Instantly share code, notes, and snippets.

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

Last active December 20, 2021 02:46
Star You must be signed in to star a gist
Sage code to compute the integral of the square root of the Todd class on a hyperkähler variety
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 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() # 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() # 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^ 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]))
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 # 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 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()
31/967680*c2^3 - 11/241920*c2*c4 + 1/60480*c6```

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

### 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!