Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.
problem Determining when Dolbeault DGAs on complex nilmanifolds are formal.
method Using quasi-isomorphisms and cohomology connections, proving formality conditions.
result Formality of Dolbeault DGAs on complex nilmanifolds is restricted to tori.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying ∂∂-lemma but with non-vanishing Aeppli-Bott-Chern-Massey product. Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
Establishes Poincaré's lemma for formal manifolds.
problem Developing smooth relative Lie algebra homologies and cohomologies.
method Theory of formal manifolds and formal Lie groups.
result Poincaré's lemma for de Rham complexes with formal functions and generalized functions.
The study shows strong formality in certain complex manifolds.
problem Investigating strong formality in complex manifolds.
method Adapting s-strong formality from Fernandez and Muñoz to the pluripotential setting. result Compact Kähler manifolds and generalized complete intersections are strongly formal.
Inspired by a work of Kapranov, we define the notion of Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via complex analytic approach, as in the case of usual complex manifolds and their Dolbeault…
Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.
problem Formality and higher Aeppli-Bott-Chern-Massey products on complex manifolds.
method Introduce and study bigraded formality and Aeppli-Bott-Chern-Massey products, showing non-trivial pullbacks on blow-ups.
result Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to blow-ups under certain conditions.
Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.
problem Extending holomorphic structures to complex manifolds with boundary.
method Analyzing formally integrable almost complex structures and their extensions to holomorphic structures.
result Holomorphic structures can be extended to a neighborhood of a strictly pseudoconvex boundary.
A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.
We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent…
A conjecture of Hirschowitz's predicts that a globally generated vector bundle W on a compact complex manifold A satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle W. By applying Cartan's eq…
Study geometric formal metrics and Massey products on Kähler manifolds with torsion.
problem Interplay between geometrically-Bott-Chern-formal metrics and SKT metrics on Kähler manifolds.
method Analyzing nilmanifolds and Kähler solvmanifolds, proving conditions for existence of SKT metrics and Massey products.
result Any Kähler solvmanifold is geometrically formal, and explicit constructions of lattices with non-vanishing Massey products.
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
This paper formalizes manifolds in positive characteristic varieties.
problem Establishing l-adic formal manifold structures on positive characteristic varieties.
method Develops and proves the existence of l-adic formal manifold structures and abelianized Galois symmetries.
result Proves l-adic homotopic equivalence and l-local lifting for simply-connected varieties.
We study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.
Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.
problem Classical aspects of N=(2,2) supersymmetric sigma models with Hermitian symmetric target spaces. method Reformulation using Gross-Neveu formalism, proposing two types of equivalent Lagrangians.
result Proposed two types of equivalent Lagrangians for maximal isotropic Grassmannians, making either supersymmetry or geometry manifest.
Formal Normal Form created for special CR singularities.
problem CR singularities in complex manifolds.
method Formal Normal Form construction for real-smooth submanifolds.
result A new Formal Normal Form established.
Formal normal form created for real-smooth hypersurfaces.
problem Real-smooth hypersurfaces in complex spaces.
method Iterative normalization procedure.
result Formal normal form constructed for a large class of hypersurfaces.
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.
In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.
We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane B on a GC manifold (X,J), we construct a formal (pointed) groupoid $…
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
Abstract study of HKT manifolds, proving Hodge theory and formality properties.
problem Analyzing Hodge theory and formality in HKT manifolds.
method Study of Dolbeault operators and Laplacians, proving relations and properties.
result Formality of differential graded algebra for certain HKT manifolds.
In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongatio…
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
Study of nonlinear PDEs using derived geometry and BV formalism.
problem Understanding non-linear PDEs via derived geometric methods.
method Derived enhancement of de Rham complex, algebro-geometric techniques, BV formalism.
result Natural derived enhancement of de Rham complex for nonlinear PDEs.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
The paper extends Newlander-Nirenberg theorem to domains with C2 boundary.
problem Extending Newlander-Nirenberg theorem to domains with C2 boundary. method Analyzing formally integrable complex structures on domains with C2 boundary. result Existence of global holomorphic coordinate systems on the closure of a bounded strictly pseudoconvex domain.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an A∞-minimal model of this DGA. When only the cohomology is needed to determine the rational homotopy type, then the DGA is called formal. By a theorem of Miller, a co…
Newlander-Nirenberg theorem extended to complex b-manifolds.
problem Characterizing complex b-manifolds.
method Involutive splitting of b-tangent bundle, formal local invariants, singular coordinate change.
result Complex b-manifolds have a single local model.
The differential system for minimal Lagrangian surfaces in a 2C-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,C), and the minimal Lagrangian surfaces arise as th…
A conjecture about rational curves' formal principle and convergence proved for Goursat type families.
problem Formal principle and convergence for rational curves of Goursat type.
method Natural ODEs and Cartan connections constructed by Doubrov-Komrakov-Morimoto.
result The conjecture is proved for rational curves of Goursat type.
Study on properties of special Kähler metrics and their interplay.
problem Properties and interplay of Strong Kähler with torsion and astheno-Kähler metrics.
method Analyzes families of astheno-Kähler nilmanifolds, studies complex blowups, and investigates interplay between metrics.
result Existence of astheno-Kähler metrics is not preserved by blowup, and some metrics are geometrically Bott-Chern formal.
\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality.…
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.
problem Compute cohomologies of complex solvmanifolds.
method Build finite-dimensional double subcomplexes and decompose them into indecomposable ones.
result Characterize the ∂∂ˉ-Lemma property and compute triple ABC-Massey products. We obtain a formal obstruction, i.e. a necessary condition for the existence of polarised complex deformations of Kähler-Ricci solitons. This obstruction is expressed in terms of the harmonic part of the variation of the complex structure.
The paper explores connections between dg manifolds and homotopy Lie algebras.
problem Understanding the relationship between dg manifolds and homotopy Lie algebras.
method Study of formal exponential maps, Atiyah classes, and Kapranov L-infinity algebras.
result Existence of formal exponential maps linked to vanishing of Atiyah classes.
We adapt continuous time random walk (CTRW) formalism to describe asset price evolution and discuss some of the problems that can be treated using this approach. We basically focus on two aspects: (i) the derivation of the price distribution from high-frequency data, and (ii) the inverse problem, obtaining information …
This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of ∣2∣-graded parabolic geometries of some particular type. We call them k-Dirac complexes. More explicitly, we will show that each k-Dirac complex arises as…
Develops neural networks for learning physics of complex systems by enforcing thermodynamics principles.
problem Learning physics of complex systems from incomplete experimental data.
method Integrates port-metriplectic formalism with neural networks to enforce thermodynamics principles.
result Neural networks can learn physics of complex systems by parts, reducing learning burden.
Kontsevich's formality the…
Resurgence of Joyce structures gauged to a standard form using gauge transformations.
problem Resurgence of Joyce structures in complex hyperkähler geometry.
method Gauge transformations and Borel transforms to show resurgent behavior.
result Established the resurgent behavior of infinitesimal gauge transformations for Joyce structures.
We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal…