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We give the complete classification of Mukai pairs of dimension 4 and rank 2 with Picard number one, that is, pairs (X, E) where X is a Fano 4-fold with Picard number one, and E is an ample vector bundle of rank 2 on X with c1(X) = c1(E). Equivalently, the present paper completes the classification of ruled Fano 5-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g = 2 over an algebraically closed field k of characteristic p ≥ 0, and let FX : X → X be the absolute Frobenius morphism. Let MsX(r, d) be the moduli space of stable vector bundles of rank r and degree d on X. We study the Frobenius stratification of MsX(3, 0) in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each nonempty Frobenius stratum in the case in which (p, g) = (3, 2). [ABSTRACT FROM AUTHOR]

Abstract. For any (n - 1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis- Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar questions are also discussed for 2n-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasi-toric manifolds and moment angle complexes. [ABSTRACT FROM AUTHOR]