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Sorin Dumitrescu

Publications and source records attributed to Sorin Dumitrescu.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the existence of holomorphic curves in compact quotients of $\mathrm{SL}(2,\mathbb C)$

We prove the existence of a pair $(Σ,\, Γ)$, where $Σ$ is a compact Riemann surface with $\text{genus}(Σ)\, \geq\, 2$, and $Γ\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $Σ\, \longrightarrow\, {\mathrm SL}(2, \mathbb C)/Γ$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.

math.AG↗
arXiv2021

Uniformization of branched surfaces and Higgs bundles

Given a compact Riemann surface $Σ$ of genus $g_Σ\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $Σ$ with $\text{degree}(D)\, <\, 2(g_Σ-1)$, there is a unique cone metric on $Σ$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2πn_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $Σ$ parametrized by a nonempty open subset of $H^0(Σ,\,K_Σ^{\otimes 2}\otimes{\mathcal O}_Σ(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [Hi1], for the case $D\,=\, 0$.

math.DG↗
arXiv2021

On certain Tannakian categories of integrable connections over Kaehler manifolds

Given a compact Kaehler manifold X, it is shown that pairs of the form (E, D), where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on $E$, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

math.AG↗
arXiv2021

Holomorphic $\mathfrak{sl}(2,\mathbb C)$-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $Σ$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_Σ$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Σ$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

math.AG↗
arXiv2020

Holomorphic bundles trivializable by proper surjective holomorphic map

Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $φ^* E$ is trivial for some surjective holomorphic map $φ$, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

math.AG↗
arXiv2020

Holomorphic GL(2)-geometry on compact complex manifolds

We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact Kähler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact Kähler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact Kähler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric).

math.DG↗
arXiv2020

Branched projective structures, branched SO(3,C)-opers and logarithmic connections on jet bundle

We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S\, =\, \sum_{i=1}^d x_i$, where $x_i \,\in\, X$ are distinct points. After defining branched ${\rm SO}(3,{\mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${\rm SO}(3,{\mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)\otimes {\mathcal O}_X(S))$ singular over $S$, satisfying certain natural geometric conditions.

math.AG↗
arXiv2020

Projective structures on Riemann surface and natural differential operators

We investigate the holomorphic differential operators on a Riemann surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otimes n}, F)= F\otimes (J^k(E\otimes {\mathcal L}^{\otimes n}))^*$.

math.AG↗
arXiv2020

Deformation theory of holomorphic Cartan geometries

Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.

math.DG↗
arXiv2020

Fujiki class $\mathcal C$ and holomorphic geometric structures

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal C$, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class $\mathcal {C}$ and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

math.DG↗
arXiv2019

Rational parallelism on complex manifolds

A Theorem of Wang in [Wa] implies that any holomorphic parallelism on a compact complex manifold M is flat with respect to some complex Lie algebra structure whose dimension coincides with that of M. We study here rational parallelisms on complex manifolds. We exhibit rational parallelisms on compact complex manifolds which are not flat with respect to any complex Lie algebra structure.

math.DG↗
arXiv2019

Parabolic SL(r)-opers

We define SL(r)-opers in the set-up of vector bundles on curves with a parabolic structure over a divisor. Basic properties of these objects are investigated.

math.AG↗
arXiv2019

Cartan geometries on complex manifolds of algebraic dimension zero

We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures.

math.DG↗
arXiv2019

Logarithmic Cartan geometry on complex manifolds

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry (the polar part is supported on the ramification locus). We also study the specific case of the logarithmic Cartan geometry with the model being the complex affine space.

math.CV↗
arXiv2019

Stability and holomorphic connections on vector bundles over LVMB manifolds

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previous property are always flat.

math.DG↗
arXiv2019

Generalized Holomorphic Cartan geometries

This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by the authors in [BD]. It generalizes to higher dimension the notion of a branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries (e.g. all compact complex projective manifolds admit branched holomorphic projective structures). At the same time, this new definition is rigid enough to enable us to classify branched holomorphic Cartan geometries on compact simply connected Calabi-Yau manifolds.

math.DG↗