Properties of GKM graphs

valency(G::AbstractGKMGraph) -> Int64

Return the valency of G, i.e. the number of flags at each vertex.

Example:

The valency of the GKM graph of $\mathbb{P}^3$ is 3, since all of the fixed points $[1:0:0:0], \dots, [0:0:0:1]$ are connected to each other via some $T$-invariant $\mathbb{P}^1$'s. For example, $[1:0:0:0]$ and $[0:1:0:0]$ are connected by $\{[x:y:0:0] : x,y\in\mathbb{C}\}$.

julia> valency(projective_space(GKMGraph, 3))
3
julia> valency(grassmannian(GKMGraph, 2, 4)) # The Grassmannian of 2-planes in C^4
4
julia> valency(flag_variety(GKMGraph, [1, 1, 1, 1])) # The variety of full flags in C^4
6

rank_torus(G::AbstractGKMGraph) -> Int64

Return the rank of the torus acting on G. That is, the rank of the character group.

Examples

By default, the torus acting on $\mathbb{P}^n$ is $(\mathbb{C}^\times)^{n+1}$, acting by rescaling the homogeneous coordinates.

julia> P3 = projective_space(GKMGraph, 3);

julia> rank_torus(P3)
4

Taking products adds the rank:

julia> H6 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 6));

julia> rank_torus(H6)
4
julia> rank_torus(H6 * P3)
8

is_compact(G::AbstractGKMGraph) -> Bool

Return true if G is compact, i.e. all flags at all vertices are associated with edges (no standalone flags).

Example

julia> G = projective_space(GKMGraph, 2)
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)
Birkhoff-Grothendieck connection for GKM graph with 3 nodes and valency 2

julia> is_compact(G)
true

julia> A3 = gkm_graph_of_toric(affine_space(NormalToricVariety, 3)) # affine space
GKM graph with 1 nodes, valency 3 and axial function:
Standalone flags:
Algorithmic connection for GKM graph with 1 nodes and valency 3

julia> is_compact(A3)
false

gkm_independence(G::AbstractGKMGraph) -> Int64

Return the maximum integer in i in 0,1,...,valency(G) such that the GKM graph $G$ is $i$-independent. The GKM graph is $i$-independent if for each vertex $v$, each $i$-tuple of flags at $v$ has linearly independent axial function values.

Example

julia> G = gkm_3d_twisted_flag()
GKM graph with 6 nodes, valency 3 and axial function:
2 -> 1 => (0, -1)
3 -> 2 => (1, 0)
4 -> 1 => (1, -2)
4 -> 3 => (-1, 1)
5 -> 2 => (1, -1)
5 -> 4 => (0, -1)
6 -> 1 => (1, -1)
6 -> 3 => (2, -1)
6 -> 5 => (1, 0)
Algorithmic connection for GKM graph with 6 nodes and valency 3

julia> gkm_independence(G)
2

is2_indep(G::AbstractGKMGraph) -> Bool

Return true if G is 2-independent, i.e. the weights of every two flags at a vertex are linearly independent.

is3_indep(G::AbstractGKMGraph) -> Bool

Return true if G is 3-independent, i.e. the weights of every three flags at a vertex are linearly independent.

Example

The weights of $\mathbb{P}^3$ at the fixed point $[1:0:0:0]$ are $\{t_i-t_0:i\in\{1, 2, 3\}\}$, which are linearly independent over $\mathbb{C}$.

julia> is3_indep(projective_space(GKMGraph, 3))
true

The variety of complete flags in $\mathbb{C}^3$ is an example of a GKM graph that is not 3-independent:

julia> G = flag_variety(GKMGraph, [1, 1, 1])
GKM graph with 6 nodes, valency 3 and axial function:
13 -> 12 => (0, 1, -1)
21 -> 12 => (1, -1, 0)
31 -> 13 => (1, 0, -1)
31 -> 21 => (0, 1, -1)
23 -> 13 => (1, -1, 0)
23 -> 21 => (1, 0, -1)
32 -> 12 => (1, 0, -1)
32 -> 31 => (1, -1, 0)
32 -> 23 => (0, 1, -1)
Birkhoff-Grothendieck connection for GKM graph with 6 nodes and valency 3

julia> is3_indep(G)
false

fano_index(G::AbstractGKMGraph) -> ZZRingElem

Return the Fano index of the GKM graph, which is the greatest common divisor of the first Chern numbers of all its edges.

Examples

julia> P2 = projective_space(GKMGraph, 2);

julia> fano_index(P2)
3

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

julia> fano_index(F3)
2

pseudo_index(G::AbstractGKMGraph) -> ZZRingElem

Return the pseudo index of the GKM graph, which is the minimum of the first Chern numbers of all its edges.

Examples

The examples below show that the pseudo-index differs from the Fano index in general.

julia> P2 = projective_space(GKMGraph, 2);

julia> fano_index(P2), pseudo_index(P2)
(3, 3)

julia> P3 = projective_space(GKMGraph, 3);

julia> fano_index(P3), pseudo_index(P3)
(4, 4)

julia> G = P2 * P3;

julia> fano_index(G), pseudo_index(G)
(1, 3)

julia> T = gkm_3d_twisted_flag();

julia> fano_index(T), pseudo_index(T)
(2, 0)

is_strictly_nef(G::AbstractGKMGraph) -> Bool

Return true if and only if the Chern numbers of all curve classes corresponding to edges of the GKM graph are strictly positive.

Examples

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

julia> print_curve_classes(F3)
13 -> 12: (1, 0), Chern number: 2
21 -> 12: (0, 1), Chern number: 2
31 -> 13: (0, 1), Chern number: 2
31 -> 21: (1, 1), Chern number: 4
23 -> 13: (1, 1), Chern number: 4
23 -> 21: (1, 0), Chern number: 2
32 -> 12: (1, 1), Chern number: 4
32 -> 31: (1, 0), Chern number: 2
32 -> 23: (0, 1), Chern number: 2

julia> is_strictly_nef(F3)
true

julia> H5 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 5))
GKM graph with 4 nodes, valency 2 and axial function:
2 -> 1 => (-1, 0, 1, 0)
3 -> 2 => (-5, -1, 0, 1)
4 -> 1 => (0, -1, -5, 1)
4 -> 3 => (1, 0, -1, 0)
Algorithmic connection for GKM graph with 4 nodes and valency 2

julia> print_curve_classes(H5)
2 -> 1: (-5, 1), Chern number: -3
3 -> 2: (1, 0), Chern number: 2
4 -> 1: (1, 0), Chern number: 2
4 -> 3: (0, 1), Chern number: 7

julia> is_strictly_nef(H5)
false

betti_numbers(G::AbstractGKMGraph) -> Vector{Int64}

Return the array betti_numbers such that betti_numbers[i+1] is the $2i$-th combinatorial Betti number for $i$ from $0$ to the valency of G, as defined in [GZ01, section 1.3].

Examples

julia> H6 = gkm_graph_of_toric(hirzebruch_surface(NormalToricVariety, 6));

julia> betti_numbers(H6)
3-element Vector{Int64}:
 1
 2
 1

index_periodic_betti(G::AbstractGKMGraph) -> Vector{Int64}

Return the index-periodic Betti numbers of a compact GKM graph.

For a GKM graph with Fano index $p$, this function computes the sums of Betti numbers grouped by their residue class modulo $p$.

Examples

julia> P2 = projective_space(GKMGraph, 2);

julia> index_periodic_betti(P2)
3-element Vector{Int64}:
 1
 1
 1

QH_ss_check_GLLXBR(G::AbstractGKMGraph)::Bool

Return true if G satisfies conditions (1) and (2) in [BGL+25, Theorem 2.2].

is_generic(G::AbstractGKMGraph, xi) -> Bool

Return true if xi is a generic element of the weight lattice lattice(G), i.e. the Euclidean pairing of xi with every flag weight of G is nonzero.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> g1, g2, g3 = gens(lattice(P2));

julia> is_generic(P2, g1)
false

julia> is_generic(P2, g1 + g2)
false

julia> is_generic(P2, g1 + 2*g2 + 4*g3)
true

xi_index(G::AbstractGKMGraph, xi, v::Int) -> Int

Return the xi-index of vertex v of G, i.e. the number of flags at v whose weight pairs negatively with xi. Requires xi to be generic at v.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> g1, g2, g3 = gens(lattice(P2));

julia> xi = g1 + 2*g2 + 4*g3;

julia> [xi_index(P2, xi, v) for v in 1:3]
3-element Vector{Int64}:
 0
 1
 2

is_index_increasing(G::AbstractGKMGraph, xi) -> Bool

Return true if xi is generic and index increasing on G. For each edge e = {v, w}, orient it as (v, w) so that the weight of Edge(v, w) pairs positively with xi; then require the xi-index at w to be strictly greater than the xi-index at v.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> g1, g2, g3 = gens(lattice(P2));

julia> is_index_increasing(P2, g1 + 2*g2 + 4*g3)
true

julia> is_index_increasing(P2, g1)
false

is_weakly_index_increasing(G::AbstractGKMGraph, xi) -> Bool

Return true if xi is generic and weakly index increasing on G. For each edge e = {v, w}, orient it as (v, w) so that the weight of Edge(v, w) pairs positively with xi; then require the xi-index at w to be greater than or equal to the xi-index at v.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> g1, g2, g3 = gens(lattice(P2));

julia> is_weakly_index_increasing(P2, g1 + 2*g2 + 4*g3)
true

julia> is_weakly_index_increasing(P2, g1)
false

generic_xi_representatives(G::AbstractGKMGraph) -> Vector

Return one representative per connected component of the set of generic directions xi in the weight lattice of G. Each representative lives in lattice(G) (as a primitive element of the lattice if lattice(G) is a free ZZ-module).

The components of the generic locus are the open chambers of the hyperplane arrangement cut out by the flag weights of G, viewed in lattice(G) tensored with QQ.

Example

On P^1 there is a single hyperplane so two chambers:

julia> P1 = projective_space(GKMGraph, 1);

julia> length(generic_xi_representatives(P1))
2

On P^2 there are 6 chambers corresponding to the 6 orderings of the three coordinates of xi:

julia> P2 = projective_space(GKMGraph, 2);

julia> reps = generic_xi_representatives(P2);

julia> length(reps)
6

julia> all(xi -> is_generic(P2, xi), reps)
true

Non-compact example: the total space of O(1) + O(-1) on P^1:

julia> T = total_space(vector_bundle_O(1, [1, -1]));

julia> length(generic_xi_representatives(T))
8

index_increasing_xi_representatives(G::AbstractGKMGraph) -> Vector

Return one representative per connected component of the generic locus on which xi is index increasing. Each representative lives in lattice(G).

Example

Every chamber of P^2 is index-increasing:

julia> P2 = projective_space(GKMGraph, 2);

julia> reps = index_increasing_xi_representatives(P2);

julia> length(reps)
6

julia> all(xi -> is_index_increasing(P2, xi), reps)
true

For the non-compact total space of O(1) + O(-1) on P^1, only some chambers are strictly index-increasing (compare the count with the 8 generic chambers):

julia> T = total_space(vector_bundle_O(1, [1, -1]));

julia> length(index_increasing_xi_representatives(T))
6

weakly_index_increasing_xi_representatives(G::AbstractGKMGraph) -> Vector

Return one representative per connected component of the generic locus on which xi is weakly index increasing. Each representative lives in lattice(G).

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> reps = weakly_index_increasing_xi_representatives(P2);

julia> length(reps)
6

julia> all(xi -> is_weakly_index_increasing(P2, xi), reps)
true

On the non-compact total space of O(1) + O(-1) on P^1, every generic chamber turns out to be weakly index-increasing even though only 6 of them are strictly index-increasing:

julia> T = total_space(vector_bundle_O(1, [1, -1]));

julia> length(weakly_index_increasing_xi_representatives(T))
8

admits_index_increasing_xi(G::AbstractGKMGraph) -> Tuple{Bool, FreeModuleElem}

Return (true, xi) if G admits some generic index-increasing direction xi in lattice(G), where xi is the first such direction found during chamber enumeration. Return (false, zero(lattice(G))) otherwise.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> ok, xi = admits_index_increasing_xi(P2);

julia> ok
true

julia> is_index_increasing(P2, xi)
true

julia> G = gkm_2d([1 0; 1 1; 0 1; -1 0; -1 -1; 0 -1]);

julia> admits_index_increasing_xi(G)
(false, (0, 0))

The same is true on the non-compact total space of O(1) + O(-1) on P^1:

julia> T = total_space(vector_bundle_O(1, [1, -1]));

julia> ok, xi = admits_index_increasing_xi(T);

julia> ok
true

julia> is_index_increasing(T, xi)
true

admits_weakly_index_increasing_xi(G::AbstractGKMGraph) -> Tuple{Bool, FreeModuleElem}

Return (true, xi) if G admits some generic weakly-index-increasing direction xi in lattice(G), where xi is the first such direction found during chamber enumeration. Return (false, zero(lattice(G))) otherwise.

Example

julia> P2 = projective_space(GKMGraph, 2);

julia> ok, xi = admits_weakly_index_increasing_xi(P2);

julia> ok
true

julia> is_weakly_index_increasing(P2, xi)
true

julia> G = gkm_2d([1 0; 1 1; 0 1; -1 0; -1 -1; 0 -1]);

julia> ok, xi = admits_weakly_index_increasing_xi(G);

julia> ok && is_weakly_index_increasing(G, xi)
true

Non-compact total space of O(1) + O(-1) on P^1:

julia> T = total_space(vector_bundle_O(1, [1, -1]));

julia> ok, xi = admits_weakly_index_increasing_xi(T);

julia> ok
true

julia> is_weakly_index_increasing(T, xi)
true