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""" Klyachko Bundles and Sheaves.
Klyachko bundles are torus-equivariant bundles on toric varieties. That is, the action of the maximal torus on the toric variety lifts to an action on the bundle. There is an equivalence of categories between [Klyachko]_ bundles and multiple filtrations (one for each ray of the fan) of a vector space. The multi-filtrations are implemented in :mod:`sage.modules.multi_filtered_vector_space`.
EXAMPLES::
sage: X = toric_varieties.dP6xdP6() sage: TX = X.sheaves.tangent_bundle() sage: Alt2TX = TX.exterior_power(2); Alt2TX Rank 6 bundle on 4-d CPR-Fano toric variety covered by 36 affine patches.
sage: K = X.sheaves.line_bundle(X.K()) sage: antiK = X.sheaves.line_bundle(-X.K()) sage: (Alt2TX * K).cohomology(dim=True, weight=(0,0,0,0)) # long time (0, 0, 18, 0, 0)
sage: G_sum = TX + X.sheaves.trivial_bundle(2) sage: V_sum = G_sum.wedge(2) * K # long time sage: V_sum.cohomology(dim=True, weight=(0,0,0,0)) # long time (0, 0, 18, 16, 1) sage: Gtilde = G_sum.random_deformation() sage: V = Gtilde.wedge(2) * K # long time sage: V.cohomology(dim=True, weight=(0,0,0,0)) # long time (0, 0, 3, 0, 0)
REFERENCES:
.. [Klyachko] Klyachko, Aleksandr Anatolevich: Equivariant Bundles on Toral Varieties, Math USSR Izv. 35 (1990), 337-375. http://iopscience.iop.org/0025-5726/35/2/A04/pdf/0025-5726_35_2_A04.pdf
.. [BirknerIltenPetersen] Rene Birkner, Nathan Owen Ilten, and Lars Petersen: Computations with equivariant toric vector bundles, The Journal of Software for Algebra and Geometry: Macaulay2. http://msp.org/jsag/2010/2-1/p03.xhtml http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.8.2/share/doc/Macaulay2/ToricVectorBundles/html/ """
#***************************************************************************** # Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
""" Test whether ``X`` is a Klyachko bundle
INPUT:
- ``X`` -- anything.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.schemes.toric.sheaf.klyachko import is_KlyachkoBundle sage: is_KlyachkoBundle('test') False """
r""" Construct a Klyacho bundle
INPUT:
- ``toric_variety`` -- a toric variety. The base space of the bundle.
- ``multi_filtration`` -- a multi-filtered vectors space with multiple filtrations being indexed by the one-dimensional cones of the fan. Either an instance of :func:`~sage.modules.multi_filtered_vector_space.MultiFilteredVectorSpace` or something (like a dictionary of ordinary filtered vector spaces).
EXAMPLES::
sage: P1 = toric_varieties.P1() sage: v1, v2, v3 = [(1,0,0),(0,1,0),(0,0,1)] sage: F1 = FilteredVectorSpace({1:[v1, v2, v3], 3:[v1]}) sage: F2 = FilteredVectorSpace({0:[v1, v2, v3], 2:[v2, v3]}) sage: P1 = toric_varieties.P1() sage: r1, r2 = P1.fan().rays() sage: F = MultiFilteredVectorSpace({r1:F1, r2:F2}); F Filtrations N(-1): QQ^3 >= QQ^2 >= QQ^2 >= 0 >= 0 N(1): QQ^3 >= QQ^3 >= QQ^1 >= QQ^1 >= 0
You should use the :meth:`~sage.schemes.toric.sheaf.constructor.SheafLibrary.Klyachko` method to construct instances::
sage: P1.sheaves.Klyachko(F) Rank 3 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches.
sage: P1.sheaves.Klyachko({r1:F1, r2:F2}) # alternative Rank 3 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches.
The above is just a shorthand for::
sage: from sage.schemes.toric.sheaf.klyachko import Bundle sage: Bundle(P1, F) Rank 3 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches. """ # try to construct a MultiFilteredVectorSpace multi_filtration, base_ring=base_ring, check=check) multi_filtration = multi_filtration.change_ring(base_ring)
r""" A toric bundle using Klyachko's representation.
.. warning::
You should always use the :func:`Bundle` factory function to construct instances.
INPUT:
- ``toric_variety`` -- a toric variety. The base space of the bundle.
- ``multi_filtration`` -- a :func:`~sage.modules.multi_filtered_vector_space.MultiFilteredVectorSpace` with index set the rays of the fan.
- ``check`` -- boolean (default: ``True``). Whether to perform consistency checks.
EXAMPLES::
sage: P1 = toric_varieties.P1() sage: r1, r2 = P1.fan().rays() sage: F = MultiFilteredVectorSpace({ ....: r1:FilteredVectorSpace(3,1), ....: r2:FilteredVectorSpace(3,0)}); F Filtrations N(-1): QQ^3 >= 0 >= 0 N(1): QQ^3 >= QQ^3 >= 0 sage: from sage.schemes.toric.sheaf.klyachko import Bundle sage: Bundle(P1, F) Rank 3 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches. """ raise ValueError('the index set of the multi-filtration must be' ' all rays of the fan.') raise ValueError('multi-filtration must be exhaustive') raise ValueError('multi-filtration must be separating')
r""" Return the base toric variety.
OUTPUT:
A toric variety.
EXAMPLES::
sage: X = toric_varieties.P2() sage: V = X.sheaves.tangent_bundle(); V Rank 2 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches. sage: V.variety() is X True """
r""" Return the base field.
OUTPUT:
A field.
EXAMPLES::
sage: T_P2 = toric_varieties.P2().sheaves.tangent_bundle() sage: T_P2.base_ring() Rational Field """
r""" Return the generic fiber of the vector bundle.
OUTPUT:
A vector space over :meth:`base_ring`.
EXAMPLES::
sage: T_P2 = toric_varieties.P2().sheaves.tangent_bundle() sage: T_P2.fiber() Vector space of dimension 2 over Rational Field """
r""" Return the rank of the vector bundle.
OUTPUT:
Integer.
EXAMPLES::
sage: T_P2 = toric_varieties.P2().sheaves.tangent_bundle() sage: T_P2.rank() 2 """
r""" Return a string representation.
OUTPUT:
String.
EXAMPLES::
sage: toric_varieties.P2().sheaves.tangent_bundle() Rank 2 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches. """
r""" Return the filtration associated to the ``ray``.
INPUT:
- ``ray`` -- Integer, a `N`-lattice point, a one-dimensional cone, or ``None`` (default). Specifies a ray of the fan of the toric variety, either via its index or its generator.
OUTPUT:
The filtered vector space associated to the given ``ray``. If no ray is specified, all filtrations are returned.
EXAMPLES::
sage: TX = toric_varieties.dP6().sheaves.tangent_bundle() sage: TX.get_filtration(0) QQ^2 >= QQ^1 >= 0 sage: TX.get_filtration([-1, -1]) QQ^2 >= QQ^1 >= 0 sage: TX.get_filtration(TX.variety().fan(1)[0]) QQ^2 >= QQ^1 >= 0 sage: TX.get_filtration() Filtrations N(-1, -1): QQ^2 >= QQ^1 >= 0 N(-1, 0): QQ^2 >= QQ^1 >= 0 N(0, -1): QQ^2 >= QQ^1 >= 0 N(0, 1): QQ^2 >= QQ^1 >= 0 N(1, 0): QQ^2 >= QQ^1 >= 0 N(1, 1): QQ^2 >= QQ^1 >= 0 """ raise ValueError('not a one-dimensional cone') else:
r""" Return the vector subspace ``E^\alpha(i)``.
- ``ray`` -- Integer, a `N`-lattice point, a one-dimensional cone, or ``None`` (default). Specifies a ray of the fan of the toric variety, either via its index or its generator.
- ``i`` -- integer. The filtration degree.
OUTPUT:
A subspace of the :meth:`fiber` vector space. The defining data of a Klyachko bundle.
EXAMPLES::
sage: TX = toric_varieties.dP6().sheaves.tangent_bundle() sage: TX.get_degree(0, 1) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [0 1] """
r""" Return the intersection of the filtered subspaces.
INPUT:
- ``sigma`` -- a cone of the fan of the base toric variety.
- ``i`` -- integer. The filtration degree.
OUTPUT:
Let the cone be spanned by the rays `\sigma=\langle r_1,\dots, r_k\rangle`. This method returns the intersection
.. MATH::
\bigcap_{r\in \{r_1,\dots,r_k\}} E^{r}(i)
EXAMPLES::
sage: X = toric_varieties.P2() sage: fan = X.fan() sage: V = X.sheaves.tangent_bundle() sage: V.filtration_intersection(fan(1)[0], 1) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] sage: V.filtration_intersection(fan(2)[0], 1) Vector space of degree 2 and dimension 0 over Rational Field Basis matrix: [] """
r""" Return the vector subspace `E^\alpha(m)`.
INPUT:
- ``alpha`` -- a ray of the fan. Can be specified by its index (an integer), a one-dimensional cone, or a `N`-lattice point.
- ``m`` -- tuple of integers or `M`-lattice point. A point in the dual lattice of the fan.
OUTPUT:
The subspace $E^\alpha(\alpha m)$ of the filtration indexed by the ray $\alpha$ and at the filtration degree $\alpha * m$
EXAMPLES::
sage: X = toric_varieties.P2() sage: M = X.fan().dual_lattice() sage: V = X.sheaves.tangent_bundle() sage: V.E_degree(X.fan().ray(0), (1,0)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] sage: V.E_degree(X.fan(1)[0], (1,0)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] sage: V.E_degree(0, (1,0)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] """ else: raise ValueError('does not determine one-dimensional cone')
def E_intersection(self, sigma, m): r""" Return the vector subspace ``E^\sigma(m)``.
See [Klyachko]_, equation 4.1.
INPUT:
- ``sigma`` -- a cone of the fan of the base toric variety.
- ``m`` -- tuple of integers or `M`-lattice point. A point in the dual lattice of the fan. Must be immutable.
OUTPUT:
The subspace `E^\sigma(m)`
EXAMPLES::
sage: X = toric_varieties.P2() sage: fan = X.fan() sage: V = X.sheaves.tangent_bundle() sage: V.E_intersection(fan(1)[0], (1,0)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] sage: V.E_intersection(fan(2)[0], (-1,1)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [0 1]
For the empty cone, this is always the whole vector space::
sage: V.E_intersection(fan(0)[0], (1,0)) Vector space of dimension 2 over Rational Field """
def E_quotient(self, sigma, m): r""" Return the vector space quotient `E_\sigma(m)`.
See [Klyachko]_, equation 4.1.
INPUT:
- ``sigma`` -- a cone of the fan of the base toric variety.
- ``m`` -- tuple of integers or `M`-lattice point. A point in the dual lattice of the fan. Must be immutable.
OUTPUT:
The subspace `E_\sigma(m)`
EXAMPLES::
sage: X = toric_varieties.P2() sage: fan = X.fan() sage: M = fan.dual_lattice() sage: cone = fan(1)[0] sage: V = X.sheaves.tangent_bundle() sage: m = M(1, 0) sage: m.set_immutable() sage: V.E_quotient(cone, m) Vector space quotient V/W of dimension 1 over Rational Field where V: Vector space of dimension 2 over Rational Field W: Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] sage: V.E_quotient(fan(2)[0], (-1,1)) Vector space quotient V/W of dimension 0 over Rational Field where V: Vector space of dimension 2 over Rational Field W: Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] """
def E_quotient_projection(self, sigma, tau, m): r""" Return the projection map `E_\sigma(m) \to E_\tau(m)` where `\sigma` is a face of `\tau`.
INPUT:
- ``sigma`` -- a cone of the fan of the base toric variety.
- ``tau`` -- a cone of the fan containing ``sigma``.
- ``m`` -- tuple of integers or `M`-lattice point. A point in the dual lattice of the fan. Must be immutable.
OUTPUT:
The restriction map
.. MATH::
E_\sigma(m) \to E_\tau(m)
EXAMPLES::
sage: P3 = toric_varieties.P(3) sage: rays = [(1,0,0), (0,1,0), (0,0,1)] sage: F1 = FilteredVectorSpace(rays, {0:[0], 1:[2], 2:[1]}) sage: F2 = FilteredVectorSpace(3, 0) sage: r = P3.fan().rays() sage: V = P3.sheaves.Klyachko({r[0]:F1, r[1]:F2, r[2]:F2, r[3]:F2}) sage: tau = Cone([(1,0,0), (0,1,0)]) sage: sigma = Cone([(1,0,0)]) sage: M = P3.fan().dual_lattice() sage: m = M(2,1,0) sage: m.set_immutable() sage: V.E_quotient(sigma, m) Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] sage: V.E_quotient(tau, m) Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] sage: V.E_quotient_projection(sigma, tau, m) Vector space morphism represented by the matrix: [1 0] [0 1] Domain: Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] Codomain: Vector space quotient V/W of dimension 2 over Rational Field where V: Vector space of dimension 3 over Rational Field W: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [0 1 0] """ raise ValueError('the cone sigma is not a face of the cone tau')
r""" Return the "cohomology complex" `C^*(m)`
See [Klyachko]_, equation 4.2.
INPUT:
- ``m`` -- tuple of integers or `M`-lattice point. A point in the dual lattice of the fan. Must be immutable.
OUTPUT:
The "cohomology complex" as a chain complex over the :meth:`base_ring`.
EXAMPLES::
sage: P3 = toric_varieties.P(3) sage: rays = [(1,0,0), (0,1,0), (0,0,1)] sage: F1 = FilteredVectorSpace(rays, {0:[0], 1:[2], 2:[1]}) sage: F2 = FilteredVectorSpace(rays, {0:[1,2], 1:[0]}) sage: r = P3.fan().rays() sage: V = P3.sheaves.Klyachko({r[0]:F1, r[1]:F2, r[2]:F2, r[3]:F2}) sage: tau = Cone([(1,0,0), (0,1,0)]) sage: sigma = Cone([(1, 0, 0)]) sage: M = P3.fan().dual_lattice() sage: m = M(1, 1, 0); m.set_immutable() sage: V.cohomology_complex(m) Chain complex with at most 2 nonzero terms over Rational Field
sage: F = CyclotomicField(3) sage: P3 = toric_varieties.P(3).change_ring(F) sage: V = P3.sheaves.Klyachko({r[0]:F1, r[1]:F2, r[2]:F2, r[3]:F2}) sage: V.cohomology_complex(m) Chain complex with at most 2 nonzero terms over Cyclotomic Field of order 3 and degree 2 """ else:
r""" Return the bundle cohomology groups.
INPUT:
- ``degree`` -- ``None`` (default) or an integer. The degree of the cohomology group.
- ``weight`` -- ``None`` (default) or a tuple of integers or a `M`-lattice point. A point in the dual lattice of the fan defining a torus character. The weight of the cohomology group.
- ``dim`` -- Boolean (default: ``False``). Whether to return vector spaces or only their dimension.
OUTPUT:
The cohomology group of given cohomological ``degree`` and torus ``weight``.
* If no ``weight`` is specified, the unweighted group (sum over all weights) is returned.
* If no ``degree`` is specified, a dictionary whose keys are integers and whose values are the cohomology groups is returned. If, in addition, ``dim=True``, then an integral vector of the dimensions is returned.
EXAMPLES::
sage: V = toric_varieties.P2().sheaves.tangent_bundle() sage: V.cohomology(degree=0, weight=(0,0)) Vector space of dimension 2 over Rational Field sage: V.cohomology(weight=(0,0), dim=True) (2, 0, 0) sage: for i,j in cartesian_product((list(range(-2,3)), list(range(-2,3)))): ....: HH = V.cohomology(weight=(i,j), dim=True) ....: if HH.is_zero(): continue ....: print('H^*i(P^2, TP^2)_M({}, {}) = {}'.format(i,j,HH)) H^*i(P^2, TP^2)_M(-1, 0) = (1, 0, 0) H^*i(P^2, TP^2)_M(-1, 1) = (1, 0, 0) H^*i(P^2, TP^2)_M(0, -1) = (1, 0, 0) H^*i(P^2, TP^2)_M(0, 0) = (2, 0, 0) H^*i(P^2, TP^2)_M(0, 1) = (1, 0, 0) H^*i(P^2, TP^2)_M(1, -1) = (1, 0, 0) H^*i(P^2, TP^2)_M(1, 0) = (1, 0, 0) """ raise NotImplementedError('sum over weights is not implemented') else:
""" Compare ``self`` and ``other``.
.. warning::
This method tests whether the underlying representation is the same. Use :meth:`is_isomorphic` to test for mathematical equivalence.
INPUT:
- ``other`` -- anything.
OUTPUT:
Boolean.
EXAMPLES::
sage: X = toric_varieties.P2() sage: V1 = X.sheaves.trivial_bundle(1) sage: V2 = X.sheaves.trivial_bundle(2) sage: V2 == V1 False sage: V2 == V1+V1 True
sage: T_X = X.sheaves.tangent_bundle() sage: O_X = X.sheaves.trivial_bundle(1) sage: T_X + O_X == O_X + T_X False """ return NotImplemented
return richcmp_not_equal(lr, rx, op)
""" Test whether two bundles are isomorphic.
INPUT:
- ``other`` -- anything.
OUTPUT:
Boolean.
EXAMPLES::
sage: X = toric_varieties.P2() sage: T_X = X.sheaves.tangent_bundle() sage: O_X = X.sheaves.trivial_bundle(1) sage: T_X + O_X == O_X + T_X False sage: (T_X + O_X).is_isomorphic(O_X + T_X) Traceback (most recent call last): ... NotImplementedError """
""" Return the sum of two vector bundles.
INPUT:
- ``other`` -- a Klyachko bundle over the same base.
OUTPUT:
The direct sum as a new Klyachko bundle.
EXAMPLES::
sage: X = toric_varieties.P2() sage: V1 = X.sheaves.trivial_bundle(1) sage: V2 = X.sheaves.trivial_bundle(2) sage: V2.direct_sum(V1) Rank 3 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches.
sage: V1 = X.sheaves.trivial_bundle(1) sage: V2 = X.sheaves.trivial_bundle(2) sage: V2 == V1 + V1 True """ raise ValueError('the bundles must be over the same base toric variety')
""" Return the sum of two vector bundles.
INPUT:
- ``other`` -- a Klyachko bundle over the same base.
OUTPUT:
The tensor product as a new Klyachko bundle.
EXAMPLES::
sage: X = toric_varieties.P2() sage: OX = X.sheaves.trivial_bundle(1) sage: X.sheaves.tangent_bundle().tensor_product(OX) Rank 2 bundle on 2-d CPR-Fano toric variety covered by 3 affine patches. sage: OX == OX * OX True """ raise ValueError('the bundles must be over the same base toric variety')
""" Return the `n`-th exterior power.
INPUT:
- ``n`` -- integer.
OUTPUT:
The `n`-th exterior power `\wedge_{i=1}^n V` of the bundle `V` as a new Klyachko bundle.
EXAMPLES::
sage: X = toric_varieties.P2_123() sage: TX = X.sheaves.tangent_bundle() sage: antiK = X.sheaves.line_bundle(-X.K()) sage: TX.exterior_power(2) == antiK True sage: TX.wedge(2) == antiK # alias True """
""" Return the `n`-th symmetric power.
INPUT:
- ``n`` -- integer.
OUTPUT:
The `n`-th symmetric power as a new Klyachko bundle.
EXAMPLES::
sage: P1 = toric_varieties.P1() sage: H = P1.divisor(0) sage: L = P1.sheaves.line_bundle(H) sage: (L+L).symmetric_power(2) Rank 3 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches. sage: (L+L).symmetric_power(2) == L*L+L*L+L*L True """
""" Return the dual bundle.
OUTPUT:
The dual bundle as a new Klyachko bundle.
EXAMPLES::
sage: P1 = toric_varieties.P1() sage: H = P1.divisor(0) sage: L = P1.sheaves.line_bundle(H) sage: L.dual() Rank 1 bundle on 1-d CPR-Fano toric variety covered by 2 affine patches. sage: L.dual() == P1.sheaves.line_bundle(-H) True """
""" Return a generic torus-equivariant deformation of the bundle.
INPUT:
- ``epsilon`` -- an element of the base ring. Scales the random deformation.
OUTPUT:
A new Klyachko bundle with randomly perturbed moduli. In particular, the same Chern classes.
EXAMPLES::
sage: P1 = toric_varieties.P1() sage: H = P1.divisor(0) sage: V = P1.sheaves.line_bundle(H) + P1.sheaves.line_bundle(-H) sage: V.cohomology(dim=True, weight=(0,)) (1, 0) sage: Vtilde = V.random_deformation() sage: Vtilde.cohomology(dim=True, weight=(0,)) (1, 0) """ |