR0674165

    1982

  4. [4]

    Birkar, P

    C. Birkar, P. Cascini, D. Hacon and J. Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405--468; MR2601039

    work page 2010

  5. [5]

    Birkar and Z

    C. Birkar and Z. Hu, Polarized pairs, log minimal models, and Zariski decompositions, Nagoya Math. J. 215 (2014), 203--224; MR3263528

    work page 2014

  6. [6]

    Boucksom, On the volume of a line bundle, Internat

    S. Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), no. 10, 1043--1063; MR1945706

    work page 2002

  7. [7]

    Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann

    S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. \'Ecole Norm. Sup. (4) 37 (2004), no. 1, 45--76; MR2050205

    work page 2004

  8. [8]

    Boucksom, J.-P

    S. Boucksom, J.-P. Demailly, M. Paun and T. Peternell, The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), no. 2, 201--248; MR3019449

    work page 2013

Show all 40 references
  1. [9]

    Boucksom, P

    S. Boucksom, P. Eyssidieux, V. Guedji, and A. Zeriahi, Monge-Amp\`ere equations in big cohomology classes, Acta Math. 205 (2010), no. 2, 199--262; MR2746347

    2010

  2. [10]

    Boucksom, C

    S. Boucksom, C. Favre and M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), no. 2, 279--308; MR2475816

    2009

  3. [11]

    Cao, A remark on compact K\" ahler manifolds with nef anticanonical bundles and its applications

    J. Cao, A remark on compact K\" ahler manifolds with nef anticanonical bundles and its applications. arXiv preprint arXiv:1305.4397 (2013)

    2013 arXiv

  4. [12]

    Chen, Admissible Hermitian–Yang–Mills connections over normal varieties

    X. Chen, Admissible Hermitian–Yang–Mills connections over normal varieties. Mathematische Annalen (2025): 1-37

    2025

  5. [13]

    T. C. Collins and V. Tosatti, K\"ahler currents and null loci, Invent. Math. 202 (2015), no. 3, 1167--1198; MR3425388

    2015

  6. [14]

    T. C. Collins and V. Tosatti, Restricted volumes on K\"ahler manifolds, Ann. Fac. Sci. Toulouse Math. (6) 31 (2022), no. 3, 907--947; MR4452256

    2022

  7. [15]

    O. Das, C. D. Hacon and M. P aun, On the 4-dimensional minimal model program for K\"ahler varieties, Adv. Math. 443 (2024), Paper No. 109615, 68 pp.; MR4719824

    2024

  8. [16]

    O. Das, C. Hacon, and J. I. Yáñez, MMP for Generalized Pairs on K\" ahler 3-folds. arXiv preprint arXiv:2305.00524 (2023)

    2023 arXiv

  9. [17]

    Demailly, Regularization of closed positive currents and intersection theory, J

    J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361--409; MR1158622

    1992

  10. [18]

    Demailly, Complex Analytic and Differential Geometry

    J-.P. Demailly, Complex Analytic and Differential Geometry. Book available at https://www-fourier.ujf-grenoble.fr/ demailly/documents.html

  11. [19]

    Demailly, Analytic Methods in Algebraic Geometry, Higher Education Press, Surveys of Modern Mathematics, Vol

    J.-P. Demailly, Analytic Methods in Algebraic Geometry, Higher Education Press, Surveys of Modern Mathematics, Vol. 1, 2010, 231 pages. Book available at https://www-fourier.ujf-grenoble.fr/ demailly/documents.html

    2010

  12. [20]

    S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987), no. 1, 231--247; MR0885784

    1987

  13. [21]

    Filip and V

    S. Filip and V. Tosatti, Smooth and rough positive currents, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2981--2999; MR3959103

    2018

  14. [22]

    Fujino, Basepoint-free theorem of Reid-Fukuda type for quasi-log schemes, Publ

    O. Fujino, Basepoint-free theorem of Reid-Fukuda type for quasi-log schemes, Publ. Res. Inst. Math. Sci. 52 (2016), no. 1, 63--81; MR3452046

    2016

  15. [23]

    D. Greb, S. Kebekus and T. Peternell, Movable curves and semistable sheaves, Int. Math. Res. Not. IMRN 2016 , no. 2, 536--570; MR3493425

    2016

  16. [24]

    D. Greb, S. Kebekus, T. Peternell, B. Taji, The Miyaoka-Yau inequality and uniformisation of canonical models, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 52 (2019), no. 6, 1487--1535; MR4061021

    2019

  17. [25]

    Greb and M

    D. Greb and M. Toma, Compact moduli spaces for slope-semistable sheaves, Algebr. Geom. 4 (2017), no. 1, 40--78; MR3592465

    2017

  18. [26]

    Guedj and A

    V. Guedj and A. Z\'eriahi, Degenerate complex Monge-Amp\`ere equations , EMS Tracts in Mathematics, 26, Eur. Math. Soc., Z\"urich, 2017; MR3617346

    2017

  19. [27]

    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; MR3568336

    2016

  20. [28]

    Hacon, and M

    C. Hacon, and M. Paun, On the Canonical Bundle Formula and Adjunction for Generalized Kaehler Pairs. (2024), arXiv preprint arXiv:2404.12007

    2024 arXiv

  21. [29]

    Hironaka, Flattening theorem in complex-analytic geometry, Amer

    H. Hironaka, Flattening theorem in complex-analytic geometry, Amer. J. Math. 97 (1975), 503--547; MR0393556

    1975

  22. [30]

    Höring, and T

    A. Höring, and T. Peternell, (2016). Minimal models for Kähler threefolds. (2016), Inventiones mathematicae, 203(1), 217-264

    2016

  23. [31]

    Kawamata, Flops connect minimal models, Publ

    Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419--423; MR2426353

    2008

  24. [32]

    Kobayashi, Differential geometry of complex vector bundles , Publications of the Mathematical Society of Japan Kan\^o Memorial Lectures, 15 5, Princeton Univ

    S. Kobayashi, Differential geometry of complex vector bundles , Publications of the Mathematical Society of Japan Kan\^o Memorial Lectures, 15 5, Princeton Univ. Press, Princeton, NJ, 1987 Princeton Univ. Press, Princeton, NJ, 1987; MR0909698

    1987

  25. [33]

    R. K. Lazarsfeld, Positivity in algebraic geometry. I , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48, Springer, Berlin, 2004; MR2095471

    2004

  26. [34]

    R. K. Lazarsfeld, Positivity in algebraic geometry. II , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 49, Springer, Berlin, 2004; MR2095472

    2004

  27. [35]

    Lehmann, and J

    B. Lehmann, and J. Xiao. Zariski decomposition of curves on algebraic varieties. arXiv preprint arXiv:1507.04316 (2015)

    2015 arXiv

  28. [36]

    Ma and G

    X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels , Progress in Mathematics, 254, Birkh\"auser, Basel, 2007; MR2339952

    2007

  29. [37]

    Principato, (2013)

    M. Principato, (2013). Mobile product and zariski decomposition. arXiv preprint arXiv:1301.1477

    2013 arXiv

  30. [38]

    K. K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), no. S, S 257-- S 293; MR0861491

    1986

  31. [39]

    Vu, D. V. Derivative of volumes of big cohomology classes. (2023), arXiv preprint arXiv:2307.15909

    2023 arXiv

  32. [40]

    Witt Nystr\"om, Duality between the pseudoeffective and the movable cone on a projective manifold, J

    D. Witt Nystr\"om, Duality between the pseudoeffective and the movable cone on a projective manifold, J. Amer. Math. Soc. 32 (2019), no. 3, 675--689; MR3981985

    2019

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