Abstract VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term up to homotopy We attach every VB-groupoid a cochain complex controlling its deformations discuss fundamental features, such as Morita invariance van Est theorem. Several exa...

Given a compact Kähler manifold X, there is an equivalence of categories between the completely reducible flat vector bundles on X and polystable Higgs $$(E, \theta )$$ with $$c_1(E)= 0= c_2(E)$$ (Simpson in J Am Math Soc 1(4):867–918, 1988; Corlette Differ Geom 28:361–382, Uhlenbeck Yau Commun Pure Appl 39:257–293, 1986; Donaldson Duke 54(1):231–247, 1987). We extend this to context Sasakian m...

Let M denote the moduli space of stable vector bundles of rank n and fixed determinant of degree coprime to n on a non-singular projective curve X of genus g ≥ 2. Denote by U a universal bundle on X × M . We show that, for x, y ∈ X, x 6= y, the restrictions U|{x} × M and U|{y} × M are stable and nonisomorphic when considered as bundles on X .

Let X be a nonsingular algebraic curve of genus g ≥ 3, and let Mξ denote the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3, and suppose further that n0, d0 are integers such that n0 ≥ 1 and nd0 + n0d > nn0(2g − 2). Let E be a semistable vector bundle over ...

We prove that for well-behaved small commutative rig categories (aka. symmetric bimoidal categories) R the algebraic K-theory space of the K-theory spectrum, HR, of R is equivalent to K0(π0R)× |BGL(R)| where GL(R) is the monoidal category of weakly invertible matrices over R. In particular, this proves the conjecture from [BDR] that K(ku) is the K-theory of the 2-category of complex 2-vector sp...

We describe recent work on the arithmetic properties of moduli spaces of stable vector bundles and stable parabolic bundles on a curve over a global field. In particular, we describe a connection between the period-index problem for Brauer classes over the function field of the curve and the Hasse principle for rational points on étale forms of such moduli spaces, refining classical results of ...

Let X be a compact complex manifold equipped with a hermitian metric. In this note we define an A∞-category Db ∞(X) which is a refined version of the derived category of X. This A∞-category is a mirror partner of the A∞-category F (M) constructed by Fukaya for a symplectic manifold M (see [1]). The objects of Db ∞(X) are bounded complexes of holomorphic vector bundles on X equipped with hermiti...

We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. discover examples of moduli rank 2 stable Picard one with arbitrarily many connected components.