Lie algebroid Connections, Moduli of $\mathcal{L}$--twisted Principal Objects and motives

Arjun Paul; Samit Ghosh

arxiv: 2605.18304 · v1 · pith:NVDXJG43new · submitted 2026-05-18 · 🧮 math.AG · math.DG

Lie algebroid Connections, Moduli of mathcal{L}--twisted Principal Objects and motives

Samit Ghosh , Arjun Paul This is my paper

Pith reviewed 2026-05-20 00:06 UTC · model grok-4.3

classification 🧮 math.AG math.DG

keywords Lie algebroidsmoduli spacesprincipal bundlesHiggs bundlesnon-abelian Hodge correspondenceGrothendieck ringmotivesTannakian categories

0 comments

The pith

Semiprojectivity of the L-Hodge moduli spaces yields a description of their smooth loci in the Grothendieck ring and a motivic non-abelian Hodge correspondence.

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

The paper first classifies integrable transitive algebraic Lie algebroids on a smooth complex projective variety X. It then defines L-connections and L-twisted Higgs bundles for principal G-bundles, shows that the relevant categories are neutral Tannakian when L is transitive, and constructs their moduli spaces by geometric invariant theory. The authors introduce the L-Hodge moduli space and prove that the connection moduli, the harmonic Higgs moduli, and the L-Hodge moduli are all semiprojective varieties. This semiprojectivity is used to write the class of the smooth locus of each moduli space in the Grothendieck ring of varieties and to obtain a motivic form of the non-abelian Hodge correspondence. A reader cares because the result gives an explicit algebraic invariant that relates two different kinds of moduli spaces without choosing coordinates or bases.

Core claim

For a transitive Lie algebroid L on an irreducible smooth complex projective variety X and a connected reductive group G over C, the moduli spaces of principal G-bundles equipped with integrable L-connections, of L-twisted principal G-Higgs bundles of semiharmonic type, and the associated L-Hodge moduli spaces are all semiprojective varieties. The semiprojectivity of the L-Hodge moduli spaces then produces a concrete description of the smooth locus of these moduli spaces inside the Grothendieck ring of varieties and establishes a motivic non-abelian Hodge correspondence type theorem.

What carries the argument

The L-Hodge moduli space for principal G-bundles, whose semiprojectivity supplies the Grothendieck-ring description and the motivic correspondence.

If this is right

The category of vector bundles with integrable L-connections is neutral Tannakian.
The category of L-twisted Higgs bundles of semiharmonic type is neutral Tannakian.
Moduli spaces of principal G-bundles with integrable L-connections and of L-twisted principal G-Higgs bundles are semiprojective.
The smooth loci of these moduli spaces possess well-defined classes in the Grothendieck ring of varieties.
A motivic non-abelian Hodge correspondence holds between the L-connection moduli and the L-Higgs moduli.

Where Pith is reading between the lines

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

The same semiprojectivity technique could be applied to compute explicit classes in the Grothendieck ring for concrete choices of L, X and G.
The construction suggests a route toward motivic versions of other non-abelian Hodge correspondences that currently lack algebraic invariants.
If the transitivity hypothesis can be relaxed, the framework might extend to a wider class of Lie algebroids while preserving the motivic output.

Load-bearing premise

The Lie algebroid L must be transitive on the given projective variety X with G connected and reductive, so that the Tannakian categories are neutral and the GIT and semiprojectivity arguments apply.

What would settle it

A concrete transitive Lie algebroid L on a smooth projective X for which the L-Hodge moduli space fails to be semiprojective, or for which the class of the smooth locus in the Grothendieck ring does not satisfy the expected equality between the connection side and the Higgs side.

read the original abstract

Let $X$ be an irreducible smooth complex projective variety, and let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. In this paper, we first classify integrable transitive algebraic Lie algebroids on $X$. We then introduce Higgs bundles associated to a Lie algebroid and study their moduli spaces. In particular, we show that the category of vector bundles equipped with integrable $\mathcal{L}$-connections and the category of $\mathcal{L}$-twisted Higgs bundles of semiharmonic type on $X$ are neutral Tannakian categories, provided that $\mathcal{L}$ is a transitive Lie algebroid. Using this Tannakian framework, we obtain a characterization of principal $G$-bundles with integrable $\mathcal{L}$-connections and $\mathcal{L}$-twisted principal $G$-Higgs bundles of semiharmonic type on $X$, and construct their moduli spaces via Mumford's geometric invariant theory. We further introduce the notion of the $\mathcal{L}$-Hodge moduli space for principal $G$-bundles and prove that the moduli spaces of principal $G$-bundles with integrable $\mathcal{L}$-connections, $\mathcal{L}$-twisted principal $G$-Higgs bundles of harmonic type, and the associated $\mathcal{L}$-Hodge moduli spaces are semiprojective varieties. Finally, using the semiprojectivity of the $\mathcal{L}$-Hodge moduli spaces for principal $G$-bundles, we obtain a description of smooth locus of these moduli spaces in the Grothendieck ring of varieties and establish a motivic non-abelian Hodge correspondence type theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends non-abelian Hodge to transitive Lie algebroids and claims a motivic version by reading off smooth-locus classes from semiprojectivity of the moduli spaces.

read the letter

The key point is that this paper extends non-abelian Hodge theory to transitive Lie algebroids and uses semiprojectivity of the L-Hodge moduli to describe the smooth loci of the moduli spaces in the Grothendieck ring, yielding a motivic correspondence. They start by classifying integrable transitive algebraic Lie algebroids on the projective variety X. Then they define L-twisted Higgs bundles and show that the categories of integrable L-connections and semiharmonic L-Higgs bundles are neutral Tannakian when L is transitive. This lets them characterize the principal G-bundles with these structures and construct their moduli spaces using GIT. The central technical result is that the moduli of integrable L-connections, the semiharmonic L-Higgs bundles, and the interpolating L-Hodge moduli are all semiprojective. From there they claim the class of the smooth locus in K(Var) and the motivic non-abelian Hodge statement. The semiprojectivity is the strongest part of the argument because it supplies the C* action needed for cell decompositions. The Tannakian and GIT steps look standard once the categories are set up. The classification of the algebroids is also a clean preliminary result. One place that needs scrutiny is the passage from semiprojectivity of the full moduli space to the class of its smooth locus. Semiprojectivity controls the whole space via Białynicki-Birula cells, but the smooth locus excludes singular or unstable strata. Unless those strata are unions of cells or the action preserves smoothness in a strong way, an additional argument is required to compute the class after removal. The abstract does not detail this, so the full paper should make the subtraction explicit. This paper is for algebraic geometers working on moduli spaces of Higgs bundles and their motivic properties. A reader who knows the classical non-abelian Hodge correspondence will see how the Lie algebroid version fits in. It deserves a serious referee because the results are specific and the methods are traceable to existing literature. I would send this to peer review.

Referee Report

1 major / 2 minor

Summary. The paper classifies integrable transitive algebraic Lie algebroids on an irreducible smooth complex projective variety X with connected reductive group G. It shows that the categories of vector bundles with integrable L-connections and L-twisted semiharmonic Higgs bundles are neutral Tannakian when L is transitive, uses this to characterize and construct via GIT the moduli spaces of principal G-bundles with these structures, introduces the L-Hodge moduli space, proves semiprojectivity of the moduli spaces of integrable L-connections, harmonic L-Higgs bundles and the L-Hodge space, and finally uses semiprojectivity to obtain a description of the class of the smooth loci in the Grothendieck ring of varieties together with a motivic non-abelian Hodge correspondence.

Significance. If the semiprojectivity and motivic claims hold, the work would supply a motivic lift of the non-abelian Hodge correspondence in the Lie algebroid setting and explicit classes in K(Var) for smooth loci of these moduli spaces. The Tannakian neutrality and GIT constructions rest on standard external results and the classification of transitive Lie algebroids is a useful preliminary step.

major comments (1)

[the section on the motivic non-abelian Hodge correspondence] The final step (using semiprojectivity of the L-Hodge moduli spaces to describe the class of the smooth loci in K(Var) and to establish the motivic non-abelian Hodge correspondence) does not address whether the Białynicki-Birula cells of the C^*-action intersect both the smooth and singular loci. Semiprojectivity controls the class of the whole space, but removing GIT-unstable or singular strata requires an explicit stratification or subtraction argument to justify that the classes of the smooth loci for the integrable L-connection and semiharmonic L-Higgs cases coincide; this step is load-bearing for the central motivic claim.

minor comments (2)

[Abstract] Clarify the precise relationship between 'semiharmonic type' and 'harmonic type' for the L-twisted Higgs bundles, as the abstract uses both terms.
Add explicit citations for the external results on neutral Tannakian categories and Mumford's GIT that are invoked for the moduli constructions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the motivic section of the manuscript. We address the point below.

read point-by-point responses

Referee: The final step (using semiprojectivity of the L-Hodge moduli spaces to describe the class of the smooth loci in K(Var) and to establish the motivic non-abelian Hodge correspondence) does not address whether the Białynicki-Birula cells of the C^*-action intersect both the smooth and singular loci. Semiprojectivity controls the class of the whole space, but removing GIT-unstable or singular strata requires an explicit stratification or subtraction argument to justify that the classes of the smooth loci for the integrable L-connection and semiharmonic L-Higgs cases coincide; this step is load-bearing for the central motivic claim.

Authors: We agree that the manuscript would benefit from an explicit clarification of this step. The C^*-action on the L-Hodge moduli space is realized by algebraic automorphisms of the ambient variety. The singular locus is an intrinsic closed subscheme preserved by every automorphism, so the action restricts to an algebraic action on the smooth locus. Consequently the Białynicki-Birula decomposition of the full space induces a decomposition of the smooth locus into locally closed cells whose classes can be summed separately in K(Var). The motivic correspondence is constructed at the level of the semiprojective spaces and identifies the classes of corresponding closed invariant subsets (including the singular loci, which are the loci where the underlying objects cease to be simple or stable). Subtracting the class of the singular locus from both sides therefore yields the equality of the classes of the smooth loci. In the revised manuscript we will insert a short paragraph in the motivic section spelling out the invariance of the singular locus under the C^*-action and the resulting subtraction argument. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on external Tannakian/GIT/semiprojectivity results

full rationale

The paper classifies transitive Lie algebroids, proves that the relevant categories are neutral Tannakian under the transitivity hypothesis, constructs moduli spaces via Mumford GIT, establishes semiprojectivity of the L-Hodge spaces, and then extracts the Grothendieck-ring class of the smooth locus from the semiprojectivity data. Each step invokes standard external theorems (Tannakian duality, GIT quotients, Białynicki-Birula decompositions) rather than defining any quantity in terms of itself or reducing a central claim to a self-citation chain. The motivic non-abelian Hodge statement is presented as a consequence of these independent constructions, not as a tautological renaming or fitted prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard background results in algebraic geometry and Lie theory rather than introducing new free parameters or invented entities. Transitivity of L and projectivity of X are domain assumptions.

axioms (3)

domain assumption X is an irreducible smooth complex projective variety
Stated in the opening sentence of the abstract; required for the classification and moduli constructions.
domain assumption G is a connected reductive linear algebraic group over C
Stated in the opening sentence; used for principal G-bundles and GIT quotients.
domain assumption L is a transitive Lie algebroid
Explicitly required for the Tannakian categories to be neutral and for the semiprojectivity results.

pith-pipeline@v0.9.0 · 5846 in / 1589 out tokens · 43175 ms · 2026-05-20T00:06:48.581469+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

[Bal06] V . Balaji. Lectures on principal bundles. 3rd cycle. Guanajuato (Mexique), 2006, pp.17. cel- 00392133. [↑25.] [Sor99] Christoph Sorger. Lectures on moduli of principalG-bundles over algebraic curves. Lecture given at the School on Algebraic Geometry Trieste, 26 July - 13 August

work page 2006
[2]

Lie algebroid connections on principal bundles

[↑6.] [GP25] Samit Ghosh and Arjun Paul. Lie algebroid connections on principal bundles. arXiv:2505.18821. [↑2, 10, and 11.] [Sim92] Carlos T. Simpson. Higgs bundles and local systems. Inst. Hautes ´Etudes Sci. Publ. Math. 75 (1992), 5–95. [↑2, 4, 23, 28, 29, and 30.] [Sim94] Carlos T. Simpson. Moduli of representations of the fundamental group of a smoot...

work page arXiv 1992
[3]

[↑47.] [Tor11] Pietro Tortella.Λ-modules and holomorphic Lie algebroid connections. Cent. Eur. J. Math. 10 (4) (2012) 1422–1441. [↑2.] [To11T] Pietro Tortella.Λ-modules and holomorphic Lie algebroids. PhD thesis, Scuola Internazionale Superiore di Studi Avanzati,

work page 2012
[4]

Deligne, J.S

[↑23.] [DMOS] P . Deligne, J.S. Milne, A.Ogus and K.Shih. Hodge Cycles, Motives, and Shimura Varieties. Lect. Notes in Math, 900 (1982), Heidelberg, Springer. [↑23.] [No76] M.V .Nori. On the representation of the fundamental group. Compositio Math, 33 (1976), 29-

work page 1982
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Bernstein

[↑25.] [Be83] J. Bernstein. Course onD-modules. Harvard, 1983-1984. [Not cited.] [Ha77] Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, vol. 52, Springer- Verlag, New York,

work page 1983
[6]

Mumford, John Fogarty and Frances Kirwan

[↑30.] [Mu65] D. Mumford, John Fogarty and Frances Kirwan. Geometric Invariant theory. Springer Verlag, New York (1965). [↑37 and 43.] [HV15] Tam ´as Hausel, F. Rodriguez-Villegas. Cohomology of Large semiprojective hyperkahler vari- eties. Ast´erisque 2015 (370) (2015) 113 -

work page 1965
[7]

Mechanical Control Systems on Lie algebroids

[↑3 and 51.] [CM04] Jorge Cortes, Eduardo Martinez. Mechanical Control Systems on Lie algebroids. IMA J. Math.Control Inf. 21(4) (2004) 457-492. [↑2.] [LM12] Serge Lazzarini, Thierry Masson. Connections on Lie algebroids and on derivation-based noncommutative geometry. J. Geom. Phys. 62 (2) (2012) 387–402. [↑2.] [Hi87] N.J. Hitchin. The self-duality equat...

work page 2004
[8]

Moduli spaces of flat Lie algebroid connections

[↑2.] ON SEMIPROJECTIVE MODULI OFL-TWISTED PRINCIPAL OBJECTS AND MOTIVES 53 [Bo02] Philip Boalch. Quasi-Hamiltonian geometry of meromorphic connections. Duke Math. J. 139 (2002) 369–405. [↑2.] [BS12] Christopher L. Bremer, Daniel S. Sage. Moduli spaces of irregular singular connections. Int. Math. Res. Not. 2013 (8) (2012) 1800–1872. [↑2.] [Li1] Krizka Li...

work page internal anchor Pith review Pith/arXiv arXiv 2002
[9]

Semiprojectivity of the moduli of principalG–bundles withλ– connections

[↑2.] [RS24] Sumit Roy, Anoop Singh. Semiprojectivity of the moduli of principalG–bundles withλ– connections. arxiv.org/abs/2310.15652v2 [↑3.] [Bor87] A. Borel (et al.). Algebraic D-modules (Perspectives in Mathematics; Vol. 2).Academic Press, 1987[↑17 and 26.] [GLP18] Marco Gualtieri, Songhao Li, and Brent Pym. The Stokes groupoids.Journal f¨ ur die rein...

work page arXiv 1987
[10]

[↑12.] [AO24] David Alfaya and Andr ´e Oliveira

doi:10.1007/s002080100273. [↑12.] [AO24] David Alfaya and Andr ´e Oliveira. Lie algebroid connections, twisted Higgs bun- dles and motives of moduli spaces.J. Geom. Phys., 201:Paper No. 105195, 55,

work page doi:10.1007/s002080100273
[11]

[↑2, 3, 5, 6, 9, 15, 45, 46, 47, 48, 50, and 51.] [Ati57] M

doi:10.1016/j.geomphys.2024.105195. [↑2, 3, 5, 6, 9, 15, 45, 46, 47, 48, 50, and 51.] [Ati57] M. F. Atiyah. Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc., 85:181– 207,

work page doi:10.1016/j.geomphys.2024.105195 2024
[12]

[↑3, 9, and 17.] [ABKA25] David Alfaya,Indranil Biswas, Pradip Kumar and Anoop Singh

doi:10.2307/1992969. [↑3, 9, and 17.] [ABKA25] David Alfaya,Indranil Biswas, Pradip Kumar and Anoop Singh. A criterion for holomorphic Lie algebroid connections.Journal of Algebra, Volume 681, 2025, Pages 343-366,ISSN 0021-8693. [↑9 and 10.] [Bi26] Indranil Biswas. Holomorphic Lie Algebroid Connections on Holomorphic Principal Bundles on Compact Riemann S...

work page doi:10.2307/1992969 2025
[13]

2010 , author =

doi:10.1017/CBO9780511711985. [↑16, 37, and 43.] [Sch08] Alexander H. W. Schmitt. Geometric Invariant Theory and Decorated Principal Bundles.Eu- ropean Mathematical Society Publishing House, Z¨ urich. ISBN 978-3-03719-065-4.[↑6.] [GL21] Marco Gualtieri, Kevin Luk. Log Picard Algebroids and Meromorphic Line Bundles. In- ternational Mathematics Research Not...

work page doi:10.1017/cbo9780511711985 2021
[14]

Franco and P

[↑2, 3, 14, and 15.] [FT17] E. Franco and P . Tortella. Moduli spaces ofΛ-modules on abelian varieties.Advances in Math- ematics, 318, 459-496, 2017,.[↑14.] [Mor07] A. Moroianu. Lectures on K ¨ahler Geometry (London Mathematical Society Student Texts). Cambridge University Press, 2007:v-viii.[↑17.] [Fer02] Rui Loja Fernandes. Lie Algebroids, Holonomy and ...

work page 2017

[1] [1]

[Bal06] V . Balaji. Lectures on principal bundles. 3rd cycle. Guanajuato (Mexique), 2006, pp.17. cel- 00392133. [↑25.] [Sor99] Christoph Sorger. Lectures on moduli of principalG-bundles over algebraic curves. Lecture given at the School on Algebraic Geometry Trieste, 26 July - 13 August

work page 2006

[2] [2]

Lie algebroid connections on principal bundles

[↑6.] [GP25] Samit Ghosh and Arjun Paul. Lie algebroid connections on principal bundles. arXiv:2505.18821. [↑2, 10, and 11.] [Sim92] Carlos T. Simpson. Higgs bundles and local systems. Inst. Hautes ´Etudes Sci. Publ. Math. 75 (1992), 5–95. [↑2, 4, 23, 28, 29, and 30.] [Sim94] Carlos T. Simpson. Moduli of representations of the fundamental group of a smoot...

work page arXiv 1992

[3] [3]

[↑47.] [Tor11] Pietro Tortella.Λ-modules and holomorphic Lie algebroid connections. Cent. Eur. J. Math. 10 (4) (2012) 1422–1441. [↑2.] [To11T] Pietro Tortella.Λ-modules and holomorphic Lie algebroids. PhD thesis, Scuola Internazionale Superiore di Studi Avanzati,

work page 2012

[4] [4]

Deligne, J.S

[↑23.] [DMOS] P . Deligne, J.S. Milne, A.Ogus and K.Shih. Hodge Cycles, Motives, and Shimura Varieties. Lect. Notes in Math, 900 (1982), Heidelberg, Springer. [↑23.] [No76] M.V .Nori. On the representation of the fundamental group. Compositio Math, 33 (1976), 29-

work page 1982

[5] [5]

Bernstein

[↑25.] [Be83] J. Bernstein. Course onD-modules. Harvard, 1983-1984. [Not cited.] [Ha77] Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, vol. 52, Springer- Verlag, New York,

work page 1983

[6] [6]

Mumford, John Fogarty and Frances Kirwan

[↑30.] [Mu65] D. Mumford, John Fogarty and Frances Kirwan. Geometric Invariant theory. Springer Verlag, New York (1965). [↑37 and 43.] [HV15] Tam ´as Hausel, F. Rodriguez-Villegas. Cohomology of Large semiprojective hyperkahler vari- eties. Ast´erisque 2015 (370) (2015) 113 -

work page 1965

[7] [7]

Mechanical Control Systems on Lie algebroids

[↑3 and 51.] [CM04] Jorge Cortes, Eduardo Martinez. Mechanical Control Systems on Lie algebroids. IMA J. Math.Control Inf. 21(4) (2004) 457-492. [↑2.] [LM12] Serge Lazzarini, Thierry Masson. Connections on Lie algebroids and on derivation-based noncommutative geometry. J. Geom. Phys. 62 (2) (2012) 387–402. [↑2.] [Hi87] N.J. Hitchin. The self-duality equat...

work page 2004

[8] [8]

Moduli spaces of flat Lie algebroid connections

[↑2.] ON SEMIPROJECTIVE MODULI OFL-TWISTED PRINCIPAL OBJECTS AND MOTIVES 53 [Bo02] Philip Boalch. Quasi-Hamiltonian geometry of meromorphic connections. Duke Math. J. 139 (2002) 369–405. [↑2.] [BS12] Christopher L. Bremer, Daniel S. Sage. Moduli spaces of irregular singular connections. Int. Math. Res. Not. 2013 (8) (2012) 1800–1872. [↑2.] [Li1] Krizka Li...

work page internal anchor Pith review Pith/arXiv arXiv 2002

[9] [9]

Semiprojectivity of the moduli of principalG–bundles withλ– connections

[↑2.] [RS24] Sumit Roy, Anoop Singh. Semiprojectivity of the moduli of principalG–bundles withλ– connections. arxiv.org/abs/2310.15652v2 [↑3.] [Bor87] A. Borel (et al.). Algebraic D-modules (Perspectives in Mathematics; Vol. 2).Academic Press, 1987[↑17 and 26.] [GLP18] Marco Gualtieri, Songhao Li, and Brent Pym. The Stokes groupoids.Journal f¨ ur die rein...

work page arXiv 1987

[10] [10]

[↑12.] [AO24] David Alfaya and Andr ´e Oliveira

doi:10.1007/s002080100273. [↑12.] [AO24] David Alfaya and Andr ´e Oliveira. Lie algebroid connections, twisted Higgs bun- dles and motives of moduli spaces.J. Geom. Phys., 201:Paper No. 105195, 55,

work page doi:10.1007/s002080100273

[11] [11]

[↑2, 3, 5, 6, 9, 15, 45, 46, 47, 48, 50, and 51.] [Ati57] M

doi:10.1016/j.geomphys.2024.105195. [↑2, 3, 5, 6, 9, 15, 45, 46, 47, 48, 50, and 51.] [Ati57] M. F. Atiyah. Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc., 85:181– 207,

work page doi:10.1016/j.geomphys.2024.105195 2024

[12] [12]

[↑3, 9, and 17.] [ABKA25] David Alfaya,Indranil Biswas, Pradip Kumar and Anoop Singh

doi:10.2307/1992969. [↑3, 9, and 17.] [ABKA25] David Alfaya,Indranil Biswas, Pradip Kumar and Anoop Singh. A criterion for holomorphic Lie algebroid connections.Journal of Algebra, Volume 681, 2025, Pages 343-366,ISSN 0021-8693. [↑9 and 10.] [Bi26] Indranil Biswas. Holomorphic Lie Algebroid Connections on Holomorphic Principal Bundles on Compact Riemann S...

work page doi:10.2307/1992969 2025

[13] [13]

2010 , author =

doi:10.1017/CBO9780511711985. [↑16, 37, and 43.] [Sch08] Alexander H. W. Schmitt. Geometric Invariant Theory and Decorated Principal Bundles.Eu- ropean Mathematical Society Publishing House, Z¨ urich. ISBN 978-3-03719-065-4.[↑6.] [GL21] Marco Gualtieri, Kevin Luk. Log Picard Algebroids and Meromorphic Line Bundles. In- ternational Mathematics Research Not...

work page doi:10.1017/cbo9780511711985 2021

[14] [14]

Franco and P

[↑2, 3, 14, and 15.] [FT17] E. Franco and P . Tortella. Moduli spaces ofΛ-modules on abelian varieties.Advances in Math- ematics, 318, 459-496, 2017,.[↑14.] [Mor07] A. Moroianu. Lectures on K ¨ahler Geometry (London Mathematical Society Student Texts). Cambridge University Press, 2007:v-viii.[↑17.] [Fer02] Rui Loja Fernandes. Lie Algebroids, Holonomy and ...

work page 2017