The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.
problem Formal geometry of dg manifolds.
method Construction of Fedosov dg foliation and homotopy contractions.
result Isomorphism of Cartan and noncommutative calculi.
We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show …
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
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.
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.
problem Vertical isomorphisms of Fedosov dg manifolds associated with Lie pairs.
method Construction of Fedosov dg manifolds via splitting and connection, proving unique isomorphisms using iteration formula.
result Existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds.
Study Hochschild cohomology of dg manifolds linked to integrable distributions.
problem Understanding Hochschild cohomology of dg manifolds associated with integrable distributions.
method Analyzing the Hochschild cohomology of (F[1],dF) and relating it to the algebra of functions on leaf space. result Established a canonical isomorphism between the Hochschild cohomology of (F[1],dF) and the algebra of functions on leaf space. This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
For every Lie pair (L,A) of algebroids we construct a dg-manifold structure on the Z-graded manifold M=L[1]⊕L/A such that the inclusion ι:A[1]→M and the projection p:M→L[1] are morphisms of dg-manifolds. The vertical tangent bundle TpM then inherit…
The paper studies Atiyah and Todd classes for DG manifolds derived from integrable distributions.
problem Understanding Atiyah and Todd classes for DG manifolds.
method Analyzing DG manifolds (F[1],dF) corresponding to integrable distributions F. result Atiyah and Todd classes of DG manifolds are identical to those of Lie pairs $(T_{\mathbb{K}} M, F).
Defines Floer homology with DG coefficients for symplectic manifolds.
problem Computing Floer homology with DG coefficients for symplectic manifolds.
method Develops DG Floer toolset, defines spectral invariants, and proves Viterbo isomorphism theorem.
result Establishes almost existence of contractible periodic orbits on cotangent bundles.
Curved L∞ spaces form a category of fibrant objects.
problem Understanding the structure of curved L∞ spaces. method Proving L∞ spaces over dg manifolds form a category of fibrant objects. result Transitive L∞ algebroids over dg manifolds also form a category of fibrant objects. We find a minimal differential graded (dg) operad whose generic representations in Rn are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to Rn which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar constru…
This paper studies Hopf algebras from dg manifolds.
problem Understanding Hopf algebras from the perspective of dg manifolds.
method Analyzes the universal enveloping algebra of Lie algebra objects in homotopy categories of dg modules.
result The universal enveloping algebra of the Lie algebra object is a Hopf algebra.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities.…
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. Given any pair (L,A) of Lie algebroids, we construct a differential graded manifold (L[1]⊕L/A,Q), which we call Fedosov dg manifold. We prove that the cohomological vector field Q constructed on L[1]⊕L/A by the Fedosov iteration method arises as a byproduct of the Poincaré--Birkhoff--Witt map establ…
Develops theory of differential graded schemes for derived stacks.
problem Creating a theory for derived stacks using dg schemes.
method Formulates dg schemes as homotopy sites, equates to stacks on dg algebras.
result Infinity category of stacks represented by dg schemes is derived schemes.
In this article we show that some of the recent results of Marino, Moore, and Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten simple type) are of `superconformal simple type' -- can be understood using a simple mathema…
Develops Morse homology with DG coefficients for manifolds and spaces.
problem Homology with DG coefficients for manifolds and spaces.
method Derived local systems, DG modules, twisting cocycles, Morse trajectories.
result Isomorphic to DG Tor and Ext functors, recovers homology of total space of fibrations.
New algebraic structure derived from Kähler manifolds.
problem Understanding algebraic structures on differential forms.
method Introducing L∞[1] R-algebras and proving linearization theorems. result Induced L∞[1] R-algebra structures on Γ(L) are linearizable under certain conditions. 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…
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved L∞[1]-algebras. result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
Generalizes Riemann-Hilbert correspondence for curved local systems.
problem Higher Riemann-Hilbert correspondence with scalar curvature.
method Equivalence of dg-categories of curved local systems, graded vector bundles, and representations.
result Equivalence of dg-enhancements of twisted sheaves categories.
Letters discuss results on Courant algebroids, including classification and reduction.
problem Understanding and classifying Courant algebroids.
method Analyzes properties of Courant algebroids, including exact and transitive ones, and describes them in terms of symplectic dg manifolds.
result Provides a canonical generating Dirac operator and relates CAs to Poisson-Lie T-duality.
New L∞ liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical L∞ liftings of Buchweitz-Flenner semiregularity maps. The study extends a theorem to bundles on manifolds with boundaries.
problem Extending a theorem to bundles on manifolds with boundaries.
method Using recent anomaly results by Brüning, Ma, and Zhang, the theorem is generalized for a general flat bundle with a unimodular restriction to the boundary.
result An analogous statement for a general flat bundle is proven.
Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference eq…
This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singulari…
Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the …
New instanton homology theories for 3-manifolds and bundles.
problem Defining instanton homology for a class of 3-manifolds and bundles.
method Functorial construction for 4-manifold cobordisms, algebraic construction of homology theories for dg-modules.
result Isomorphic to Floer's instanton homology for admissible bundles, calculates I∞ in all cases it is defined. Study proves orientability of specific hypersurfaces in positive Ricci curvature manifolds.
problem Proving orientability of min-max hypersurfaces in manifolds with positive Ricci curvature.
method Analyzes Almgren-Pitts width and uses index 1 minimal hypersurfaces with multiplicity 1.
result Extends previous results to dimensions n+1≥8. We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field Q admits a structure of L-infinity algebra with the Lie derivative LQ as unary …
We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ∙A∨⊗L/A) carries a canonical L∞[1] algebra structure whose una…
We prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold. In a previous article, dg-ga/9710032, we proved transversality using an extension of the hol…
Constructs dg categories from surfaces using Khovanov homology.
problem Categorify quantum topology using surfaces and Khovanov homology.
method Constructs dg categories from surfaces using structures in Khovanov homology.
result Unified perspective on various categorified quantum topology constructions.
New approach to principal groupoid bundles with connections using dg-Lie groupoids.
problem Developing a new perspective on principal bundles with connections.
method Using dg-Lie groupoids and additional adjustment data for Lie groupoids.
result Adjusted connections provide a global formulation of curved Yang-Mills-Higgs theories.
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…
New dg-algebras link graph colorings to sheaves.
problem Linking graph colorings to sheaves for Legendrian surfaces.
method Generalized Casals-Murphy dg-algebra to non-commutative coefficients and computed Legendrian contact dg-algebra.
result Rank r representations of dg-algebras correspond to colorings of faces in Grassmannian.
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We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in th…
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …
This paper contains a thorough investigation of invariant distributions supported on limit sets of discrete groups acting convex cocompactly on symmetric spaces of negative curvature. It can be considered as a continuation of math.DG/9810146. Based on this investigation we provide proofs of the Hodge theoretic results …
We describe an A∞-quasi-equivalence of dg-categories between the first authors' PA ---the category of category of prefect A0-modules with flat Z-connection, corresponding to the de Rham dga A of a compact manifold M--- and the dg-category of \emph{infinity-local syst…
This paper shows a category equivalence between Lie group representations and Lie algebra representations.
problem Understanding the relationship between Lie group representations and Lie algebra representations.
method Establishes an equivalence between two categories of modules and representations.
result The equivalence between DG-algebra of singular chains on Lie groups and DG-Lie algebra representations.
Solved Gromoll-Walschap's conjecture on negative curvature manifolds.
problem Existence of Riemannian foliations on compact manifolds of negative curvature.
method Provided a proof for the conjecture.
result Gromoll-Walschap's conjecture is proven to be true.
The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of L∞-algebras for deformations. result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.