We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “holomorphic affine connections”
No holomorphic affine connections on compact manifolds with algebraic dimension zero.
problem Existence of holomorphic affine connections on complex manifolds with algebraic dimension zero.
method Proved the non-existence of holomorphic affine connections on compact manifolds with finite fundamental group and algebraic dimension zero.
result Compact complex manifolds with finite fundamental group and algebraic dimension zero admit no holomorphic affine connections.
Study holomorphic affine connections on non-Kähler manifolds.
problem Investigate geometric structures on non-Kähler compact complex manifolds.
method Prove properties of holomorphic affine connections on Calabi-Yau manifolds and compact complex manifolds of algebraic dimension one.
result Holomorphic affine structures on Calabi-Yau manifolds with polystable tangent bundles are locally homogeneous.
We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if is a generic connection on a princ…
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
Study of Tannakian categories for integrable connections on Kaehler manifolds.
problem Understanding Tannakian categories for integrable connections on Kaehler manifolds.
method Analyzing pairs (E, D) where E is a trivial holomorphic vector bundle and D is an integrable holomorphic connection.
result The pro-algebraic affine group scheme uniquely determines the isomorphism class of compact Riemann surfaces.
Classifies meromorphic affine connections on complex surfaces.
problem Investigating uniformization in higher dimensions with singularities.
method Extending work on holomorphic connections, classifying meromorphic connections on compact surfaces.
result Classification of meromorphic affine connections on compact complex surfaces.
Study on holomorphic structures on complex manifolds with specific properties.
problem Holomorphic geometric structures on complex manifolds with vanishing first Chern class.
method Proves properties of holomorphic geometric structures on compact complex manifolds.
result Holomorphic geometric structures are locally homogeneous for certain manifolds.
This paper is a review of the twistor theory of irreducible G-structures and affine connections. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections, while the situation with holonomies of…
Local models for special Kähler structures in 2D computed without essential singularities.
problem Computing holonomy of special Kähler structures in 2D.
method Constructing local models assuming no essential singularities in the holomorphic cubic form.
result Computed holonomy of the flat symplectic connection.
The study constructs differential systems on Riemann surfaces and explores their monodromy properties.
problem Constructing holomorphic differential systems with specific monodromy properties.
method Exploring the monodromy of holomorphic differential systems on Riemann surfaces.
result Holomorphic maps from Riemann surfaces to quotient spaces exist without factoring through elliptic curves.
Constructs moduli spaces for complex affine and dilation surfaces.
problem Classifying and understanding moduli spaces of complex surfaces.
method Using Veech's ideas, constructs holomorphic affine bundles and covering spaces.
result Moduli spaces of dilation surfaces are orbifold K(G,1) where G is the framed mapping class group.
Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.
problem Holomorphic projective structures and bundles on surfaces.
method Generalization of principal bundle of projective 2-frames to branched projective structures.
result Affine spaces of branched projective structures with given branching classes.
Complex manifolds with specific geometric structures have infinite fundamental groups.
problem Characterizing complex manifolds with holomorphic Cartan geometries.
method Analyzing the fundamental group of complex manifolds with holomorphic Cartan geometries of algebraic type.
result Compact complex manifolds of algebraic dimension zero with holomorphic Cartan geometries of algebraic type have infinite fundamental groups.
Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, φ), consisting of a flat vector bundle E over M and a flat nonzero section φ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional redu…
Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfacesmath.AG
Study connections on complex Riemann surfaces for Lie algebroid structures.
problem Investigating connections on holomorphic Lie algebroid structures on Riemann surfaces.
method Analyzing equivariant holomorphic Lie algebroid connections on holomorphic principal bundles over compact Riemann surfaces.
result Every holomorphic principal G-bundle admits an equivariant holomorphic Lie algebroid connection under certain conditions.
Holonomy of affine surfacesmath.AG
Identifies holonomy of affine surfaces via meromorphic connections.
problem Holonomy of complex affine surfaces.
method Moduli space of complex affine surfaces and meromorphic connections.
result Holonomy map is a submersion for non-finite-area translation surfaces.
Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
We prove that any simply connected special Kaehler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we show that a classical result of Calabi and Pogorelov on parabolic spheres implies …
Establishes jet transversality for regular maps from flexible manifolds.
problem Transversality for regular maps in algebraic geometry.
method Algebraic version of Forstnerič's theorem for holomorphic maps.
result Genericity theorems for regular maps of maximal ranks.
Affirmative answer to flat holomorphic Cartan geometries on complex tori.
problem Whether all flat holomorphic Cartan geometries on complex tori are translation invariant.
method Using complex affine Lie groups, we show that all holomorphic Cartan geometries on complex tori are translation invariant.
result Holomorphic Cartan geometries on complex tori are translation invariant.
Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.
problem Characterizing holomorphic structures on Oeljeklaus-Toma manifolds.
method Proving local homogeneity for various holomorphic geometric structures.
result Holomorphic geometric structures on Oeljeklaus-Toma manifolds are locally homogeneous.
We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and th…
Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the po…
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…
Holomorphic map connects Hitchin components to character varieties.
problem Complex affine spheres and their properties.
method Mapping class group equivariant holomorphic map from Hitchin components to character varieties.
result Holomorphic map includes holonomies of SL(3,C)-opers.
Study of flows on complex manifolds with holomorphic properties.
problem Global rigidity of transversely holomorphic Anosov flows on smooth compact manifolds.
method Analyzing the integrability of unstable and stable distributions, proving uniqueness in low dimensions.
result For topologically transitive flows, they are either orbit equivalent to a hyperbolic automorphism or geodesic flow.
The study explores extensions of Kähler manifolds and their properties.
problem Analyzing extensions of Kähler manifolds and their properties.
method Formulating conditions for affine bundles and studying their properties.
result Established a relation to adapted complex structures and good complexifications.
Study numerically flat bundles on Fujiki manifolds using algebraic groups.
problem Characterize numerically flat principal bundles on Fujiki manifolds.
method Analyzes holomorphic principal bundles and their quotient structures, proving equivalence of conditions involving numerically flat ad bundles and nef line bundles.
result Establishes equivalence among numerically flat ad bundles, nef line bundles, and degree inequalities for reductions of structure groups.
Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
Affine vector fields on pseudo-Kähler manifolds are symplectic.
problem Characterize affine vector fields on compact pseudo-Kähler manifolds.
method Two proofs provided, showing affine vector fields are symplectic and discuss properties of Lie derivatives.
result Affine vector fields on compact pseudo-Kähler manifolds are symplectic.
Special complex manifoldsmath.DG
We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition …
Generalizes results for lambda-connections and Higgs bundles.
problem Understanding the Bialynicki-Birula stratification of lambda-connections.
method Analyzes the Bialynicki-Birula decomposition and its relation to Morse and partial oper stratifications.
result Fibers of the Morse and partial oper stratifications are transverse at the base point and are half-dimensional affine spaces.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the `twistor…
Limits of Blaschke metricsmath.DG
Study Blaschke metrics to compactify Hitchin component.
problem Compactify the SL(3,R)-Hitchin component.
method Study degeneration of Blaschke metrics on equivariant affine spheres.
result Establish closure in projectivized geodesic currents space.
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…
The paper explores anti-hyperbolicity for hyperkähler varieties.
problem Anti-hyperbolicity of hyperkähler varieties.
method Exploring various examples and criteria for meromorphic and holomorphic dominability by C^m.
result Generalizing known results about K3 surfaces to hyperkähler manifolds.
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
New affine Szabó connections defined on smooth manifolds.
problem Defining and characterizing affine Szabó connections on manifolds.
method Introduced affine Szabó connections and proved their equivalence to cyclic parallelism on 2D affine manifolds.
result Characterization of locally homogeneous affine Szabó surfaces.
Affine connections become simple spaces locally.
problem Understanding affine connections and their properties.
method Combinatorial and geometric argument.
result Affine connections are locally affine spaces.
The paper proves conditions for minimal surfaces to be holomorphic and stable.
problem Conditions for stable minimal surfaces to be holomorphic.
method Developed a method of constructing variations to prove the equivalence.
result Holomorphicity and stability conditions for minimal surfaces.
We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic…
The paper characterizes flat affine connections on manifolds and Lie groups.
problem Characterizing flat affine connections on manifolds and Lie groups.
method New characterization through affine representations of automorphisms.
result Existence of a Lie group with a flat affine bi-invariant connection.
We show that the moduli space of holomorphic vector bundles on that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on , called the space of twistor sections. We then use this c…
On d-Holomorphic Connectionsmath.AG
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
The study compares Kähler and Riemannian normal coordinates on manifolds.
problem Understanding the differences and similarities between Kähler and Riemannian normal coordinates.
method Developed an algorithm to calculate the difference between Kähler and Riemannian normal coordinates as a universal power series in curvature tensor and its derivatives.
result The difference between Kähler and Riemannian normal coordinates is a universal power series in curvature tensor and its derivatives.
Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.
problem Understanding the dynamics of automorphism groups on cubic surfaces.
method Analyzing holomorphic automorphisms and character varieties.
result Several open questions about the dynamics of automorphism groups.