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Sorin Dumitrescu

Publications and source records attributed to Sorin Dumitrescu.

At least 19 recordsLinked to original sources
arXiv2026

Opers on transversely holomorphic foliations

For real codimension two smooth foliations, we study the transversely complex structures, transversely complex projective structures and transverse opers associated to the foliation. We prove a uniqueness theorem for the transversely holomorphic ${\mathbb C}{\mathbb P}^1$--bundle naturally associated with a transversely complex projective structures. A similar uniqueness theorem is proved for the transversely holomorphic filtered ${\mathbb C}{\mathbb P}^{r-1}$--bundle naturally associated to a transverse ${\rm PGL}(r,{\mathbb C})$--oper.

math.DG↗
arXiv2026

Infinitesimal deformations of parabolic connections and parabolic opers

We compute the infinitesimal deformations of quadruples $(X, S, E_*, D)$, where $(X, S)$ is a compact Riemann surface with $n$ marked points, $E_*$ is a parabolic vector bundle on $X$ with parabolic structure over $S$, and $D$ is a parabolic connection on $E_*$. Using it we compute the infinitesimal deformations of $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$. It is shown that the monodromy map, from the moduli space of triples $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$, to the SL(r, C)-character variety of X - S, is an immersion.

math.AG↗
arXiv2026

A local Lorentzian Ferrand-Obata theorem for conformal vector fields

For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere conformally flat. The main theorem can be viewed as a local version of the Lorentzian Lichnerowicz conjecture in the real-analytic setting. The key result is an optimal improvement of the local normal forms for conformal vector fields of [FM13], which focused on non-linearizable singularities. This article is primarily concerned with essential linearizable singularities, and the proofs include global arguments which rely on the compactness assumption.

math.DG↗
arXiv2026

Non-Kähler Calabi-Yau manifolds and holomorphic geometric structures

We study holomorphic geometric structures on non-Kähler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally homogeneous. Moreover, if the geometric structure is rigid, then the Vaisman manifold must be a Kodaira manifold. The proof uses a Beauville-Bogomolov type decomposition from [Is] together with a weak form of Bochner principle for Vaisman Calabi-Yau manifolds that we prove here. Other results show that a compact complex manifold with self-dual holomorphic tangent bundle bearing a rigid holomorphic geometric structure of affine type have infinite fundamental group. We prove the same result for compact complex manifolds with trivial canonical line bundle having semistable holomorphic tangent bundle, with respect to some Gauduchon metric. We exhibit (non-Kähler) compact complex simply connected manifolds with trivial canonical line bundle that admit non-closed holomorphic one-forms.

math.DG↗
arXiv2025

Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry

We define a class of nonsingular holomorphic foliations on compact complex tori which generalizes (in higher codimension) the turbulent foliations of codimension one constructed by Ghys. For those smooth turbulent foliations we prove that all transversely holomorphic Cartan geometries are flat. We also establish a uniqueness result for the transversely holomorphic Cartan geometries.

math.DG↗
arXiv2024

Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,Θ)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,Θ)$ is isomorphic to $(ρ^* E_H,\,ρ^* Θ)$ for every $ρ\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $Δ$ singular on $D$ such that $Θ$ is preserved by the connection $Δ$.

math.CV↗
arXiv2024

Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures

Let $X$ be a compact complex manifold such that its canonical bundle $K_X$ is numerically trivial. Assume additionally that $X$ is Moishezon or $X$ is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville-Bogomolov type decomposition theorem for $X$. We deduce that holomorphic geometric structures of affine type on $X$ are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless $X$ is an étale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex $n$-folds of algebraic dimension at least $n-1$. Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either $X$ is Fujiki, $X$ is a threefold, or $X$ is of algebraic dimension at most one.

math.DG↗
arXiv2024

Holomorphic geometric structures on Oeljeklaus-Toma manifolds

We prove that any holomorphic geometric structure of affine type on an Oeljeklaus- Toma manifold is locally homogeneous. For locally conformal Kähler Oeljeklaus-Toma manifolds we prove that all holomorphic geometric structures, and also all holomorphic Cartan geometries, on them are locally homogeneous.

math.DG↗
arXiv2023

Principal bundles with holomorphic connections over a Kaehler Calabi-Yau manifold

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in \cite{BD} for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds.

math.DG↗
arXiv2023

On the monodromy of holomorphic differential systems

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $Σ_g$ of genus $g\geq 2$, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian \cite{BDHH} or some cocompact Kleinian subgroup $$Γ\subset \text{SL}(2, \mathbb C)$$ as in \cite{BDHH2}. As a consequence, there exist holomorphic maps from $Σ_g$ to the quotient space $\text{SL}(2, \mathbb C)/ Γ$, where $Γ\subset \text{SL}(2, \mathbb C)$ is a cocompact lattice, that do not factor through any elliptic curve \cite{BDHH2}. This answers positively a question of Ghys in \cite{Gh}; the question was also raised by Huckleberry and Winkelmann in \cite{HW}. Then we prove that when $M$ is a Riemann surface, a Torelli type theorem holds for the affine group scheme over $\mathbb C$ obtained from the category of holomorphic connections on {\it 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 $M$, we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over $M$. If $\varpi$ (respectively, $Θ$) stands for the affine group scheme over $\mathbb C$ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism $v:{\mathcal O}(Θ)\longrightarrow {\mathcal O}(\varpi)$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals.

math.DG↗
arXiv2023

Holomorphic projective connections on compact complex threefolds

We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection.

math.DG↗
arXiv2023

Parabolic opers and differential operators

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.

math.AG↗
arXiv2022

Nonabelian Hodge theory for Fujiki class $\mathcal C$ manifolds

The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal C$ manifolds.

math.AG↗
arXiv2022

Branched SL(r,C)-opers

We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.

math.AG↗
arXiv2022

The monodromy map from differential systems to character variety is generically immersive

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb C$ and $\mathfrak g$ its Lie algebra. We study the monodromy map from the space of $\mathfrak g$-differential systems on a compact connected Riemann surface $Σ$ of genus $g \,\geq\, 2$ to the character variety of $G$-representations of the fundamental group of $Σ$. If the complex dimension of $G$ is at least three, we show that the monodromy map is an immersion at the generic point.

math.AG↗
arXiv2022

Deformation Theory of Holomorphic Cartan Geometries, II

In this continuation of \cite{BDS}, we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

math.DG↗
arXiv2022

On the monodromy map for the logarithmic differential systems

We study the monodromy map for logarithmic $\mathfrak g$-differential systems over an oriented surface $S_0$ of genus $g$, with $\mathfrak g$ being the Lie algebra of a complex reductive affine algebraic group $G$. These logarithmic $\mathfrak g$-differential systems are triples of the form $(X, D,Φ)$, where $(X, D) \in {\mathcal T}_{g,d}$ is an element of the Teichmüller space of complex structures on $S_0$ with $d \geq 1$ ordered marked points $D\subset S_0= X$ and $Φ$ is a logarithmic connection on the trivial holomorphic principal $G$-bundle $X \times G$ over $X$ whose polar part is contained in the divisor $D$. We prove that the monodromy map from the space of logarithmic $\mathfrak g$-differential systems to the character variety of $G$-representations of the fundamental group of $S_0\setminus D$ is an immersion at the generic point, in the following two cases: A) $g \geq 2$, $d \geq 1$, and $\dim_{\mathbb C}G \geq d+2$; B) $g=1$ and $\dim_{\mathbb C}G \geq d$. The above monodromy map is nowhere an immersion in the following two cases: 1) $g=0$ and $d \geq 4$; 2) $g\geq 1$ and $\dim_{\mathbb C}G < \frac{d+3g-3}{g}$. This extends to the logarithmic case the main results in \cite{CDHL}, \cite{BD} dealing with nonsingular holomorphic $\mathfrak g$-differential systems (which corresponds to the case of $d\,=\,0$).

math.AG↗