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Sage code to compute the integral of the square root of the Todd class on a hyperkähler variety
from sage.schemes.chow.all import *
# dimension 6
X = ChowScheme(6, ["c2", "c4", "c6"], [2, 4, 6])
X.chowring().inject_variables()
td = Sheaf(X, 6, 1 + sum(X.gens())).todd_class()
integrand = (td._logg()/2)._expp().by_degrees()[6]
# O'Grady 6
values = {c2^3: 30720, c2*c4: 7680, c6: 1920}
print(integrand.reduce([m - values[m] for m in values]))
# Kum^3
values = {c2^3: 30208, c2*c4: 6784, c6: 448}
print(integrand.reduce([m - values[m] for m in values]))
# dimension 10
X = ChowScheme(10, ["c2", "c4", "c6", "c8", "c10"], [2, 4, 6, 8, 10])
X.chowring().inject_variables()
td = Sheaf(X, 10, 1 + sum(X.gens())).todd_class()
integrand = (td._logg()/2)._expp().by_degrees()[10]
# O'Grady 10
values = {c2^5: 127370880, c2^3*c4: 53071200, c2^2*c6: 12383280, c2*c8: 1791720, \
c2*c4^2: 22113000, c4*c6: 5159700, c10: 176904}
print(integrand.reduce([m - values[m] for m in values]))
# K3^[5]
values = {c2^5: 126867456, c4*c2^3: 52697088, c4^2*c2: 21921408, c6*c2^2: 12168576, \
c6*c4: 5075424, c8*c2: 1774080, c10: 176256}
print(integrand.reduce([m - values[m] for m in values]))
# expressions taken from Propositions 19 and 21 of Sawon's PhD thesis
def K3n(n): return (n+3)^n / (4^n * factorial(n))
def Kum_n(n): return (n+1)^(n+1) / (4^n * factorial(n))
def todd_root(n):
"""Determine the square root of the Todd class of a hyperkaehler
Returns the square root of the Todd class expressed in terms of Chern numbers
together with the Chern numbers as monomials in a polynomial ring.
"""
def modified_bernoulli(n):
if n == 0: return ln(2) / 2
if n % 2 == 1: return 0
return bernoulli(n) / (2 * n**2 * gamma(n))
def exp(t, precision=n):
return sum([t**i / factorial(i) for i in range(precision + 1)])
R = PolynomialRing(QQ, ["ch" + str(2*i) for i in range(1, n+1)])
# (1) truncated power series expansion of square root of Todd class
result = prod([exp(-modified_bernoulli(2*k) * factorial(2*k) \
* R.gen(k-1)) for k in range(1, n+1)])
# (2) now collect degree 2*n part
# not really Chern numbers but didn't have better name
chern_numbers = WeightedIntegerVectors(n, range(1, n+1))
chern_numbers = [prod([R.gen(i)**c for (i, c) in enumerate(C)]) for C in chern_numbers]
result = sum([result.monomial_coefficient(ch) * ch for ch in chern_numbers])
# (3) from ch_i to c_i
S = PolynomialRing(QQ, ["c" + str(2*i) for i in range(1, n+1)])
morphism = []
for k in range(2, 2*n+1, 2):
# go from the Chern character to Chern classes using symmetric functions
Sym = SymmetricFunctions(QQ)
terms = Sym.elementary()(Sym.powersum()[k]).monomial_coefficients().items()
# select only the even Chern classes on a hyperkaehler
terms = [(E, c) for (E, c) in terms if all([i % 2 == 0 for i in E])]
expression = 1 / factorial(k) \
* sum([c * prod([S.gen(i // 2 - 1) for i in E]) for (E, c) in terms])
morphism.append(expression)
result = R.hom(morphism, S)(result)
# (4) list the Chern numbers that can appear
chern_numbers = WeightedIntegerVectors(n, range(1, n+1))
chern_numbers = [prod([S.gen(i)**c for (i, c) in enumerate(C)]) for C in chern_numbers]
return (result, chern_numbers)
print("O'Grady 6-fold")
(square_root, chern_numbers) = todd_root(3)
square_root.parent().inject_variables()
values = {c2^3: 30720, c2*c4: 7680, c6: 1920}
value = sum([square_root.monomial_coefficient(c) * values[c] for c in values])
assert value == Kum_n(3)
print(value, "\n")
print("O'Grady 10-fold")
(square_root, chern_numbers) = todd_root(5)
square_root.parent().inject_variables()
values = {c2^5: 127370880, c2^3*c4: 53071200, c2^2*c6: 12383280, c2*c8: 1791720, \
c2*c4^2: 22113000, c4*c6: 5159700, c10: 176904}
value = sum([square_root.monomial_coefficient(c) * values[c] for c in values])
assert value == K3n(5)
print(value, "\n")
print("Generalised Kummer variety of dimension 6-fold: computation by hand")
(square_root, chern_numbers) = todd_root(3)
square_root.parent().inject_variables()
values = {c2^3: 30208, c2*c4: 6784, c6: 448}
value = sum([square_root.monomial_coefficient(c) * values[c] for c in values])
assert value == Kum_n(3)
print(value, "\n")
@8d1h

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@8d1h 8d1h commented Jul 7, 2021

Have you ever tried the package chow? This can be done as simple as

sage: from sage.schemes.chow.all import ChowScheme, Sheaf
sage: X = ChowScheme(6, ["c2", "c4", "c6"], [2,4,6])
sage: td = Sheaf(X, 6, 1+sum(X.gens())).todd_class()
sage: (td._logg()/2)._expp().by_degrees()[6]
31/967680*c2^3 - 11/241920*c2*c4 + 1/60480*c6

(and of course there are similar functionalities in Macaulay2 and Julia)

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@pbelmans pbelmans commented Jul 7, 2021

Dang, I expected that some functionality must've existed somewhere but not that it would've been so easy! I indeed briefly thought of using Macaulay2, but just wrote it from scratch in Sage as I find that more pleasant usually. But chow really is the best of both worlds, I had seen it before but forgot about it as I hadn't used it yet (which will change now for sure).

Thanks!

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@8d1h 8d1h commented Jul 7, 2021

Glad to help! (long time admirer of your awesome blogs)

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