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.
Foundations laid for formal manifolds in differential geometry.
problem No specific problem stated; focuses on formal manifolds.
method Introducing formal manifolds, developing their theory, and proving finite products.
result Established a fully faithful contravariant functor and finite products in the category of formal manifolds.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
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.
Formal manifolds with non-negative Ricci curvature have formal covers.
problem Formality of manifolds with non-negative Ricci curvature.
method Study of universal covers and formal properties.
result Closed non-orientable manifolds with non-negative Ricci curvature are formal.
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.
Defines formal exponentials for graded manifolds and linearizes QP-manifolds.
problem Formal exponentials and linearizations of QP-manifolds.
method Definition of formal exponential maps, Grothendieck connections, and connections on tangent bundles.
result Linearizes QP-manifolds at points, giving formal tangent spaces L∞-algebra structures. 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.
Research on formality problem for special holonomy manifolds.
problem Formality problem for manifolds with special holonomy.
method Using intersection Massey products to establish formality.
result Recent results on formality of Joyce's G_2-manifolds.
Extended equivariant BV formalism to manifolds with boundaries.
problem Handling manifolds with boundaries in equivariant BV formalism.
method Extension of AKSZ theories to manifolds with boundaries.
result Successfully applied to manifolds with boundaries.
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.
Paper applies Newman-Penrose formalism to ACM manifolds.
problem Classifying compact ACM manifolds with η-Einstein metrics. method Newman-Penrose formalism applied to ACM manifolds.
result Classification of compact ACM manifolds with η-Einstein metrics. Non-formal G2 manifold found with holonomy.
problem Existence of non-formal G2 manifolds.
method Construction method of compact torsion-free G2 manifolds.
result Found a compact, simply connected G2 manifold that is non-formal.
We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2=3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2≤2 and …
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. New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.
problem Identifying non-formal Sasaki-Einstein 7-manifolds and their submanifolds.
method Construction of new examples and analysis of fibre bundles, total spaces, and Sasaki-Einstein structures.
result Examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …
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.
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…
Compact non-formal G2 manifold with b1=1.
problem Constructing a compact manifold with specific properties.
method Developed a method of resolution for orbifolds.
result First Betti number b1=1 for a compact non-formal G2 manifold. The formal principle holds for certain globally generated vector bundles on Fano manifolds and smooth rational curves.
problem Proving the formal principle for globally generated vector bundles on compact complex manifolds.
method Applying Cartan's equivalence method to a differential system on the universal family of the Douady space.
result The formal principle is true for Fano manifolds and smooth rational curves under specific conditions.
We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is b2<2. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number b2≥2, which is formal. Therefore, such an example does not adm…
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
The abstract manifold cannot have uniformly quasiregular self-maps.
problem Characterizing uniformly quasiregularly elliptic manifolds.
method Introducing conformally formal manifolds and proving their properties.
result The abstract manifold is not uniformly quasiregularly elliptic.
The article confirms Joyce's examples of G2-holonomy are formal spaces.
problem Formality of special holonomy manifolds and nearly Kähler manifolds.
method Analyzing rational homotopy theory and cohomology algebra.
result Joyce's examples of G2-holonomy are formal spaces.
A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to …
This paper shows hypercommutative algebras on Calabi-Yau manifolds are formal.
problem Formality of hypercommutative algebras on Calabi-Yau manifolds.
method Using purity of mixed Hodge structures.
result The canonical hypercommutative algebra on compact Calabi-Yau manifolds is formal.
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.
The paper extends the formal manifold theorem to higher dimensions and characterizes A∞-minimal models for certain differential graded algebras.
problem Characterizing formal differential graded algebras and their A∞-minimal models. method Expanding the formal manifold theorem and proving properties of A∞-minimal models for specific cases. result The de Rham complex of certain differential graded algebras has A∞-minimal models with specific non-trivial terms. We construct examples of non-formal simply connected and compact oriented manifolds of any dimension bigger or equal to 7.
A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.
We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using th…
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.
Using the concept of s-formality we are able to extend the bounds of a Theorem of Miller and show that a compact k-connected 4k+3- or 4k+4-manifold with b_{k+1}=1 is formal. We study k connected n-manifolds, n= 4k+3, 4k+4, with a hard Lefschetz-like property and prove that in this case if b_{k+1}=2, then the manifold i…
The paper proves conditions for 2-torus manifolds to be equivariantly formal.
problem Characterizing 2-torus manifolds as equivariantly formal.
method Proving 2-torus manifolds are equivariantly formal under specific conditions.
result 2-torus manifolds are equivariantly formal if and only if the action is locally standard and all faces of the orbit space are mod 2 acyclic.
Quantizes functions on Kähler manifolds without formal deformation.
problem Deforming smooth functions on Kähler manifolds to non-formal quantization.
method Using Fedosov connections and prequantum line bundles.
result Quantizable functions form a sheaf of twisted differential operators.
The paper examines metrics on foliated manifolds that have special geometric properties.
problem Characterizing metrics on foliated manifolds with specific harmonic properties.
method Examining the properties of bundle-like metrics on foliated manifolds.
result The interior product of basic harmonic forms is basic harmonic under certain conditions.
We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connec…
We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the L…
A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…
For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
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.
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…
We study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.
Let (M, π ) be a Poisson manifold. A Poisson submanifold P∈M gives rise to an algebroid AP→P, to which we associate certain chomology groups which control formal deformations of π around P . Assuming that these groups vanish, we prove that π is formally rigid around P , i.e. any other Poisson struct…
We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension m and with first Betti n…
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.