The compactification from the eleven-dimensional Ho\v{r}ava-Witten orbifold to five-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is reviewed, as specific $SU(4)$ vector bundle leading "heterotic standard model" in observable sector. Within context of strongly coupled M-theory, formalism for consistent hidden-sector bundles associated with single line presented, and introduced...

De nition 3.1. Let X be a complex manifold. A holomorphic vector bundle of rank k over E is a complex manifold E and a holomorphic map : E ! X such that makes E ! X into a complex vector bundle of rank k, and E admits holomorphic trivializations, i.e. there’s an open cover U of X trivializing E such that for each U 2 U, there’s a biholomorphic map ' : EjU ! U C commuting with projection to U th...

In this paper we derive a list of all the possible indecomposable normalized rank–two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi-Yau threefolds, say V , of Picard number ρ = 1. For any such bundle E, if it exists, we find the projective invariants of the curves C ⊂ V which are the zero–locus of general global sections of E...

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on given K\"ahler manifold. goal paper give an $\varepsilon$-regularity theorem for flow. To establish theorem, provide scale invariant monotone quantity. As critical point quantity, define self-shrinker solution Liouville type se...

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is substitute for Narasimhan-Atiyah-Bott K\"ahler form that realizes Chern class determinant-of-cohomology line bundle on moduli space bundles curve. As replacement we use explicit realisation Atiyah this bundle, based theory trace complex due to B...