Pith. sign in

REVIEW 2 major objections 6 minor 53 references

Rational Morita equivalence for holomorphic Poisson modules

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper proves that every locally free Poisson module on a klt Poisson projective variety is rationally Morita equivalent to one of four explicit elementary objects: a flat holomorphic sheaf, a meromorphic flat connection, a co-Higgs…

desk verdict A genuinely new framework for Poisson modules, with a central but repairable gap in the proof of the main theorem. read the letter →

arxiv 1908.02325 v2 pith:V2VVEV6L submitted 2019-08-06 math.AG math.DGmath.SG

Maurício Corrêa This is my paper · ORCID
classification math.AGmath.DGmath.SG MSC 17B6353D1770G4537F75
keywords PoissonmodulesrationalMoritaequivalencekltvarietiesco-Higgssheavesmeromorphicflatconnectionsholomorphicfoliationssymplecticsl2-Poisson
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

Poisson modules are vector bundles with a flat connection that differentiates along the Hamiltonian vector fields of a Poisson structure. This paper introduces rational Morita equivalence, a birational weakening of the usual Morita equivalence: two Poisson varieties are equivalent when a common normal variety maps to both by dominant Poisson morphisms. On projective varieties with mild singularities (klt varieties whose Poisson structure is either generically symplectic or has canonical foliation singularities), the main theorem states that every locally free Poisson module is rationally Morita equivalent to one of four explicit objects: a flat holomorphic sheaf, a sheaf with a meromorphic flat connection, a co-Higgs sheaf, or a meromorphic co-Higgs sheaf with an integrability condition. This matters because it reduces arbitrary Poisson modules to objects whose birational geometry is already accessible. As an application, rank-two $\mathfrak{sl}_2$-Poisson modules are classified up to these normal forms, and the induced symplectic foliation on their projectivization is described.

What carries the argument

The paper's central object is rational Morita equivalence: a roof $(S,\varrho)$ of a normal variety with a possibly meromorphic Poisson bivector and two dominant Poisson morphisms to $(X,\sigma_1)$ and $(Y,\sigma_2)$. Its load-bearing identity is the pushforward condition $f_*\varrho=\sigma_1$ and $h_*\varrho=\sigma_2$ that makes the roof a Poisson correspondence. The proof is carried by the symplectic foliation $\mathcal{F}_\sigma$ of the Poisson bivector: a klt Poisson variety has either generically symplectic structure or a foliation with canonical singularities, which allows a quasi-étale cover $Z=W\times Y\to X$ over which the pulled-back foliation becomes a product. The decisive dichotomy is that a flat Poisson connection $\nabla:\mathcal{E}\to T_X\otimes\mathcal{E}$ either lands in $T_{\mathcal{F}_\sigma}\otimes\mathcal{E}$, giving a flat partial connection, or else projects to a section of $\mathcal{N}_{\mathcal{F}_\sigma}\otimes\operatorname{End}(\mathcal{E})$ with $\varphi\wedge\varphi=0$, i.e. a co-Higgs field. For rank-two $\mathfrak{sl}_2$-modules the concrete machinery is the trace-free connection matrix $\nabla=\delta+\begin{pmatrix} v_1 & v_2 \\ v_0 & -v_1 \end{pmatrix}$ and the induced Poisson bivector $\Sigma=\sigma+(v_0+2v_1z+v_2z^2)\wedge\frac{\partial}{\partial z}$ on $\mathbb{P}(\mathcal{E})$, which encodes the symplectic foliation of the projectivized bundle.

What would settle it

Compute the pushforward of the pulled-back Poisson bivector along a degree-two quasi-étale cover of the sort used in Theorem 4.8: if the result is twice the original bivector rather than the original, the cover is not a Poisson morphism and the claimed Morita reduction fails at that step.

Watch

Extended reading notes

Core claim

Theorem 1.1 (restated as Theorem 4.8) is the central claim: given a locally free Poisson module $(\mathcal{E}, \nabla)$ on a klt Poisson projective variety $(X,\sigma)$, there is a rationally Morita equivalent model on which $(\mathcal{E},\nabla)$ takes one of four forms: (a) a flat holomorphic sheaf on a transcendental Poisson variety; (b) a meromorphic flat connection on a generically symplectic variety; (c) a co-Higgs sheaf (a sheaf with a field $\varphi\in H^0(X,T_X\otimes\operatorname{End}(\mathcal{E}))$ satisfying $\varphi\wedge\varphi=0$) on a variety with trivial Poisson structure; or (d) a meromorphic co-Higgs sheaf $(\mathcal{E}_0,\psi)$ on a transcendental Poisson variety $(Y,\sigma_0)$, equipped with a rational map $\zeta:Y\dashrightarrow B$ with $\dim B=\dim \mathcal{F}_{\sigma_0}$, whose co-Higgs field $\psi$ is tangent to the pullback tangent sheaf and is annihilated by a meromorphic Poisson connection $D_0$ with $D_0(\psi)=0$. The proof reduces the symplectic foliation $\mathcal{F}_\sigma$ to a product over a quasi-étale cover, then applies a dichotomy: a flat Poisson connection either factors through the tangent sheaf of the foliation, producing a flat partial connection, or it induces a section $\varphi$ of $\mathcal{N}_{\mathcal{F}_\sigma}\otimes \operatorname{End}(\mathcal{E})$ satisfying $\varphi\wedge\varphi=0$. The four cases of the theorem come from combining this dichotomy with whether the pushed-forward Poisson structure is zero, generically symplectic, or transcendental.

Load-bearing premise

The reduction rests on the assumption that the quasi-étale cover that simplifies the symplectic foliation is a Poisson morphism, so that the pulled-back Poisson structure pushes forward to the original one.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, invariants of flat sheaves, meromorphic connections, and co-Higgs sheaves become invariants of arbitrary Poisson modules on klt Poisson projective varieties, up to rational Morita equivalence.
  • For rank-two meromorphic $\mathfrak{sl}_2$-Poisson modules, the normal forms show that the geometry is controlled by a triple of rational vector fields satisfying the equations $\delta(v_0)=v_0\wedge v_1$, $\delta(v_1)=2v_0\wedge v_2$, and $\delta(v_2)=v_1\wedge v_2$, linking the theory to transversely projective foliations.
  • The induced symplectic foliation on $\mathbb{P}(\mathcal{E})$ for a rank-two $\mathfrak{sl}_2$-Poisson module is either a two-dimensional pullback of a curve foliation, a codimension-one Riccati foliation, or the pullback of a foliation generated by a rational Poisson vector field and the symplectic foliation; in dimension $2k+1$ the induced map to the base is Poisson.
  • The proof yields functors from the category of Poisson modules on $(X,\sigma)$ either to co-Higgs bundles on the base $Y$ (when the pushed-forward Poisson structure is trivial) or to meromorphic connections along the degeneracy divisor (when it is generically symplectic), so categorical questions about Poisson modules can be transferred to those better-studied categories.

Reading between the lines

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

  • The dichotomy in the proof—flat connection either tangent to the symplectic foliation or inducing a co-Higgs field—suggests that the category of Poisson modules may admit a semiorthogonal decomposition into a flat part and a co-Higgs part, mirroring the four cases of Theorem 1.1.
  • One testable extension is to apply the rank-two normal form to a concrete Poisson threefold: write an explicit $\mathfrak{sl}_2$-Poisson connection, extract the triple $(v_0,v_1,v_2)$, and verify which of the cases in Corollary 5.11 is realized; this would give a computational check of the structure theorem.
  • The methods seem adaptable to reflexive coherent Poisson modules, not just locally free ones, because the key co-Higgs field construction uses a codimension-at-least-two singular set and flatness rather than local freeness; proving the theorem in that setting would broaden its scope to torsion-free sheaves.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper introduces a birational notion of rational Morita equivalence for Poisson modules on complex normal projective Poisson varieties and proves a structural dichotomy theorem (Theorem 1.1, proved as Theorem 4.8): every locally free Poisson module on a klt Poisson projective variety is rationally Morita equivalent to at least one of four normal forms: (a) a flat holomorphic sheaf on a transcendental Poisson variety; (b) a meromorphic flat connection on a generically symplectic variety; (c) a co-Higgs sheaf on a variety with trivial Poisson structure; or (d) a meromorphic co-Higgs sheaf on a transcendental Poisson variety with a Poisson-flatness condition. The proof passes through Druel's structural result for foliations with canonical singularities, a quasi-étale product cover, and Polishchuk's pushforward of Poisson modules. The paper then specializes to rank-two sl2-Poisson modules, obtaining explicit triples of rational vector fields, normal forms for the connection, and a geometric description of the induced symplectic foliation on the projectivization. It also claims a leaf bijection and an isomorphism of Casimir function spaces for rational Morita equivalence.

Significance. If the main theorem is made correct, the paper gives a useful unifying framework: holomorphic flat connections, meromorphic flat connections, and co-Higgs sheaves appear as different manifestations of Poisson modules, and the connection to Druel's and Polishchuk's results is natural. The rank-two normal forms and the projective-foliation interpretation are concrete and potentially useful. The paper is not circular and does not fit parameters; it relies transparently on external theorems. However, the central proof has a normalization gap in the construction of the quasi-étale span, and Proposition 4.2 contains an overclaim about leaf bijections. These issues are technically significant, but the main theorem is plausibly repairable, so the paper deserves a major revision rather than rejection.

major comments (2)
  1. [Section 4, proof of Theorem 4.8] After invoking [22, Proposition 8.14], the proof asserts: "Since f: Z→X is a quasi-étale cover there is a Poisson bivector σ̃∈H^0(Z,∧²T_Z) such that f_*σ̃=σ." For a finite morphism of degree d, the natural pullback satisfies f_*(f^*σ)=d·σ on the étale locus, so the unnormalized assertion is false when d>1. This is load-bearing: under Definition 4.1, the span (X←Z→Y) is a rational Morita equivalence only if f is a Poisson morphism, and without f_*σ̃=σ the subsequent pushforward (π2)_*(f^*E,∇̃) need not define a Poisson module on Y. The gap is repairable by taking σ̃=(1/d)f^*σ and lifting the connection as (1/d)f^*∇; flatness is then preserved, since both the Poisson differential and the quadratic curvature term rescale consistently. The paper should carry out this normalization explicitly. As written, the proof is incomplete at the central reduction, and the issue propagates to all cases of Theorem 4.8 and to Corollary 5.11.
  2. [Section 4, Proposition 4.2(1)] The claimed bijection between leaves of the symplectic foliations is asserted without proof and is not a formal consequence of the equality f^{-1}F_{σ1}=h^{-1}F_{σ2}. For example, take S=P^1 with the zero Poisson structure and let f=h be the double cover z↦z². Then f and h are dominant Poisson morphisms and f^{-1}F=h^{-1}F is the point foliation, but the induced map on the set of leaves is two-to-one over all but one point of the target, so there is no bijection of leaves. The statement should either be proved under additional hypotheses (for instance étale or submersion hypotheses) or removed from Proposition 4.2.
minor comments (6)
  1. [Section 4, Definition 4.1] The definition says "h_*ρ = σ1" for the second arrow; this should presumably be "h_*ρ = σ2".
  2. [Section 4, proof of Theorem 4.8] In the sentence "E1 is rationally Morita equivalent to ...", the symbol E1 should be (E,∇); as written it is an undefined notation.
  3. [Section 5, proof of Corollary 5.12] In the co-Higgs case the text says "This shows the part (c)," but the dimension-two pullback foliation corresponds to part (a), not part (c).
  4. [Theorem 1.1(d) and Corollary 5.12(d)] The notation "TY|B" should be clarified as the relative tangent sheaf T_{Y/B} (or T_{Y/B}⊗ something); as written it is ambiguous.
  5. [Corollary 5.11(c)] The phrase "a rational vector field v tangent to (Y,0)" is unclear; presumably v is a rational vector field on Y with the zero Poisson structure, but the notation (Y,0) is not introduced.
  6. [Definition 5.1] The displayed type "∇:E→E⊗TX(D)" should be balanced with the usage elsewhere, which suggests ∇:E→TX⊗E(D); if the tensor factors are intentionally ordered differently, the convention should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.8 is derived from external structure theorems (Druel, Polishchuk, Kaledin, Wang), with the author's own [15] cited only as an illustrative example.

full rationale

The paper's main reduction in Theorem 4.8 is not circular. The proof invokes Druel's Proposition 8.14 from [22] to obtain a quasi-étale cover f:Z=W×Y→X with π2^{-1}K=f^{-1}Fσ and a transcendental foliation K on Y. This is an external, load-bearing structure theorem whose assumptions (KFσ≃O_X and Fσ having canonical singularities) do not include the target classification of Poisson modules. The subsequent dichotomy — either the Poisson connection factors through TFσ, giving a flat partial connection, or it induces a non-trivial co-Higgs field φ with φ∧φ=0 — is a direct case distinction from the definition of Poisson connection, not a restatement of the conclusion. Polishchuk's Proposition 4.3 is used to push forward Poisson modules along morphisms with f_*O≃O, again an external result. Wang's [48, Corollary 3.3] supplies the partial connection D0 and the identity D0(φ)=0; this is an independent input. The four cases (a)–(d) are not fitted to the data or defined in terms of the conclusion; they are outcomes of the case analysis on the pushed-forward Poisson structure σ2=(π2)_*σ̃. The author's own prior work [15] appears only in an explanatory example about nilpotent co-Higgs bundles and is not used in the proof of Theorem 4.8 or in the derivation of the rank-two corollaries. The skeptical concern about the pushforward identity f_*σ̃=σ being potentially unnormalized on a quasi-étale cover is a mathematical correctness issue in the proof, not an instance of circularity: it does not show that the theorem's conclusion is equivalent to its inputs by construction. There is no fitted parameter renamed as a prediction, no self-citation chain used to forbid alternatives, and no known empirical pattern repackaged under new coordinates. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No fitted parameters appear in this mathematical paper. The central claim rests on several external theorems: Druel's quasi-étale splitting for foliations with canonical singularities, Wang's result on partial connections, Polishchuk's pushforward construction for Poisson modules, and the transcendental-foliation framework. The paper also implicitly assumes that pushforwards of Poisson bivectors along finite covers behave as stated, which is not generally true without a degree normalization.

assumptions (5)
  • domain assumption The symplectic foliation Fσ has canonical singularities and K_{Fσ} ≅ O_X, enabling the use of Druel's splitting theorem.
    Used at the start of the proof of Theorem 4.8 to obtain the quasi-étale cover Z=W×Y→X. The canonical singularity condition is part of the paper's definition of klt Poisson variety; the triviality of K_Fσ is asserted without proof but follows from the top power of the Poisson bivector on the regular locus.
  • domain assumption [22, Proposition 8.14] (Druel): for a foliation with canonical singularities and numerically trivial canonical class, there exist a quasi-étale cover and a fibration splitting the foliation as a pullback of a transcendental foliation.
    This external theorem provides the crucial reduction from X to W×Y→Y. The paper does not verify all hypotheses in detail.
  • domain assumption [48, Corollary 3.3] (Wang): a partial connection and a Poisson connection induce a Poisson connection on N K^*⊗End(E0) such that D0(φ)=0 for a co-Higgs field φ.
    Used in the proof of Theorem 4.8(d) to obtain D0(φ)=0. The hypotheses are not checked in the text.
  • domain assumption Polishchuk's pushforward (Proposition 4.3): if f:X→Y is a morphism with f_*O_X≅O_Y, then a Poisson structure on X descends to Y and Poisson modules push forward.
    Stated without proof; used to define σ2=(π2)_*σ̃ and to push forward Poisson modules in the main construction.
  • domain assumption Definition of transcendental Poisson variety (Definition 4.6): no positive-dimensional algebraic subvariety through a general point is tangent to the symplectic foliation.
    This notion from [3,36] is used to label case (a) and to extract the foliation K in the non-algebraic case.
invented entities (1)
  • Rational Morita equivalence relation
    purpose: Weak birational notion of equivalence for Poisson modules and Poisson varieties, used to state the classification theorem.
    This is a new definition introduced by the paper, not a physical entity. It lacks independent falsifiable handles outside the paper; its utility must be judged by the validity of the theorems proved with it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rational Morita equivalence for holomorphic Poisson modules." pith.science (2026). https://pith.science/paper/V2VVEV6L

@misc{pith2026190802325,
  author       = {Pith},
  title        = {Pith review of: Rational Morita equivalence for holomorphic Poisson modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2VVEV6L}},
  note         = {Machine review of arXiv:1908.02325}
}
abstract

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foliations. Moreover, we describe the geometry of the symplectic foliation induced by the Poisson connection on the projectivization of the Poisson module.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 52 canonical work pages

  1. [1]

    Atiyah, Complex analytic connections in fibre bundles

    M.F. Atiyah, Complex analytic connections in fibre bundles . Transactions of the Amer. Math. Soc. 85 (1957), 181-207

    work page 1957

  2. [2]

    Araujo, S

    C. Araujo, S. Druel, On Fano foliations . Adv. Math. 238 (2013), p. 70-118

    work page 2013

  3. [3]

    Araujo, S

    C. Araujo, S. Druel, On codimension 1 del Pezzo foliations on varieties with mild sin- gularities, Math. Ann. 360 (2014), no. 3-4, 769-798

    work page 2014

  4. [4]

    Ballico, S

    E. Ballico, S. Huh, 2-Nilpotent co-Higgs structures , 159, Issue 1-2, (2019), 39-56

    work page 2019

  5. [5]

    Beauville, Vari´ et´ es Kahl´ eriennes dont la premi` ere classe de Chernest nulle , J

    A. Beauville, Vari´ et´ es Kahl´ eriennes dont la premi` ere classe de Chernest nulle , J. Dif- ferential Geometry 18 (1983), 755-782

    work page 1983

  6. [6]

    Beauville, Holomorphic symplectic geometry: a problem list

    A. Beauville, Holomorphic symplectic geometry: a problem list. In: Complex and differ- ential geometry, Springer Proc. Math., vol. 8 (2011), Sprin ger, Heidelberg, 49-63

    work page 2011

  7. [7]

    Biswas, S

    I. Biswas, S. Rayan, A vanishing theorem for co-Higgs bundles on the moduli space of bundles, Geom. Dedicata 193 (2018), 145–154

    work page 2018

  8. [8]

    Biswas, O

    I. Biswas, O. Garc ´ ıa-Prada, J. Hurtubise, and S. Rayan, Principal co-Higgs bundles on P1, Proc. Edinburgh Math. Soc. (2020), 63(2), 512-530

    work page 2020

Show all 53 references
  1. [9]

    Biswas, On the Quantization of a Transversely Symplectic Foliation

    I. Biswas, On the Quantization of a Transversely Symplectic Foliation . , Southeast Asian Bulletin of Mathematics , Vol. 30, 6, (2006), 1029-1047

    2006

  2. [10]

    Bottacin, Poisson structures on Hilbert schemes of points of a surface and integrable systems, Manuscripta math

    F. Bottacin, Poisson structures on Hilbert schemes of points of a surface and integrable systems, Manuscripta math. 97 (1998), 517-527

    1998

  3. [11]

    Bartocci and E

    C. Bartocci and E. Macri, Classification of Poisson surfaces , Commun. Contemp. Math. 7 (2005), 89-95

    2005

  4. [12]

    Bogomolov, M

    F. Bogomolov, M. McQuillan, Rational Curves on Foliated Varieties. Foliation theory in algebraic geometry . Cham: Springer. Simons Symposia, (2016), 21-51. RATIONAL MORITA EQUIV ALENCE FOR HOLOMORPHIC POISSON MODUL ES 21

    2016

  5. [13]

    Campana, M

    F. Campana, M. Paun, Foliations with positive slopes and birational stability o f orbifold cotangent bundles . Publ. math. IHES 129 (2019), 1-49

    2019

  6. [14]

    Cerveau, A Lins

    D. Cerveau, A Lins. Irreducible components of the space of holomorphic foliati ons of degree two in CP(n). Annals of Math. 143 (1996), 577–612

    1996

  7. [15]

    Dedicata 183 (2016), 25-31

    M, Corrˆ ea, Rank two nilpotent co-Higgs sheaves on complex surfaces , Geom. Dedicata 183 (2016), 25-31

    2016

  8. [16]

    A. V. Colmenares, Moduli spaces of semistable rank-2 co-Higgs bundles over P1 × P1, Q. J. Math. , 68 (2017), 1139-1162

    2017

  9. [17]

    Cousin, J

    G. Cousin, J. V. Pereira, Transversely affine foliations on projective manifolds . Math. Res. Letters, 21, (2014), 985-1014

    2014

  10. [18]

    Cousin, Connexions plates logarithmiques de rang deux sur le plan pr ojectif complexe

    G. Cousin, Connexions plates logarithmiques de rang deux sur le plan pr ojectif complexe. Phd Thesis IRMAR (2011). http://tel.archives-ouvertes.f r/tel-00779098

    2011

  11. [19]

    M. Crainic. Differentiable and algebroid cohomology, van Est isomorphi sms and char- acteristic classes . Comment. Math. Helv., Tome 78 (2003) no. 4, 681-721

    2003

  12. [20]

    Deligne, Equations differentielles a points singuliers reguliers , Lecture Notes in Math- ematics, Vol

    P. Deligne, Equations differentielles a points singuliers reguliers , Lecture Notes in Math- ematics, Vol. 163, Springer-Verlag, Berlin, 1970

    1970

  13. [21]

    Druel, Some remarks on regular foliations with numerically trivia l canonical class , EPIGA 1 (2017), Article Nr

    S. Druel, Some remarks on regular foliations with numerically trivia l canonical class , EPIGA 1 (2017), Article Nr. 4

    2017

  14. [22]

    Druel, Codimension one foliations with numerically trivial canon ical class on singular spaces, preprint arXiv:1809.06905, 2018

    S. Druel, Codimension one foliations with numerically trivial canon ical class on singular spaces, preprint arXiv:1809.06905, 2018

    2018 arXiv

  15. [23]

    Fantechi, L

    B. Fantechi, L. Gottsche, L. Illusie, S. Kleiman, N. Nit sure, A. Vistoli, Fundamental Algebraic Geometry: Grothendieck’s FGA explained , Mathematical Surveys and Mono- graphs 123 (2006), A.M.S.,

    2006

  16. [24]

    R. L. Fernandes, Connections in Poisson geometry 1: holonomy and invariants , J. Diff. Geom. 54 (2000), 303-366

    2000

  17. [25]

    B. Fu. Poisson resolutions . J. Reine Angew. Math. 587 (2005), 17-26

    2005

  18. [26]

    Ginzburg, Grothendieck groups of Poisson vector bundles , J

    V. Ginzburg, Grothendieck groups of Poisson vector bundles , J. Symplectic Geometry, 1 (2001), 121-169

    2001

  19. [27]

    D. Greb, S. Kebekus, S. J. Kov´ acs, T. Peternell, Differential forms on log canonical spaces, Inst. Hautes Etudes Sci. Publ. Math. 114 (2011), no. 1, 87-1 69

    2011

  20. [28]

    Gualtieri, B

    M. Gualtieri, B. Pym, Poisson modules and degeneracy loci , Proc. Lond. Math. Soc. (3) 107 (2013), no. 3, 627-654

    2013

  21. [29]

    Gualtieri, Branes on Poisson varieties

    M. Gualtieri, Branes on Poisson varieties . In The Many Facets of Geometry: A Tribute to Nigel Hitchin. OUP, Oxford, (2010), 368-394

    2010

  22. [30]

    Gualtieri, Generalized complex geometry

    M. Gualtieri, Generalized complex geometry . Ann. of Math. (2) 174, 1 (2011), 75-123

    2011

  23. [31]

    Hitchin, Generalized holomorphic bundles and the B-field action , Journal of Geometry and Physics 61 (2011), 352-362

    N. Hitchin, Generalized holomorphic bundles and the B-field action , Journal of Geometry and Physics 61 (2011), 352-362

    2011

  24. [32]

    Geometry and Analysis Vol. 1

    N. Hitchin, Poisson modules and generalized geometry , in “Geometry and Analysis Vol. 1”, Lizhen Ji (ed.) 403–406 (2010), Advanced Lectures in Mat hematics 17, Higher Edu- cation Press Beijing

    2010

  25. [33]

    Hitchin, Generalized Calabi-Yau manifolds , Quart

    N. Hitchin, Generalized Calabi-Yau manifolds , Quart. J. Math. Oxford, 54 (2003) 281– 308

    2003

  26. [34]

    D. B. Kaledin, Normalization of a Poisson Algebra is Poisson , Proc. Steklov Inst. Math., 264 (2009)

    2009

  27. [35]

    Koll´ ar, S

    J. Koll´ ar, S. Mori, Birational geometry of algebraic varieties . In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Camb ridge (1998). With the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original. 22 M. CORR ˆEA

    1998

  28. [36]

    Loray, J

    F. Loray, J. Pereira, F. Touzet, Singular foliations with trivial canonical class . Invent. math., 213 (2018), 1327–1380

    2018

  29. [37]

    Loray, J

    F. Loray, J. Pereira, F. Touzet, Representations of quasiprojective groups, flat connec- tions and transversely projective foliations . Journal de l’ ˆEcole polytechnique 3 (2016), 263-308

    2016

  30. [38]

    McQuillan, Canonical models of foliations

    M. McQuillan, Canonical models of foliations . Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 877-1012

    2008

  31. [39]

    Morita, Duality for modules and its applications to the theory of rin gs with minimum condition

    K. Morita, Duality for modules and its applications to the theory of rin gs with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83-142

    1958

  32. [40]

    Polishchuk, Algebraic geometry of Poisson brackets

    A. Polishchuk, Algebraic geometry of Poisson brackets. Algebraic geometr y, 7. J. Math. Sci. 84 (1997), no. 5, 1413-1444

    1997

  33. [41]

    Pym, Poisson Structures and Lie Algebroids in Complex Geometry , Phd

    B. Pym, Poisson Structures and Lie Algebroids in Complex Geometry , Phd. Thesis, University of Toronto, 2013

    2013

  34. [42]

    Ran, Deformations of holomorphic pseudo-symplectic Poisson ma nifolds

    Z. Ran, Deformations of holomorphic pseudo-symplectic Poisson ma nifolds. Adv. Math. 304 (2017), 1156-1175

    2017

  35. [43]

    Rayan, Constructing co-Higgs bundles on CP2, Q

    S. Rayan, Constructing co-Higgs bundles on CP2, Q. J. Math. 65 (2014), no. 4, 1437- 1460

    2014

  36. [44]

    Rayan, Geometry of co-Higgs bundles , D.Phil

    S. Rayan, Geometry of co-Higgs bundles , D.Phil. Thesis, Oxford (2011)

    2011

  37. [45]

    Sakai, Anti-Kodaira dimension of ruled surfaces , Sci

    F. Sakai, Anti-Kodaira dimension of ruled surfaces , Sci. Rep. Saitama Univ. 2 (1982), 1-7

    1982

  38. [46]

    Sc´ ardua,Transversely affine and transversely projective holomorphi c foliations , Ann

    B. Sc´ ardua,Transversely affine and transversely projective holomorphi c foliations , Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), no. 2, 169-204

    1997

  39. [47]

    R. L. E., Schwarzenberger, Vector bundles on the projective plane . Proc. London Math. Soc. (3) 11 (1961), 623-640

    1961

  40. [48]

    W ang, Generalized holomorphic structures , J

    Y. W ang, Generalized holomorphic structures , J. Geom. Phys. 86 (2014), 273-283

    2014

  41. [49]

    W einstein, Coisotropic calculus and Poisson groupoids

    A. W einstein, Coisotropic calculus and Poisson groupoids . J. Math. Soc. Jap. 40 (1988), 705-727

    1988

  42. [50]

    W einstein, The local structure of Poisson manifolds , J

    A. W einstein, The local structure of Poisson manifolds , J. Differential Geometry , 18 (1983), 523-557

    1983

  43. [51]

    W einstein, Symplectic groupoids and Poisson manifolds , Bulletin of the American mathematical Society, 16 (1987), no

    A. W einstein, Symplectic groupoids and Poisson manifolds , Bulletin of the American mathematical Society, 16 (1987), no. 1, 101-104

    1987

  44. [52]

    Xu, Morita equivalence of Poisson manifolds

    P. Xu, Morita equivalence of Poisson manifolds . Comm. Math. Phys. 142 (1991), 493-509

    1991

  45. [53]

    Xu, Morita equivalent symplectic groupoids

    P. Xu, Morita equivalent symplectic groupoids . In: Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), (1991) 291-311. S pringer, New York. ICEX-UFMG, Departamento de Matem ´atica, A v. Ant ˆonio Carlos, 6627, 31270-901 Belo Horizonte-MG, Brazil E-mai...

    1991

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.