On promoting the type IIA side of the N = 1 Heterotic/type IIA dual pairs of [1] to M-theory on a 'barely G 2 Manifold' of [2], by spectrum-matching we show a possible triality between Heterotic on a self-mirror Calabi-Yau, M-theory on the above 'barely G 2-Manifold' constructed from the Calabi-Yau on the type IIA side and F-theory on an elliptically fibered Calabi-Yau 4-fold fibered over a tri...

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and EG −→ X a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group EP ⊂ EG to P , such that ...

The bundle of Dirac spinors is used for describing particles with half-integer spin in general relativity and in quantum field theory. It is a special four-dimensional complex vector-bundle over the space-time manifold M . Let’s remind that the space-time manifold M itself is a four-dimensional real manifold equipped with a Minkowski type metric g of the signature (+,−,−,−). Apart from g, the s...

We give a simple interpretation of the adapted complex structure of Lempert-Szőke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise ...

Let $(M,J,g,\omega)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of Bott-Chern Laplacian on $(M,J,g,\omega)$, proving that it is still elliptic. On compact K\"ahler manifold, kernels Dolbeault and coincide. show such property does not hold when providing an explicit structure Kodaira-Thurston Furthermore, if connected $4$-manifold, denoting by $h^{1,1}_{BC}$ dimens...

Let M be a closed hyperbolic three manifold. We show that the number of genus g surface subgroups of π1(M ) grows like g.

Given a manifold M of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold W is isomorphic to a cylinder M × [0, 1] if and only if W is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of M ; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, ...

The projective space of Rn has a natural open subset: the set of lines spanned by vectors with all coordinates > 0. Such a subset can be defined more generally for any partial flag manifold of a split semisimple real algebraic group. The main result of the paper is that this subset can be defined by algebraic equalities and inequalities. Let G be a simply connected semisimple algebraic group ov...

Abstract Let $$\Gamma $$ Γ be a finite group acting on Lie G . We consider class of extensions $$1 \rightarrow \hat{G} \Gamma 1$$ 1 → G ^ defined by this action and 2-cocycle with values in the ...

A Kähler-Nijenhuis manifold is a Kähler manifold M , with metric g, complex structure J and Kähler form Ω, endowed with a Nijenhuis tensor field A that is compatible with the Poisson structure defined by Ω in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if AJ = ±JA, M is foliated by im A into non degenerate Kähler-Nijenhuis submanifolds. If A is a non degenerate...