Convexity (algebraic geometry)

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces formula_1 in quantum cohomology.These moduli spaces are smooth orbifolds whenever the target space is convex. A variety formula_2 is called convex if the pullback of the tangent bundle to a stable rational curve formula_3 has globally generated sections. Geometrically this implies the curve is free to move around formula_2 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 formula_6 in a convex set formula_7, all of the points formula_8 are contained in that set. There is a vector field formula_9 in a neighborhood formula_10 of formula_11 transporting formula_11 to each point formula_13. Since the vector bundle of formula_14 is trivial, hence globally generated, there is a vector field formula_15 on formula_14 such that the equality formula_17 holds on restriction.
Examples.
There are many examples of convex spaces, including the following.
Spaces with trivial rational curves.
If the only maps from a rational curve to formula_2 are constants maps, then the pullback of the tangent sheaf is the free sheaf formula_19 where formula_20. These sheaves have trivial non-zero cohomology, and hence they are always convex. In particular, Abelian varieties have this property since the Albanese variety of a rational curve formula_21 is trivial, and every map from a variety to an Abelian variety factors through the Albanese.
Projective spaces.
Projective spaces are examples of homogeneous spaces, but their convexity can also be proved using a sheaf cohomology computation. Recall the Euler sequence relates the tangent space through a short exact sequence
If we only need to consider degree formula_23 embeddings, there is a short exact sequence
giving the long exact sequence
since the first two formula_26-terms are zero, which follows from formula_21 being of genus formula_28, and the second calculation follows from the Riemann–Roch theorem, we have convexity of formula_29. Then, any nodal map can be reduced to this case by considering one of the components formula_30 of formula_21.
Homogeneous spaces.
Another large class of examples are homogenous spaces formula_32 where formula_33 is a parabolic subgroup of formula_34. These have globally generated sections since formula_34 acts transitively on formula_2, meaning it can take a bases in formula_37 to a basis in any other point formula_38, hence it has globally generated sections. Then, the pullback is always globally generated. This class of examples includes Grassmannians, projective spaces, and flag varieties.
Product spaces.
Also, products of convex spaces are still convex. This follows from the Kunneth theorem in coherent sheaf cohomology.
Applications.
There are many useful technical advantages of considering moduli spaces of stable curves on convex spaces. That is, the spaces formula_1 have nice geometric and deformation-theoretic properties.
Deformation theory.
The deformations of formula_3 in the Hilbert scheme of graphs formula_41 has tangent space
where formula_43 is the point in the scheme representing the map. Convexity of formula_2 gives the dimension formula below. In addition, convexity implies all infinitesimal deformations are unobstructed.
Structure.
These spaces are normal projective varieties of pure dimension
which are locally the quotient of a smooth variety by a finite group. Also, the open subvariety formula_46 parameterizing non-singular maps is a smooth fine moduli space. In particular, this implies the stacks formula_47 are orbifolds.
Boundary divisors.
The spaces formula_1 have nice boundary divisors given by
for a partition formula_50 of formula_51 and formula_52 the point lying along the intersection of two rational curves formula_53.