check_chern_numbers.py

#!/usr/bin/env python3

def binomial(n, k):
    """
    A fast way to calculate binomial coefficients by Andrew Dalke.
    See http://stackoverflow.com/questions/3025162/statistics-combinations-in-python
    """
    if 0 <= k <= n:
        ntok = 1
        ktok = 1
        for t in range(1, min(k, n - k) + 1):
            ntok *= n
            ktok *= t
            n -= 1
        return ntok // ktok
    else:
        return 0

def hypersurface(n, d):
    '''Calculate the number 

        p(n,d) := 12 \int_X c_4(End T_X) \cup h^{n-4} / \int_X h^n,

    where X is a hypersurface of degree d in n+1 dimensional projective space,
    and h is the hyperplane class restricted to X.'''

    # Chern classes of a hypersurface of degree d in P^{n+1} (modulo multiples of h)
    c0 = n
    c1 = n+1-d
    c2 = binomial(n+1,2) - d*c1
    c3 = binomial(n+1,3) - d*c2
    c4 = binomial(n+1,4) - d*c3

    return (n-1) * c1**4 \
            - (n-1) * c1**2 * c2 \
            + (n+1) * c2**2 \
            + (n-1) * c1 * c3 \
            - n * c4

def check_hypersurfaces(limit):
    positive = []
    negative = []
    for n in range(4, limit):
        # Only check for Kahler--Einstein hypersurfaces, i.e., ones of degree >= n+1
        for d in range(n+1, n+4+limit):
            chern = hypersurface(n, d)
            if chern < 0:
                negative.append({'dimension': n, 'degree': d, 'value': chern})
            else:
                positive.append({'dimension': n, 'degree': d, 'value': chern})
    return (positive, negative)

(positive, negative) = check_hypersurfaces(1000)

if negative:
    print("Negative intersections:")
    for neg in negative:
        print(neg)
else:
    print("No negative intersections.")

if positive:
    print("Positive intersections:")
    for pos in positive:
        print(pos)
else:
    print("No positive intersections.")

print("")
print("First negative intersection: ", negative[0])
print("First positive intersection: ", positive[0])