On certain Tannakian categories of integrable connections over Kähler manifolds

Biswas, Indranil; Santos, João Pedro dos; Dumitrescu, Sorin; Heller, Sebastian

doi:10.4153/s0008414x21000201

Cited by 2 publications

(1 citation statement)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where Π(X , x 0 ) and Θ(X , x 0 ) are constructed in ( 5) and ( 7) respectively. Along the lines of Proposition 3.1 of [1], we have:…”

Section: Let Now {σ I } Hmentioning

confidence: 98%

Connections on trivial vector bundles over projective schemes

Biswas,

Hô Hai,

dos Santos

2024

Comptes Rendus. Mathématique

Self Cite

View full text Add to dashboard Cite

Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain "closure" of the aforementioned Lie algebra.

show abstract

“…where Π(X , x 0 ) and Θ(X , x 0 ) are constructed in ( 5) and ( 7) respectively. Along the lines of Proposition 3.1 of [1], we have:…”

Section: Let Now {σ I } Hmentioning

confidence: 98%

Connections on trivial vector bundles over projective schemes

Biswas,

Hô Hai,

dos Santos

2024

Comptes Rendus. Mathématique

Self Cite

View full text Add to dashboard Cite

show abstract

On the monodromy of holomorphic differential systems

Biswas,

Dumitrescu,

Heller

et al. 2024

Int. J. Math.

View full text Add to dashboard Cite

First we survey and explain the strategy of some recent results that construct holomorphic [Formula: see text]-differential systems over some Riemann surfaces [Formula: see text] of genus [Formula: see text], satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl[Formula: see text]-systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup [Formula: see text] as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of [Formula: see text], preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from [Formula: see text] to the quotient space [Formula: see text], where [Formula: see text] is a cocompact lattice, that do not factor through any elliptic curve [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of [Formula: see text], preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de [Formula: see text], J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. Then we prove that when [Formula: see text] is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over [Formula: see text] obtained from the category of holomorphic connections on étale trivial holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold [Formula: see text], we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over [Formula: see text]. If [Formula: see text] (respectively, [Formula: see text]) stands for the affine group scheme over [Formula: see text] obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism [Formula: see text] of Hopf algebras. We present a description of the transpose of [Formula: see text] in terms of the iterated integrals.

show abstract

On certain Tannakian categories of integrable connections over Kähler manifolds

Cited by 2 publications

References 34 publications

Connections on trivial vector bundles over projective schemes

Connections on trivial vector bundles over projective schemes

On the monodromy of holomorphic differential systems

Contact Info

Product

Resources

About