Convexity (algebraic geometry)

rdf:langString Convexity (algebraic geometry)

rdf:langString In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces in quantum cohomology. These moduli spaces are smooth orbifolds whenever the target space is convex. A variety is called convex if the pullback of the tangent bundle to a stable rational curve has globally generated sections. Geometrically this implies the curve is free to move around infinitesimally without any obstruction. Convexity is generally phrased as the technical condition since Serre's vanishing theorem guarantees this sheaf has globally generated sections. Intuitively this means that on a neighborhood of a point, with a vector field in that neighborhood, the local parallel transport can be extended globally. This generalizes the idea of convexity in Euclidean geometry, where given two points in a convex set , all of the points are contained in that set. There is a vector field in a neighborhood of transporting to each point . Since the vector bundle of is trivial, hence globally generated, there is a vector field on such that the equality holds on restriction.