Convexity (algebraic geometry)

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces ${\overline {M}}_{0,n}(X,\beta )$ in quantum cohomology. These moduli spaces are smooth orbifolds whenever the target space is convex. A variety $X$ is called convex if the pullback of the tangent bundle to a stable rational curve $f:C\to X$ has globally generated sections. Geometrically this implies the curve is free to move around $X$ infinitesimally without any obstruction. Convexity is generally phrased as the technical condition

H^{1}(C,f^{*}T_{X})=0

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 $p,q$ in a convex set $C\subset \mathbb {R} ^{n}$ , all of the points $tp+(1-t)q$ are contained in that set. There is a vector field ${\mathcal {X}}_{U_{p}}$ in a neighborhood $U_{p}$ of $p$ transporting $p$ to each point $p'\in \{tp+(1-t)q:t\in \}\cap U_{p}$ . Since the vector bundle of $\mathbb {R} ^{n}$ is trivial, hence globally generated, there is a vector field ${\mathcal {X}}$ on $\mathbb {R} ^{n}$ such that the equality ${\mathcal {X}}|_{U_{p}}={\mathcal {X}}_{U_{p}}$ 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 $X$ are constants maps, then the pullback of the tangent sheaf is the free sheaf ${\mathcal {O}}_{C}^{\oplus n}$ where $n=\dim(X)$ . 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 $C$ 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

0\to {\mathcal {O}}\to {\mathcal {O}}(1)^{\oplus (n+1)}\to {\mathcal {T}}_{\mathbb {P} ^{n}}\to 0

If we only need to consider degree $d$ embeddings, there is a short exact sequence

0\to {\mathcal {O}}_{C}\to {\mathcal {O}}_{C}(d)^{\oplus (n+1)}\to f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}}\to 0

giving the long exact sequence

{\begin{aligned}0&\to H^{0}(C,{\mathcal {O}})\to H^{0}(C,{\mathcal {O}}(d)^{\oplus (n+1)})\to H^{0}(C,f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}})\\&\to H^{1}(C,{\mathcal {O}})\to H^{1}(C,{\mathcal {O}}(d)^{\oplus (n+1)})\to H^{1}(C,f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}})\to 0\end{aligned}}

since the first two $H^{1}$ -terms are zero, which follows from $C$ being of genus $0$ , and the second calculation follows from the Riemann–Roch theorem, we have convexity of $\mathbb {P} ^{n}$ . Then, any nodal map can be reduced to this case by considering one of the components $C_{i}$ of $C$ .

Homogeneous spaces

Another large class of examples are homogenous spaces $G/P$ where $P$ is a parabolic subgroup of $G$ . These have globally generated sections since $G$ acts transitively on $X$ , meaning it can take a bases in $T_{x}X$ to a basis in any other point $T_{y}X$ , 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 Künneth theorem in coherent sheaf cohomology.

Projective bundles over curves

One more non-trivial class of examples of convex varieties are projective bundles $\mathbb {P} ({\mathcal {E}})$ for an algebraic vector bundle ${\mathcal {E}}\to C$ over a smooth algebraic curve.

Applications

There are many useful technical advantages of considering moduli spaces of stable curves mapping to convex spaces. That is, the Kontsevich moduli spaces ${\overline {M}}_{0,n}(X,\beta )$ have nice geometric and deformation-theoretic properties.

Deformation theory

The deformations of $f:C\to X$ in the Hilbert scheme of graphs $\operatorname {Hom} (C,X)\subset \operatorname {Hilb} _{C\times X/\operatorname {Spec} (\mathbb {C} )}$ has tangent space

T_{\operatorname {Hom} (C,X)}()\cong H^{0}(C,f^{*}T_{X})

where $\in \operatorname {Hom} (C,X)$ is the point in the scheme representing the map. Convexity of $X$ gives the dimension formula below. In addition, convexity implies all infinitesimal deformations are unobstructed.

Structure

These spaces are normal projective varieties of pure dimension

\dim({\overline {M}}_{0,n}(X,\beta ))=\dim(X)+\int _{\beta }c_{1}(T_{X})+n-3

which are locally the quotient of a smooth variety by a finite group. Also, the open subvariety ${\overline {M}}_{0,n}^{*}(X,\beta )$ parameterizing non-singular maps is a smooth fine moduli space. In particular, this implies the stacks ${\overline {\mathcal {M}}}_{0,n}(X,\beta )$ are orbifolds.