Convexity (algebraic geometry)

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces M ¯ 0 , n ( X , β ) {\displaystyle {\overline {M}}_{0,n}(X,\beta )} in quantum cohomology.[1]: §1 [2][3] These moduli spaces are smooth orbifolds whenever the target space is convex. A variety  X {\displaystyle X} is called convex if the pullback of the tangent bundle to a stable rational curve f : C → X {\displaystyle f:C\to X} has globally generated sections.[2] Geometrically this implies the curve is free to move around  X {\displaystyle X} infinitesimally without any obstruction. Convexity is generally phrased as the technical condition

H 1 ( C , f ∗ T X ) = 0 {\displaystyle 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 {\displaystyle p,q} in a convex set  C ⊂ R n {\displaystyle C\subset \mathbb {R} ^{n}} , all of the points  t p + ( 1 − t ) q {\displaystyle tp+(1-t)q} are contained in that set. There is a vector field  X U p {\displaystyle {\mathcal {X}}_{U_{p}}} in a neighborhood  U p {\displaystyle U_{p}} of  p {\displaystyle p} transporting  p {\displaystyle p} to each point  p ′ ∈ { t p + ( 1 − t ) q : t ∈ [ 0 , 1 ] } ∩ U p {\displaystyle p'\in \\{tp+(1-t)q:t\in [0,1]\\}\cap U_{p}} . Since the vector bundle of  R n {\displaystyle \mathbb {R} ^{n}} is trivial, hence globally generated, there is a vector field  X {\displaystyle {\mathcal {X}}} on  R n {\displaystyle \mathbb {R} ^{n}} such that the equality  X | U p = X U p {\displaystyle {\mathcal {X}}|_{U_{p}}={\mathcal {X}}_{U_{p}}} holds on restriction.

## Examples

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There are many examples of convex spaces, including the following.

### Spaces with trivial rational curves

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If the only maps from a rational curve to  X {\displaystyle X} are constants maps, then the pullback of the tangent sheaf is the free sheaf  O C ⊕ n {\displaystyle {\mathcal {O}}_{C}^{\oplus n}} where  n = dim ⁡ ( X ) {\displaystyle 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 {\displaystyle C} is trivial, and every map from a variety to an Abelian variety factors through the Albanese.[4]

### Projective spaces

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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 → O → O ( 1 ) ⊕ ( n + 1 ) → T P n → 0 {\displaystyle 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 {\displaystyle d} embeddings, there is a short exact sequence

0 → O C → O C ( d ) ⊕ ( n + 1 ) → f ∗ T P n → 0 {\displaystyle 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

0 → H 0 ( C , O ) → H 0 ( C , O ( d ) ⊕ ( n + 1 ) ) → H 0 ( C , f ∗ T P n ) → H 1 ( C , O ) → H 1 ( C , O ( d ) ⊕ ( n + 1 ) ) → H 1 ( C , f ∗ T P n ) → 0 {\displaystyle {\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 {\displaystyle H^{1}} -terms are zero, which follows from  C {\displaystyle C} being of genus  0 {\displaystyle 0} , and the second calculation follows from the Riemann–Roch theorem, we have convexity of  P n {\displaystyle \mathbb {P} ^{n}} . Then, any nodal map can be reduced to this case by considering one of the components  C i {\displaystyle C_{i}} of  C {\displaystyle C} .

### Homogeneous spaces

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Another large class of examples are homogenous spaces  G / P {\displaystyle G/P} where  P {\displaystyle P} is a parabolic subgroup of  G {\displaystyle G} . These have globally generated sections since  G {\displaystyle G} acts transitively on  X {\displaystyle X} , meaning it can take a bases in  T x X {\displaystyle T_{x}X} to a basis in any other point  T y X {\displaystyle T_{y}X} , hence it has globally generated sections.[3] Then, the pullback is always globally generated. This class of examples includes Grassmannians, projective spaces, and flag varieties.

### Product spaces

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Also, products of convex spaces are still convex. This follows from the Künneth theorem in coherent sheaf cohomology.

### Projective bundles over curves

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One more non-trivial class of examples of convex varieties are projective bundles  P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} for an algebraic vector bundle  E → C {\displaystyle {\mathcal {E}}\to C} over a smooth algebraic curve[3]pg 6.

## Applications

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

### Deformation theory

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

T Hom ⁡ ( C , X ) ( [ f ] ) ≅ H 0 ( C , f ∗ T X ) {\displaystyle T_{\operatorname {Hom} (C,X)}([f])\cong H^{0}(C,f^{*}T_{X})}   [1]

where  [ f ] ∈ Hom ⁡ ( C , X ) {\displaystyle [f]\in \operatorname {Hom} (C,X)} is the point in the scheme representing the map. Convexity of  X {\displaystyle X} gives the dimension formula below. In addition, convexity implies all infinitesimal deformations are unobstructed.[5]

### Structure

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These spaces are normal projective varieties of pure dimension

dim ⁡ ( M ¯ 0 , n ( X , β ) ) = dim ⁡ ( X ) + ∫ β c 1 ( T X ) + n − 3 {\displaystyle \dim({\overline {M}}_{0,n}(X,\beta ))=\dim(X)+\int _{\beta }c_{1}(T_{X})+n-3}   [3]

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

### Boundary divisors

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The moduli spaces  M ¯ 0 , n ( X , β ) {\displaystyle {\overline {M}}_{0,n}(X,\beta )} have nice boundary divisors for convex varieties  X {\displaystyle X} given by

D ( A , B ; β 1 , β 2 ) = M ¯ 0 , A ∪ { ∙ } ( X , β 1 ) × X M ¯ 0 , B ∪ { ∙ } ( X , β 2 ) {\displaystyle D(A,B;\beta _{1},\beta _{2})={\overline {M}}_{0,A\cup \\{\bullet \\}}(X,\beta _{1})\times _{X}{\overline {M}}_{0,B\cup \\{\bullet \\}}(X,\beta _{2})}   [3]

for a partition  A ∪ B {\displaystyle A\cup B} of  [ n ] {\displaystyle [n]} and  { ∙ } {\displaystyle \\{\bullet \\}} the point lying along the intersection of two rational curves  C = C 1 ∪ C 2 {\displaystyle C=C_{1}\cup C_{2}} .

## See also

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* Stable curve
* Moduli space
* Gromov–Witten invariant
* Quantum cohomology
* Moduli of curves

## References

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1. ^ a b Kontsevich, Maxim (1995). "Enumeration of Rational Curves Via Torus Actions". In Dijkgraaf, Robbert H.; Faber, Carel F.; van der Geer, Gerard B. M. (eds.). The Moduli Space of Curves. Progress in Mathematics. Vol. 129\. Boston: Birkhäuser. pp. 335–368\. arXiv:hep-th/9405035. doi:10.1007/978-1-4612-4264-2_12. ISBN 978-1-4612-8714-8. S2CID 16131978.
2. ^ a b Kontsevich, Maxim; Manin, Yuri. "Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry" (PDF). p. 9\. Archived (PDF) from the original on 2009-11-28.
3. ^ a b c d e Fulton, W.; Pandharipande, R. (1997-05-17). "Notes on stable maps and quantum cohomology". pp. 6, 12, 29, 31. arXiv:alg-geom/9608011.
4. ^ "ag.algebraic geometry - Is there any rational curve on an Abelian variety?". MathOverflow. Retrieved 2020-02-28.
5. ^ Maulik, Davesh. "Lectures on Donaldson-Thomas Theory" (PDF). p. 2\. Archived (PDF) from the original on 2020-03-01.

## External links

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* Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry
* Notes on Stable Maps and Quantum Cohomology arXiv:alg-geom/9608011