def TrivialBundle(X, rank=1):
r"""
Return the trivial bundle of rank ``r``.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``rank`` -- the rank of the bundle.
OUTPUT:
The trivial bundle as a Klyachko bundle.
EXAMPLES::
sage: P2 = toric_varieties.P2()
sage: from sage.schemes.toric.sheaf.constructor import TrivialBundle
sage: I3 = TrivialBundle(P2, 3); I3
Rank 3 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches.
sage: I3.cohomology(weight=(0,0), dim=True)
(3, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict(
[ray, FilteredVectorSpace(rank, 0, base_ring=base_ring)]
for ray in X.fan().rays())
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def TangentBundle(X):
r"""
Construct the tangent bundle of a toric variety.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
OUTPUT:
The tangent bundle as a Klyachko bundle.
EXAMPLES::
sage: dP7 = toric_varieties.dP7()
sage: from sage.schemes.toric.sheaf.constructor import TangentBundle
sage: TangentBundle(dP7)
Rank 2 bundle on 2-d CPR-Fano toric variety covered by 5 affine patches.
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
base_ring = X.base_ring()
fan = X.fan()
filtrations = dict()
from sage.modules.filtered_vector_space import FilteredVectorSpace
for i, ray in enumerate(fan.rays()):
F = FilteredVectorSpace(fan.rays(), {0: range(fan.nrays()), 1: [i]})
filtrations[ray] = F
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def LineBundle(X, D):
"""
Construct the rank-1 bundle `O(D)`.
INPUT:
- ``X`` -- a toric variety. The base space of the bundle.
- ``D`` -- a toric divisor.
OUTPUT:
The line bundle `O(D)` as a Klyachko bundle of rank 1.
EXAMPLES::
sage: X = toric_varieties.dP8()
sage: D = X.divisor(0)
sage: from sage.schemes.toric.sheaf.constructor import LineBundle
sage: O_D = LineBundle(X, D)
sage: O_D.cohomology(dim=True, weight=(0,0))
(1, 0, 0)
"""
if not is_ToricVariety(X):
raise ValueError('not a toric variety')
from sage.modules.free_module import VectorSpace
base_ring = X.base_ring()
filtrations = dict([
X.fan().ray(i),
FilteredVectorSpace(1, D.function_value(i), base_ring=base_ring)
] for i in range(X.fan().nrays()))
import klyachko
return klyachko.Bundle(X, filtrations, check=True)
def _repr_(self):
r"""
Return as string representation of ``self``.
OUTPUT:
A string.
EXAMPLES::
sage: rays = [(1,0), (1,1), (1,2), (-1,-1)]
sage: F1 = FilteredVectorSpace(rays, {0:[1, 2], 2:[3]})
sage: F2 = FilteredVectorSpace(rays, {0:[1, 2], oo:[3]})
sage: F3 = FilteredVectorSpace(rays, {oo:[3]})
sage: F4 = FilteredVectorSpace(rays, {0:[3]})
sage: MultiFilteredVectorSpace({'a':F1, 'b':F2, 'c': F3, 'd': F4})
Filtrations
a: QQ^2 >= QQ^1 >= QQ^1 >= 0
b: QQ^2 >= QQ^1 >= QQ^1 >= QQ^1
c: QQ^1 >= QQ^1 >= QQ^1 >= QQ^1
d: QQ^1 >= 0 >= 0 >= 0
sage: MultiFilteredVectorSpace(123, base_ring=RR)
Unfiltered RR^123
"""
if not self._filt:
F = FilteredVectorSpace(self.dimension(),
base_ring=self.base_ring())
return 'Unfiltered ' + repr(F)
rows = []
support = self.support()
min_deg, max_deg = self.min_degree(), self.max_degree()
for key in sorted(self.index_set()):
F = self.get_filtration(key)
r = [str(key)] + F._repr_degrees(min_deg, max_deg - 1)
rows.append(r)
from sage.misc.table import table
t = table(rows)
w = t._widths()
lines = ['Filtrations']
for r in rows:
s = ' '
s += r[0].rjust(w[0]) + ': '
s += ' >= '.join(r[i].center(w[i]) for i in range(1, len(w)))
lines.append(s)
return '\n'.join(lines)
def _repr_(self):
r"""
Return as string representation of ``self``.
OUTPUT:
A string.
EXAMPLES::
sage: rays = [(1,0), (1,1), (1,2), (-1,-1)]
sage: F1 = FilteredVectorSpace(rays, {0:[1, 2], 2:[3]})
sage: F2 = FilteredVectorSpace(rays, {0:[1, 2], oo:[3]})
sage: F3 = FilteredVectorSpace(rays, {oo:[3]})
sage: F4 = FilteredVectorSpace(rays, {0:[3]})
sage: MultiFilteredVectorSpace({'a':F1, 'b':F2, 'c': F3, 'd': F4})
Filtrations
a: QQ^2 >= QQ^1 >= QQ^1 >= 0
b: QQ^2 >= QQ^1 >= QQ^1 >= QQ^1
c: QQ^1 >= QQ^1 >= QQ^1 >= QQ^1
d: QQ^1 >= 0 >= 0 >= 0
sage: MultiFilteredVectorSpace(123, base_ring=RR)
Unfiltered RR^123
"""
if not self._filt:
F = FilteredVectorSpace(self.dimension(),
base_ring=self.base_ring())
return 'Unfiltered ' + repr(F)
rows = []
support = self.support()
min_deg, max_deg = self.min_degree(), self.max_degree()
for key in sorted(self.index_set()):
F = self.get_filtration(key)
r = [str(key)] + F._repr_degrees(min_deg, max_deg-1)
rows.append(r)
from sage.misc.table import table
t = table(rows)
w = t._widths()
lines = ['Filtrations']
for r in rows:
s = ' '
s += r[0].rjust(w[0]) + ': '
s += ' >= '.join(r[i].center(w[i]) for i in range(1, len(w)))
lines.append(s)
return '\n'.join(lines)