Chow rings

Summary

struct ChowRing

A Type used to encode various Chow ring data.

Fields

parent :: Any
ring :: Singular.PolyRing{Singular.n_Q}
chi :: AbstractVector{Int}
point :: Union{Singular.spoly{Singular.n_Q},UndefInitializer}
todd :: Union{Singular.spoly{Singular.n_Q},UndefInitializer}
_R :: Singular.PolyRing{Singular.n_Q}
_inclusion :: Singular.SAlgHom{Singular.Rationals}

chow_ring(Q::Quiver, d::AbstractVector{Int}, theta::AbstractVector{Int}=canonical_stability(Q, d); chi::AbstractVector{Int}=extended_gcd(d)[2])

Compute the Chow ring of the moduli space of theta-semistable representations of Q with dimension vector d, for a choice of linearization a.

This method of the function chow_ring also returns the ambient ring $R$ and the inclusion morphism.

Input

• Q::Quiver: a quiver.
• d::AbstractVector{Int}: a dimension vector.
• theta::AbstractVector{Int}: a stability parameter. Default is canonical_stability(Q, d).
• chi: a linearization. Default is the extended gcd of extended_gcd(d)[2].

Output

A tuple containing:

• the Chow ring of the moduli space,
• the polynomial ring above it,
• the inclusion map $\iota : A \to R$.

Examples

The Chow ring for the projective line has two generators:

julia> Q = kronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]);

julia> CH = chow_ring(M);

julia> QuiverTools.gens(QuiverTools.quotient_ideal(CH))
2-element Vector{Singular.spoly{Singular.n_Q}}:
 x21
 x11^2

The Chow ring for our favourite 6-fold has, in this implementation, 16 generators:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> CH = chow_ring(M); I = QuiverTools.quotient_ideal(CH);

julia> length(QuiverTools.gens(I))
16

chow_ring(M::QuiverModuliSpace; chi::Union{AbstractVector{Int},UndefInitializer}=undef)

Compute the Chow ring of the moduli space M for the given linearization chi.

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.
• chi::AbstractVector{Int}: a choice of linearization for the trivial line bundle. Default is extended_gcd(M.d)[2].

Output

• the Chow ring of the moduli space.

chow_ring(F::Bundle)

point_class(M::QuiverModuliSpace)

Compute the point class of the moduli space M.

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.

Output

• the point class of the moduli space, as a polynomial in its Chow ring.

Examples

A projective 7-fold:

julia> Q = kronecker_quiver(8);

julia> M = QuiverModuliSpace(Q, [1, 1]);

julia> chow_ring(M; chi=[1, 0]); point_class(M)
x21^7

Our favourite 6-fold:

julia> Q = kronecker_quiver(3);

julia> M = QuiverModuliSpace(Q, [2, 3]);

julia> point_class(M)
x23^2

todd_class(M::QuiverModuliSpace)

Compute the Todd class of the moduli space M.

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.

Output

• the Todd class of the moduli space, as a polynomial in its Chow ring.

Examples

The Todd class of our favourite 3-Kronecker quiver moduli:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> todd_class(M)
-17//8*x12*x21 + x21^2 + 823//360*x12*x22 - 823//1080*x22^2 + 553//1080*x21*x23 - 77//60*x22*x23 + x23^2 + 5//12*x12 - 3//2*x21 + 9//8*x23 + 1

chern_class_line_bundle(M::QuiverModuliSpace, eta::AbstractVector{Int})

Compute the first Chern class of the line bundle L(eta).

This is given by $L(eta) = \bigoplus_{i \in Q_0} \det(U_i)^{-eta_i}$.

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.
• eta::AbstractVector{Int]: a choice of linearization for the trivial line bundle.

Output

• the first Chern class of the line bundle L(eta) as a polynomial.

Examples

The line bundles $\mathcal{O}(i)$ on the projective line:

julia> Q = kronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]);

julia> l = chern_class_line_bundle(M, [1, -1])
-x11

The line bundle corresponding to the canonical stability condition on our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> chern_class_line_bundle(M, [9, -6])
-3*x21

chern_character_line_bundle(M::QuiverModuliSpace, eta::AbstractVector{Int})

Compute the Chern character of the line bundle L(eta).

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.
• eta::AbstractVector{Int}: a choice of linearization for the trivial line bundle.

Output

• the Chern character of the line bundle L(eta).

Examples

Some line bundles on the projective line:

julia> Q = kronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]);

julia> chern_character_line_bundle(M, [1, -1])
-x11 + 1

Some Chern characters for our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> chern_character_line_bundle(M, [3, -2])
1//720*x21^6 - 1//120*x21^5 + 1//24*x21^4 - 1//6*x21^3 + 1//2*x21^2 - x21 + 1

integral(M::QuiverModuliSpace, f)

Computes the integral of f according to the Hirzebruch-Riemann-Roch theorem.

In other words, it computes the Euler characteristic of the vector bundle whose Chern character is f.

Input

• M::QuiverModuliSpace: a moduli space of representations of a quiver.
• f: the Chern character in to integrate.

Output

• the integral of f.

Examples

The integral of $\mathcal{O}(i)$ on the projective line for some is.

julia> Q = kronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]);

julia> L = chern_character_line_bundle(M, [1, -1]);

julia> [integral(M, L^i) for i in 0:5]
6-element Vector{Singular.n_Q}:
 1
 2
 3
 4
 5
 6

Hilbert series for the 3-Kronecker quiver as in our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> L = chern_character_line_bundle(M, [3, -2]);

julia> [integral(M, L^i) for i in 0:5]
6-element Vector{Singular.n_Q}:
 1
 20
 148
 664
 2206
 5999

Summary

mutable struct Bundle

An abstract bundle on a quiver moduli. It is represented in practice by its Chern character.

Fields

parent :: ChowRing
rank :: Int
chern_character :: Singular.spoly{Singular.n_Q}
chern_class :: Dict{Int,Singular.spoly{Singular.n_Q}}
teleman_weights :: Dict{<:HNTypes,Vector{Int}}

chern_character(F::Bundle)

chern_class(F::Bundle)

chern_class(F::Bundle, k)

chern_classes(F::Bundle)

degree(F::Bundle)

Compute the degree of the bundle F. If rank(F) is larger than $1$, returns the degree of the determinant of F.

Input

• F::Bundle: a bundle.

Output

• the degree of F.

Example

The degrees of canonical bundles on the first projective spaces:

julia> d = [1, 1]; Pn = map(i -> QuiverModuliSpace(kronecker_quiver(i + 1), d), 1:5);

julia> omega = map(canonical_bundle, Pn);

julia> omega = map(dual, omega);

julia> map(degree, omega)
5-element Vector{Singular.spoly{Singular.n_Q}}:
 2
 9
 64
 625
 7776

An example from [arXiv:2411.15125]:

julia> Q = Quiver("1-2,1-3,2---3"); d = [1, 1, 1]; a = [1, 1, -1];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=a);

julia> F = dual(canonical_bundle(M))
Bundle of rank 1

julia> chern_class(F)
2*x31 + 1

julia> degree(F)
56

rank(F::Bundle)

teleman_weights(F::Bundle)

line_bundle(M::QuiverModuliSpace, eta::AbstractVector{Int})

Construct the descent of L(eta) on the quiver moduli space M.

Input

• M::QuiverModuliSpace: a quiver moduli space.
• eta::AbstractVector{Int}: a dimension vector.

Output

• the line bundle on M with Chern class chern_class_line_bundle(M, eta).

Examples

The ample generator of the Picard group of our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> chow_ring(M; chi=[-1, 1]);

julia> H = line_bundle(M, [3, -2]);

julia> chern_class(H)
-x21

julia> teleman_weights(H)
Dict{HNType{2}, Vector{Int64}} with 7 entries:
  [[2, 2], [0, 1]]         => [-30]
  [[2, 1], [0, 2]]         => [-40]
  [[1, 0], [1, 2], [0, 1]] => [-40]
  [[1, 0], [1, 3]]         => [-135]
  [[1, 0], [1, 1], [0, 2]] => [-105]
  [[1, 1], [1, 2]]         => [-5]
  [[2, 0], [0, 3]]         => [-90]

zero_sheaf(M::QuiverModuliSpace)

structure_sheaf(M::QuiverModuliSpace)

canonical_bundle(M::QuiverModuliSpace)

Compute the canonical bundle on the quiver moduli space M.

If $d$ is $theta$-coprime and amply stable, the canonical bundle is described in [Proposition 4.2, MR4352662].

This function computes both the Chern character and the Teleman weights of the canonical bundle.

Input

• M::QuiverModuliSpace: a quiver moduli space.

Output

• the canonical bundle on M.

Example

On various projective spaces:

julia> d = [1, 1]; Pn = map(i -> QuiverModuliSpace(kronecker_quiver(i + 1), d), 1:5);

julia> omega = map(canonical_bundle, Pn);

julia> map(chern_class, omega)
5-element Vector{Singular.spoly{Singular.n_Q}}:
 2*x11
 3*x11
 4*x11
 5*x11
 6*x11

julia> map(teleman_weights, omega)
5-element Vector{Dict{HNType{2}, Vector{Int64}}}:
 Dict([[1, 0], [0, 1]] => [8])
 Dict([[1, 0], [0, 1]] => [18])
 Dict([[1, 0], [0, 1]] => [32])
 Dict([[1, 0], [0, 1]] => [50])
 Dict([[1, 0], [0, 1]] => [72])

On our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> omega = canonical_bundle(M);

julia> chern_class(omega)
3*x21

julia> QuiverTools.teleman_weights(omega)
Dict{HNType{2}, Vector{Int64}} with 7 entries:
  [[2, 2], [0, 1]]         => [90]
  [[2, 1], [0, 2]]         => [120]
  [[1, 0], [1, 2], [0, 1]] => [120]
  [[1, 0], [1, 3]]         => [405]
  [[1, 0], [1, 1], [0, 2]] => [315]
  [[1, 1], [1, 2]]         => [15]
  [[2, 0], [0, 3]]         => [270]

universal_bundle(M:::QuiverModuliSpace, i::Int)

Compute the i-th universal bundle of M.

This function computes both the Chern classes and the Teleman weights of the universal bundle.

To use this method, the moduli space M must have a linearization. If not, this method will initialize the Chow ring of M by calling chow_ring(M) and it will use the default linearization.

Input

• M::QuiverModuliSpace: a quiver moduli space.
• i::Int: the universal bundle on the i-th vertex of the quiver.

Output

• the i-th universal bundle on M.

Example

On the projective line:

julia> Q = kronecker_quiver(2); d = [1, 1];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=[1, 0]);

julia> u1, u2 = universal_bundle(M, 1), universal_bundle(M, 2);

julia> map(chern_class, [u1, u2])
2-element Vector{Singular.spoly{Singular.n_Q}}:
 x11 + 1
 x21 + 1

julia> map(teleman_weights, [u1, u2])
2-element Vector{Dict{HNType{2}, Vector{Int64}}}:
 Dict([[1, 0], [0, 1]] => [0])
 Dict([[1, 0], [0, 1]] => [-4])

On our favourite 6-fold:

julia> Q = kronecker_quiver(3); M = QuiverModuliSpace(Q, [2, 3]);

julia> chow_ring(M; chi=[-1, 1]);

julia> u1, u2 = universal_bundle(M, 1), universal_bundle(M, 2);

julia> map(chern_character, [u1, u2])
2-element Vector{Singular.spoly{Singular.n_Q}}:
 1//6*x12*x21 + 1//2*x21^2 + 1//12*x12*x22 - 1//36*x22^2 - 7//72*x21*x23 - 1//120*x22*x23 - 1//720*x23^2 - x12 + x21 - 1//2*x23 + 2
 2//3*x12*x21 + 1//2*x21^2 - 1//6*x12*x22 - 1//2*x21*x22 + 1//12*x22^2 + 1//24*x21*x23 + 1//180*x22*x23 + x21 - x22 + 3

julia> w = map(teleman_weights, [u1, u2]);

julia> w[1]
Dict{HNType{2}, Vector{Int64}} with 7 entries:
  [[2, 2], [0, 1]]         => [15, 15]
  [[2, 1], [0, 2]]         => [20, 20]
  [[1, 0], [1, 2], [0, 1]] => [25, 15]
  [[1, 0], [1, 3]]         => [90, 45]
  [[1, 0], [1, 1], [0, 2]] => [60, 45]
  [[1, 1], [1, 2]]         => [5, 0]
  [[2, 0], [0, 3]]         => [45, 45]

julia> w[2]
Dict{HNType{2}, Vector{Int64}} with 7 entries:
  [[2, 2], [0, 1]]         => [15, 15, 0]
  [[2, 1], [0, 2]]         => [20, 10, 10]
  [[1, 0], [1, 2], [0, 1]] => [15, 15, 10]
  [[1, 0], [1, 3]]         => [45, 45, 45]
  [[1, 0], [1, 1], [0, 2]] => [45, 30, 30]
  [[1, 1], [1, 2]]         => [5, 0, 0]
  [[2, 0], [0, 3]]         => [30, 30, 30]

dual(F::Bundle)

Compute the dual bundle of F.

Input

• F::Bundle: a bundle.

Output

• the dual bundle of F.

Example

On the projective line:

julia> Q = kronecker_quiver(2); d = [1, 1]; a = [1, 0];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=a);

julia> td = Bundle(M, todd_class(M))
Bundle of rank 1

julia> chern_character(dual(td))
-x21 + 1

On our favourite 6-fold:

julia> Q, d = kronecker_quiver(3), [2, 3];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=[-1, 1]);

julia> td = Bundle(M, todd_class(M));

julia> chern_character(td)
-17//8*x12*x21 + x21^2 + 823//360*x12*x22 - 823//1080*x22^2 + 553//1080*x21*x23 - 77//60*x22*x23 + x23^2 + 5//12*x12 - 3//2*x21 + 9//8*x23 + 1


julia> tdd = dual(td)
Bundle of rank 1

julia> chern_character(tdd)
17//8*x12*x21 + x21^2 + 823//360*x12*x22 - 823//1080*x22^2 + 553//1080*x21*x23 + 77//60*x22*x23 + x23^2 + 5//12*x12 + 3//2*x21 - 9//8*x23 + 1

exterior_power(F::Bundle, k::Int)

Compute the k-th exterior power of F.

Input

• F::Bundle: a bundle.
• k::Int: the degree of the exterior power.

Output

• the k-th exterior power of F.

Example

On the projective line:

julia> Q = kronecker_quiver(2); d = [1, 1]; a = [1, 0];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=a);

julia> td = Bundle(M, todd_class(M))
Bundle of rank 1

julia> F = 3*td
Bundle of rank 3

julia> chern_character(F)
3*x21 + 3

julia> W = map(i -> exterior_power(F, i), 0:4);

julia> map(w -> (rank(w), chern_character(w)), W)
5-element Vector{Tuple{Int64, Singular.spoly{Singular.n_Q}}}:
 (1, 1)
 (3, 3*x21 + 3)
 (3, 6*x21 + 3)
 (1, 3*x21 + 1)
 (0, 0)

symmetric_power(F::Bundle, k::Int)

Compute the k-th symmetric power of F.

Input

• F::Bundle: a bundle.
• k::Int: the degree of the symmetric power.

Output

• the k-th symmetric power of F.

Example

On the projective line:

julia> Q = kronecker_quiver(2); d = [1, 1]; a = [1, 0];

julia> M = QuiverModuliSpace(Q, d); chow_ring(M; chi=a);

julia> td = Bundle(M, todd_class(M));

julia> F = 3*td
Bundle of rank 3

julia> chern_character(F)
3*x21 + 3

julia> W = map(i -> symmetric_power(F, i), 0:4);

julia> map(w -> (rank(w), chern_character(w)), W)
5-element Vector{Tuple{Int64, Singular.spoly{Singular.n_Q}}}:
 (1, 1)
 (3, 3*x21 + 3)
 (6, 12*x21 + 6)
 (10, 30*x21 + 10)
 (15, 60*x21 + 15)

det(F::Bundle)

Compute the determinant of F. This is the top exterior power of F.