In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, convexity is a restrictive technical condition for
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
originally introduced to analyze Kontsevich
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s
in
quantum cohomology In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, t ...
.
These moduli spaces are smooth
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
s whenever the target space is convex. A variety
is called convex if the pullback of the tangent bundle to a stable
rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
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 In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the exist ...
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
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
can be extended globally. This generalizes the idea of
convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, 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.
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
are constants maps, then the pullback of the tangent sheaf is the free sheaf
where
. These sheaves have trivial non-zero cohomology, and hence they are always convex. In particular,
Abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
have this property since the
Albanese variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
Precise statement
The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to ...
of a rational curve
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 In mathematics, the Euler sequence is a particular exact sequence of sheaves on ''n''-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (n+1)-fold sum of the dual of the Serre ...
relates the tangent space through a short exact sequence
:
If we only need to consider degree
embeddings, there is a short exact sequence
:
giving the long exact sequence
:
since the first two
-terms are zero, which follows from
being of genus
, and the second calculation follows from the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
, we have convexity of
. Then, any nodal map can be reduced to this case by considering one of the components
of
.
Homogeneous spaces
Another large class of examples are homogenous spaces
where
is a parabolic subgroup of
. These have globally generated sections since
acts transitively on
, meaning it can take a bases in
to a basis in any other point
, hence it has globally generated sections.
Then, the pullback is always globally generated. This class of examples includes
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
s, 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.
Projective bundles over curves
One more non-trivial class of examples of convex varieties are projective bundles
for an algebraic vector bundle
over a smooth algebraic curve
pg 6.
Applications
There are many useful technical advantages of considering moduli spaces of stable curves mapping to convex spaces. That is, the
Kontsevich moduli space
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
s
have nice geometric and deformation-theoretic properties.
Deformation theory
The deformations of
in the Hilbert scheme of graphs
has tangent space
:
where
is the point in the scheme representing the map. Convexity of
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
parameterizing non-singular maps is a smooth fine moduli space. In particular, this implies the stacks
are
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
s.
Boundary divisors
The moduli spaces
have nice boundary divisors for convex varieties
given by
:
for a partition
of