Cobord.jl

Nemo.elem_type(::Type{CobordRing}) = CobordRingElem
Nemo.parent_type(::Type{CobordRingElem}) = CobordRing

Base.parent(x::CobordRingElem) = x.parent
deepcopy(x::CobordRingElem) = parent(x)(deepcopy(x.f))

function Base.show(io::IO, R::CobordRing)
  print(io, "Cobordism Ring")
  if R.base isa Singular.N_FField
    print(io, " with parameters $(tuple(Singular.transcendence_basis(R.base)...))")
  end
end

(R::CobordRing)(f::ChRingElem) = begin
  parent(f) == R.R && return CobordRingElem(R, R.n, f)
  n = length(gens(parent(f)))
  R[n]
  CobordRingElem(R, R.n, f.f(gens(R.R)[1:n]...))
end
(R::CobordRing)(n::RingElement) = (R[1]; CobordRingElem(R, R.n, R.R(n))) # fallback method
(R::CobordRing)() = R(0)
zero(R::CobordRing) = R(0)
one(R::CobordRing) = R(1)
(R::CobordRing)(x::CobordRingElem) = (@assert parent(x) == R; x)
# get the n-th generator, extending the ring if necessary
function Base.getindex(R::CobordRing, n::Int)
  if R.n < n
    while R.n < n
      R.n = max(2R.n, 1) # double the number of generators
    end
    R.R = graded_ring(R.base, ["[P$i]" for i in 1:R.n], collect(1:R.n))[1]
  end
  R(gens(R.R)[n])
end
Base.getindex(x::CobordRingElem, n::Int) = x.parent(x.f[n])

mul!(c::CobordRingElem, a::CobordRingElem, b::CobordRingElem) = (ab = a * b; c.n = ab.n; c.f = ab.f; c)
add!(c::CobordRingElem, a::CobordRingElem, b::CobordRingElem) = (ab = a + b; c.n = ab.n; c.f = ab.f; c)
addeq!(a::CobordRingElem, b::CobordRingElem) = (ab = a + b; a.n = ab.n; a.f = ab.f; a)

Base.exp(c::CobordRingElem) = parent(c)(exp(c.f))
Base.log(c::CobordRingElem) = parent(c)(log(c.f))

(R::CobordRing)(X::Variety) = R(chern_numbers(X))

@doc Markdown.doc"""
    cobordism_class(X::Variety)
Construct the cobordism class of a variety $X$."""
cobordism_class(X::Variety; base::Ring=QQ) = cobordism_ring(base=base)(X)
cobordism_class(X::AbsVariety; base::Ring=X.base) = (if base==Singular.QQ base=QQ end; cobordism_ring(base=base)(X))

function (R::CobordRing)(c::Dict{T, U}) where {T <: Partition, U <: RingElement}
  P = collect(keys(c))
  length(P) == 0 && return R(0)
  n = sum(P[1])
  @assert all(x -> sum(x) == n, P)
  R[n]
  p = gens(R.R)[1:n]
  v = _to_prod_Pn(n, c)
  R(sum(v[i] * prod(p[d] for d in λ) for (i, λ) in enumerate(partitions(n))))
end

function expressify(x::CobordRingElem; context = nothing)
  expressify(x.f, context = context)
end

for T in [:Int, :Rational, :fmpz, :fmpq, :n_Q, :n_transExt]
  @eval promote_rule(::Type{CobordRingElem}, ::Type{$T}) = CobordRingElem
end

-(x::CobordRingElem) = x.parent(-x.f)
+(x::CobordRingElem, y::CobordRingElem) = (@assert parent(x) == parent(y); R = parent(x); R(R(x.f).f + R(y.f).f))
-(x::CobordRingElem, y::CobordRingElem) = (@assert parent(x) == parent(y); R = parent(x); R(R(x.f).f - R(y.f).f))
*(x::CobordRingElem, y::CobordRingElem) = (@assert parent(x) == parent(y); R = parent(x); R(R(x.f).f * R(y.f).f))
==(x::CobordRingElem, y::CobordRingElem) = (@assert parent(x) == parent(y); R = parent(x); R(x.f).f == R(y.f).f)
^(x::CobordRingElem, n::Int) = x.parent(x.f^n)

function dim(x::CobordRingElem)
  n = total_degree(x.f)
  x == x[n] || error("the element is not equidimensional")
  n
end

@doc Markdown.doc"""
    chern_numbers(x::CobordRingElem)
    chern_numbers(x::CobordRingElem, P::Vector{<:Partition})
    chern_numbers(x::CobordRingElem; nonzero::Bool)
Compute all the Chern numbers of a cobordism class $[X]$ as a dictionary of
$\lambda\Rightarrow c_\lambda(X)$, or only those corresponding to partitions in
a given vector. Use the argument `nonzero=true` to display only the non-zero
ones.
"""
chern_numbers(x::CobordRingElem)
  
function chern_numbers(x::CobordRingElem, P::Vector{<:Partition}=partitions(dim(x)); nonzero::Bool=false)
  d = dim(x)
  d == -1 && error("the cobordism class is trivial")
  Pd = partitions(d)
  f = λ -> (ans = repeat([0], x.n); for i in λ if i <= x.n ans[i] += 1 end end; ans)
  c = Nemo.change_base_ring(parent(x).base, _chern_numbers_of_prod_Pn(d)[[i for (i, λ) in enumerate(Pd) if λ in P], :]) * Nemo.matrix(parent(x).base, [coeff(x.f, f(λ)) for λ in Pd][:, :])
  Dict([λ => c[i] for (i, λ) in enumerate(P) if !nonzero || c[i] != 0])
end

function euler(x::CobordRingElem)
  p = Partition([dim(x)])
  chern_numbers(x, [p])[p]
end

function _exp_to_partition(v::Vector)
  Partition(vcat([repeat([n], v[n]) for n in length(v):-1:1]...))
end

@doc Markdown.doc"""
    integral(x::CobordRingElem, t::ChRingElem)
Compute the integral of an expression in terms of the Chern classes over a
cobordism class.
"""
function integral(x::CobordRingElem, t::ChRingElem)
  t = t[dim(x)]
  c = chern_numbers(x)
  ans = sum(c[_exp_to_partition(v)] * (a isa n_Q ? QQ(a) : a) for (a, v) in zip(Nemo.coefficients(t.f), Nemo.exponent_vectors(t.f)))
end

# take advantage of the multiplicativity
function todd(x::CobordRingElem)
  f = x.f.f
  n = length(gens(parent(f)))
  Nemo.evaluate(f, repeat([1], n))
end
function signature(x::CobordRingElem)
  f = x.f.f
  n = length(gens(parent(f)))
  Nemo.evaluate(f, [iseven(k) ? 1 : 0 for k in 1:n])
end
function a_hat_genus(x::CobordRingElem)
  f = x.f.f
  n = length(gens(parent(f)))
  Nemo.evaluate(f, [iseven(k) ? ((-1)^(k÷2) * factorial(ZZ(k)) // (factorial(ZZ(k÷2))^2 * ZZ(2)^2k)) : QQ() for k in 1:n])
end
chi(p::Int, x::CobordRingElem) = integral(x, chi(p, variety(dim(x))))

function _generic_product_chern_numbers(m::Int, n::Int)
  cache = get_attribute(Omega, :generic_product_chern_numbers)
  if cache == nothing
    cache = Dict{Tuple{Int, Int}, Dict{Partition, Dict{Tuple{Partition, Partition}, fmpz}}}()
    set_attribute!(Omega, :generic_product_chern_numbers => cache)
  end
  mn = (m, n)
  if !(mn in keys(cache))
    R, cd = graded_ring(ZZ, vcat(["c$i" for i in 1:m], ["d$i" for i in 1:n]), vcat(collect(1:m), collect(1:n)))
    c, d = cd[1:m], cd[m+1:end]
    TX, TY = 1 + sum(c), 1 + sum(d)
    C = (TX * TY)[1:m+n]
    pp = partitions(m+n)
    ans = Dict{Partition, Dict{Tuple{Partition, Partition}, fmpz}}()
    for p in pp
      ans[p] = Dict{Tuple{Partition, Partition}, fmpz}()
      Cp = prod(C[i] for i in p).f
      for (c,e) in zip(Nemo.coefficients(Cp), Nemo.exponent_vectors(Cp))
        if transpose(collect(1:m)) * e[1:m] == m
          p1 = Partition([i for i in m:-1:1 for _ in 1:e[i]])
          p2 = Partition([i for i in n:-1:1 for _ in 1:e[m+i]])
          ans[p][(p1,p2)] = c
        end
      end
    end
    cache[mn] = ans
  end
  cache[mn]
end
# no testing if the inputs are valid, for internal use only
function _product_chern_numbers(x::Dict{<:Partition, U}, y::Dict{<:Partition, V}) where {U, V}
  m = sum(collect(keys(x))[1])
  n = sum(collect(keys(y))[1])
  c = _generic_product_chern_numbers(m, n)
  ans = Dict{Partition, Any}()
  for k in keys(c)
    ans[k] = sum(c[k][pq] * x[pq[1]] * y[pq[2]] for pq in keys(c[k]))
  end
  ans
end

function _product_chern_numbers(Xs::Vector)
  n = length(Xs)
  n == 1 && return Xs[1]
  _product_chern_numbers(_product_chern_numbers(Xs[1:n÷2]), _product_chern_numbers(Xs[n÷2+1:end]))
end

function _chern_numbers_of_prod_Pn(n::Int)
  B = get_attribute(Omega, :chern_numbers_of_prod_Pn)
  if B == nothing
    B = Dict{Int, Nemo.fmpq_mat}()
    set_attribute!(Omega, :chern_numbers_of_prod_Pn => B)
  end
  if !(n in keys(B))
    P = partitions(n)
    Pn = [chern_numbers(proj(k)) for k in 1:n]
    c = Dict([λ => reduce(_product_chern_numbers, [Pn[d] for d in λ]) for λ in P])
    B[n] = Nemo.matrix(QQ, [c[μ][λ] for λ in P, μ in P])
  end
  B[n]
end

function _inv_chern_numbers_of_prod_Pn(n::Int)
  B = get_attribute(Omega, :inv_chern_numbers_of_prod_Pn)
  if B == nothing
    B = Dict{Int, Nemo.fmpq_mat}()
    set_attribute!(Omega, :inv_chern_numbers_of_prod_Pn => B)
  end
  if !(n in keys(B))
    B[n] = inv(_chern_numbers_of_prod_Pn(n))
  end
  B[n]
end

function _to_prod_Pn(X::Variety)
  _to_prod_Pn(dim(X), chern_numbers(X))
end

function _to_prod_Pn(n::Int, c::Dict{T, U}) where {T <: Partition, U <: RingElement}
  F = parent(sum(values(c)))
  if F isa AbstractAlgebra.Integers F = QQ end
  c = [λ in keys(c) ? c[λ] : F(0) for λ in partitions(n), j in 1:1]
  _inv_chern_numbers_of_prod_Pn(n) * Nemo.matrix(F, c)
end

# Ellingsrud-Göttsche-Lehn
# On the cobordism class of the Hilbert scheme of a surface
@doc Markdown.doc"""
    hilb_surface(n::Int, c₁², c₂)
Construct the cobordism class of the Hilbert scheme of $n$ points on a surface
with given Chern numbers $c_1^2$ and $c_2$.
"""
function hilb_surface(n::Int, c₁²::T, c₂::U; weights=:int) where {T <: RingElement, U <: RingElement}
  HS = _H(n, c₁², c₂, weights=weights)
  coeff(HS, [n])
end
function hilb_surface(n::Int, S::Union{Variety, CobordRingElem}; weights=:int)
  @assert dim(S) == 2
  c = chern_numbers(S)
  hilb_surface(n, c[Partition([1,1])], c[Partition([2])], weights=weights)
end

@doc Markdown.doc"""
    hilb_K3(n::Int)
Construct the cobordism class of a hyperkähler variety of
$\mathrm{K3}^{[n]}$-type.
"""
function hilb_K3(n; weights=:int)
  hilb_surface(n, 0, 24, weights=weights)
end

# the generating series H(S) for a surface S
function _H(n::Int, c₁²::T, c₂::U; weights=:int) where {T <: RingElement, U <: RingElement}
  F = parent(c₁² + c₂)
  if F isa AbstractAlgebra.Integers F = QQ end
  a1, a2 = Nemo.solve_left(Nemo.matrix(F, [9 3; 8 4]), Nemo.matrix(F, [c₁² c₂]))
  O = cobordism_ring(base=F)
  R, (z,) = graded_ring(O, ["z"], :truncate => n)
  HP2    = a1 == 0 ? R(1) : 1 + sum(O(hilb_P2(k, weights=weights))*z^k for k in 1:n)
  HP1xP1 = a2 == 0 ? R(1) : 1 + sum(O(hilb_P1xP1(k, weights=weights))*z^k for k in 1:n)
  exp(a1*log(HP2) + a2*log(HP1xP1))
end

@doc Markdown.doc"""
    universal_genus(n::Int; twist::Int=0)
    universal_genus(n::Int, phi::Function; twist::Int=0)
Compute the total class of the universal genus up to degree $n$, possibly with
a twist $t$, in terms of the Chern classes.

One can also compute the total class of an arbitrary genus, by specifying the
images of the projective spaces using a function `phi`.
"""
function universal_genus(n::Int, phi::Function=k -> Omega[k]; twist::Int=0)
  n == 0 && return one(phi(1))
  taylor = _taylor(n, phi)
  R, c = graded_ring(parent(taylor[1]), _parse_symbol("c", 1:n), collect(1:n), :truncate=>n)
  _genus(_logg(sum(c)), taylor, twist = QQ(twist))
end

function _taylor(n::Int, phi::Function=k -> Omega[k])
  F = parent(phi(1))
  n == 0 && return [F(1)]
  ans = push!(_taylor(n-1, phi), F(0))
  R, (h,) = graded_ring(F, ["h"], :truncate=>n)
  chTP = sum(ZZ(n+1)//factorial(ZZ(i))*h^i for i in 1:n) # ad hoc ch(proj(n).T)
  ans[n+1] = 1//(n+1)*(phi(n) - coeff(_genus(chTP, ans), [n]))
  ans
end

@doc Markdown.doc"""
    generalized_kummer(n::Int)
Construct the cobordism class of a hyperkähler variety of
$\mathrm{Kum}_{n}$-type.
"""
function generalized_kummer(n::Int; weights=:int)
  hilb_S, c₁² = hilb_P2, 9 # one can also use (hilb_P1xP1, 8)
  Rz, (z,) = graded_ring(Omega, ["z"], :truncate=>n+1)
  H = [Omega(hilb_S(k, weights=weights)) for k in 1:n+1]
  g = universal_genus(2(n+1), twist=1)
  K = coeff(log(1+sum(integral(H[k], g) * z^k for k in 1:n+1)), [n+1])[2n]
  2(n+1)^2 // c₁² * K
end

@doc Markdown.doc"""
    milnor(X::Variety)
Compute the Milnor number $\int_X\mathrm{ch}_n(T_X)$ of a variety $X$.
"""
milnor(X::Variety) = integral(ch(X.T))
milnor(x::CobordRingElem) = integral(x, ch(variety(dim(x)).T))