Cohomology

Let $X$ be a GKM variety with respect to the complex torus $T$. By [GKM98], we have the following description of its equivariant cohomology ring $H^*_T(X;\mathbb{Q})$.

Each fixed point $x\in X$ gives a ring homomorphism $H^*_T(X;\mathbb{Q})\rightarrow H^*_T(\{x\};\mathbb{Q})\cong \mathbb{Q}[\mathfrak{t}]$. We may combine these maps by taking all fixed points at once. Let $V$ be the set of fixed points of $X$ and $E$ the set of $T$-invariant rational curves. That is, $V$ are the vertices of the GKM graph and $E$ the edges. Then the map

\[ H^*_T(X;\mathbb{Q}) \longrightarrow \bigoplus_{x\in V} \mathbb{Q}[\mathfrak{t}]\]

is injective and its image consists of all $(f_x)_{x\in V}$ such that $f_{\text{src}(e)}\equiv f_{\text{dst}(e)}$ mod $w(e)$ for all edges $e\in E$, where $w(e)$ is the weight of $e$.

We may further identify $\mathbb{Q}[\mathfrak{t}]\cong \mathbb{Q}[t_1,\dots,t_r]$ where $r=\dim_{\mathbb{C}}(T)$. Hence, this package represents elements of $H^*_T(X;\mathbb{Q})$ as tuples of polynomials indexed by vertices of the GKM graph.

GKMtools.is_gkm_class — Function

is_gkm_class(c::FreeModElem{QQMPolyRingElem}, G::GKM_graph) -> Bool

Return true if the given class represents an actual cohomology class. This holds if and only if the difference between localizations at fixed points connected through an edge e is divisible by the weight of e (see above)

Examples

Standard functions for accessing cohomology classes always yield GKM classes:

julia> G = projective_space(GKM_graph, 1);

julia> is_gkm_class(point_class(G, 1), G)
true

Moreover, equivariant cohomology is a ring and a module over the coefficient ring:

julia> is_gkm_class(point_class(G, 1)^2 * point_class(G, 2), G)
true

However, it is possible to cook up non-GKM classes manually. In the example below, this is because $w(e)=t_1-t_2$, which does not divide $t_1^2 - t_2$. Here, $e$ is the unique edge of the GKM graph of $\mathbb{P}^1$.

julia> (t1, t2) = gens(G.equivariantCohomology.coeffRing);

julia> (e0, e1) = gens(G.equivariantCohomology.cohomRing);

julia> c = t1^2 * e0 + t2 * e1
t1^2*e[1] + t2*e[2]

julia> is_gkm_class(c, G)
false

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Oscar.IntersectionTheory.point_class — Function

point_class(vertexLabel::String, G::GKM_graph) -> FreeModElem{QQMPolyRingElem}

Return the equivariant Poincare dual of the fixed point with given label.

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point_class(vertex::Int, G::GKM_graph) -> FreeModElem{QQMPolyRingElem}

Return the equivariant Poincare dual of the given fixed point.

Examples

julia> P2 = projective_space(GKM_graph, 2);

julia> point_class(1, P2)
(t1^2 - t1*t2 - t1*t3 + t2*t3)*e[1]

julia> F3 = flag_variety(GKM_graph, [1, 1, 1]);

julia> point_class(1, F3)
(t1^2*t2 - t1^2*t3 - t1*t2^2 + t1*t3^2 + t2^2*t3 - t2*t3^2)*e[1]

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GKMtools.poincare_dual — Function

poincare_dual(gkmSub::GKM_subgraph) -> FreeModElem{QQMPolyRingElem}

Return the equivariant Poincare dual cohomology class of the GKM subgraph.

Example

julia> P2 = projective_space(GKM_graph, 2);

julia> P1inP2 = gkm_subgraph_from_vertices(P2, [1, 2])
GKM subgraph of:
GKM graph with 3 nodes, valency 2 and axial function:
2 -> 1 => (-1, 1, 0)
3 -> 1 => (-1, 0, 1)
3 -> 2 => (0, -1, 1)
Subgraph:
GKM graph with 2 nodes, valency 1 and axial function:
2 -> 1 => (-1, 1, 0)

julia> poincare_dual(P1inP2)
(t1 - t3)*e[1] + (t2 - t3)*e[2]

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GKMtools.weight_class — Function

weight_class(e::Edge, G::GKM_graph) -> QQMPolyRingElem

Return the weight of the edge e as an element of the coefficient ring of the equivariant cohomology theory.

Example

julia> H7 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 7))
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (1, 0, -1, 0)
3 -> 2 => (7, 1, 0, -1)
4 -> 1 => (0, 1, 7, -1)
4 -> 3 => (-1, 0, 1, 0)

julia> weight_class(Edge(3, 2), H7)
7*t1 + t2 - t4

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GKMtools.euler_class — Function

euler_class(vertex::Int64, G::GKM_graph) -> QQMPolyRingElem

Return the Euler class of the normal bundle of the fixed point v. This is the localization of point_class(vertex, G) to the given vertex.

Example

julia> F3 = flag_variety(GKM_graph, [1, 1, 1]);

julia> euler_class(1, F3)
t1^2*t2 - t1^2*t3 - t1*t2^2 + t1*t3^2 + t2^2*t3 - t2*t3^2

julia> point_class(1, F3)
(t1^2*t2 - t1^2*t3 - t1*t2^2 + t1*t3^2 + t2^2*t3 - t2*t3^2)*e[1]

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euler_class(vertexLabel::String, G::GKM_graph) -> QQMPolyRingElem

Return the euler class of the normal bundle of the fixed point with the given label.

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GKMtools.integrate_gkm_class — Function

integrate_gkm_class(class::FreeModElem{QQMPolyRingElem}, G::GKM_graph; check::Bool=true) -> QQMPolyRingElem

Integrate the GKM class, yielding an element of the coefficient ring. This checks if is_gkm_class(class,R) == true and throws an error otherwise.

Examples

julia> P2 = projective_space(GKM_graph, 2);

julia> integrate_gkm_class(point_class(1, P2), P2)
1

julia> P2inP1 = gkm_subgraph_from_vertices(P2, [1, 2]);

julia> pd = poincare_dual(P2inP1);

julia> integrate_gkm_class(pd, P2)
0

julia> integrate_gkm_class(pd^2, P2)
1

julia> (t1, t2, t3) = gens(P2.equivariantCohomology.coeffRing);

julia> integrate_gkm_class(t3 * pd^2 + (t2^2 - t1)*point_class(3, P2), P2)
-t1 + t2^2 + t3

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Nemo.integrate — Function

integrate(class::FreeModElem{QQMPolyRingElem}, G::GKM_graph, e::Edge) -> AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}

Integrate the cohomology class over the curve represented by the GKM graph edge. In mathematical notation, if the edge is $e = (p\rightarrow q)$ and the class is $c=(f_v)_{v\in X^T}$ then $\int_e c = \frac{f_p - f_q}{w(e)}$ where $w(e)$ is the weight of $e$. If is_gkm_class(class, G) == true then the result is a polynomial in the variables gens(G.equivariantCohomology.coeffRing).

Example

julia> P2 = projective_space(GKM_graph, 2);

julia> integrate(first_chern_class(P2), P2, Edge(1, 2))
3

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integrate(class::FreeModElem{QQMPolyRingElem}, G::GKM_graph) -> AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}

Integrate the cohomology class, yielding an element of the fraction field of the equivariant coefficient ring.

This uses the Atiyah–Bott localization formula: If $c = (f_v)_{v\in X^T}$ then $\int_X c = \sum_{v\in X^T}\frac{f_v}{e_T(T_vX)}$ where $e_T(T_vX)$ is the equivariant Euler class of the tangent bundle of $X$ at the fixed point $v$.

If is_gkm_class(class, G) == true, this fraction will be a polynomial.

Note

Use integrate_gkm_class() instead if you know that the class is a GKM class.

Examples

julia> G24 = flag_variety(GKM_graph, [2, 4]); # Grassmannian of 2-planes in C^4

julia> integrate(point_class(G24, 1), G24)
1

In contrast to integrate_gkm_class, we can also integrate tuples $(f_v)_{v\in X^T}$ that do not satisfy is_gkm_class(class) == true:

julia> P1 = projective_space(GKM_graph, 1);

julia> (t1, t2) = gens(P1.equivariantCohomology.coeffRing);

julia> (e0, e1) = gens(P1.equivariantCohomology.cohomRing);

julia> c = t1^2 * e0 + t2 * e1
t1^2*e[1] + t2*e[2]

julia> is_gkm_class(c, P1)
false

julia> integrate(c, P1)
(t1^2 - t2)//(t1 - t2)

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GKMtools.first_chern_class — Function

first_chern_class(G::GKM_graph) -> FreeModElem{QQMPolyRingElem}

Return the equivariant first Chern class of the GKM space, i.e., $c_1^T(TX)$ where $TX$ is the ($T$-equivariant) tangent bundle of $X$.

Examples

julia> P2 = projective_space(GKM_graph, 2);

julia> first_chern_class(P2)
(2*t1 - t2 - t3)*e[1] + (-t1 + 2*t2 - t3)*e[2] + (-t1 - t2 + 2*t3)*e[3]

julia> H3 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 3))
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (1, 0, -1, 0)
3 -> 2 => (3, 1, 0, -1)
4 -> 1 => (0, 1, 3, -1)
4 -> 3 => (-1, 0, 1, 0)

julia> first_chern_class(H3)
(-t1 - t2 - 2*t3 + t4)*e[1] + (-2*t1 - t2 - t3 + t4)*e[2] + (4*t1 + t2 - t3 - t4)*e[3] + (-t1 + t2 + 4*t3 - t4)*e[4]

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