91 lines
3.6 KiB
Python
91 lines
3.6 KiB
Python
class as_cech:
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def __init__(self, C, omega, f):
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self.curve = C
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n = C.height
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F = C.base_ring
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variable_names = 'x, y'
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for i in range(n):
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variable_names += ', z' + str(i)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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self.omega0 = omega
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self.f = f
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self.omega8 = self.omega0 - self.f.diffn()
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if self.omega0.form not in Rxyz or self.omega8.valuation() < 0:
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raise ValueError('cech cocycle not regular')
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def __repr__(self):
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return "( " + str(self.omega0)+", " + str(self.f) + " )"
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def __add__(self, other):
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C = self.curve
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omega = self.omega0
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f = self.f
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omega1 = other.omega0
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f1 = other.f
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return as_cech(C, omega + omega1, f+f1)
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def __sub__(self, other):
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C = self.curve
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omega = self.omega0
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f = self.f
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omega1 = other.omega0
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f1 = other.f
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return as_cech(C, omega - omega1, f - f1)
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def __rmul__(self, constant):
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C = self.curve
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omega = self.omega0
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f = self.f
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return as_cech(C, constant*omega, constant*f)
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def coordinates(self, threshold=10, basis = 0):
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'''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
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AS = self.curve
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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if basis == 0:
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basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
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holo_diffs = basis[0]
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coh_basis = basis[1]
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dR = basis[2]
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F = AS.base_ring
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f_products = []
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for f in coh_basis:
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f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
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print(f_products)
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product_of_fct_and_omegas = []
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fct = self.f
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product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) for omega in holo_diffs]
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V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
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coh_coordinates = V.coordinates(product_of_fct_and_omegas) #coeficients of self in the basis elts coming from cohomology of OX
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for i in range(AS.genus()):
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self -= coh_coordinates[i]*dR[i+AS.genus()]
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#We remove now from f the summands which are obviously regular at infty
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print(self, [])
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f_num = numerator(self.f.function)
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f_den = denominator(self.f.function)
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v_f_den = as_function(AS, f_den).valuation()
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for a in f_num.monomials():
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if as_function(AS, a).valuation() >= v_f_den:
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self.f.function -= f_num.monomial_coefficient(a)*a/f_den
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f_num = numerator(self.f.function)
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f_den = denominator(self.f.function)
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quo, rem = f_num.quo_rem(f_den)
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if as_function(AS, rem/f_den).valuation() >= 0:
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self.f = as_function(AS, quo)
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hol_form = self.omega0 - self.f.diffn() #now this should be a holomorphic form
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hol_form = as_form(AS, as_reduction(AS, hol_form.form))
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print('hol_form', hol_form)
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return hol_form.coordinates() + coh_coordinates
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print(self, [omega.serre_duality_pairing(self.f) for omega in holo_diffs])
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raise ValueError('I arrived at a form (omega, 0), in which omega is not regular on U0. I hoped this wouldn t happen.')
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def group_action(self, g):
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AS = self.curve
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omega = self.omega0
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f = self.f
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return as_cech(self.curve, omega.group_action(g), f.group_action(g)) |