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# pbelmans/integral-todd-root-hyperkahler-using-chow.sage

Last active December 20, 2021 02:46
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Sage code to compute the integral of the square root of the Todd class on a hyperkähler variety
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 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]))
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 # 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")

### 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!

### 8d1h commented Jul 7, 2021

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

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