This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

It is well-known that the Fundamental Identity (FI) implies that Nambu brackets are decomposable, i.e. given by a determinantal formula. We find a weaker alternative to the FI that allows for non-decomposable Nambu brackets, but still yields a Darboux-like Theorem via a Nambu-type generalization of Weinstein’s splitting principle for Poisson manifolds.

Each Lie algebra $ℱ$ of vector fields (e.g. those which are tangent to a foliation) of a smooth manifold $M$ définies, in a natural way, a spectral sequence $E_k\left(ℱ\right)$ which converges to the de Rham cohomology of $M$ in a finite number of steps. We prove e.g. that for all $k\ge 0$ there exists a foliated compact manifold with $E_k\left(ℱ\right)$ infinite dimensional.

We study the problem of the non-existence of natural transformations $J^r{J}^{s}Y\to J^s{J}^{r}Y$ of iterated jet functors depending on some geometric object on the base of Y.

Under some weak assumptions on a bundle functor $F:F{M}_{m,n}\to FM$ we prove that there is no $F{M}_{m,n}$-natural operator transforming connections on $Y\to M$ into connections on $FY\to Y$.

For a vector bundle functor $H:ℳf\to 𝒱ℬ$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $ℱ{ℳ}_{m,n}$-natural operator $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $p:Y\to M$ into connections $D\left(\Gamma \right)$ on $Hp:HY\to HM$. For a bundle functor $E:ℱ{ℳ}_{m,n}\to ℱℳ$ with some weak conditions we prove non-existence of $ℱ{ℳ}_{m,n}$-natural operators $D$ transforming connections $\Gamma$ on $(m,n)$-dimensional fibered manifolds $Y\to M$ into connections $D\left(\Gamma \right)$ on $EY\to M$.

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿Q\left(reg{T}_k^r\to K_k^r\right)$ and $C^r₀⟿Q\left(reg{T}_k^{r*}\to K_k^{r*}\right)$ over n-manifolds is proved. Some generalizations are obtained.

We generalize the concept of an $(r,s,q)$-jet to the concept of a non-holonomic $(r,s,q)$-jet. We define the composition of such objects and introduce a bundle functor ${\tilde{J}}^{r,s,q}ℱ{ℳ}_{k,l}\times ℱℳ$ defined on the product category of $(k,l)$-dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor ${\tilde{J}}_1^{r,s,q}2\text{-}ℱ{ℳ}_{k,l}\to ℱℳ$ defined on the category of $2$-fibered manifolds with $ℱ{ℳ}_{k,l}$-underlying...

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in...