Description
Given a toric vector bundle, i.e. an equivariant vector bundle on a smooth complete toric variety, this package can check the positivity of this bundle.
PositivityToricBundles can check whether a toric vector bundle is
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nef, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is nef;
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(very) ample, i.e. whether the line bundle $\mathcal{O}(1)$ on $\mathbb{P}(\mathcal E)$ is (very) ample;
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globally generated.
Additionally,
PositivityToricBundles can compute the toric Chern character of a toric vector bundle as introduced by Sam Payne.
For the computational purposes,
PositivityToricBundles uses the description of a toric vector bundles by filtrations developed by Alexander Klyachko, and relies on its implementation via the
ToricVectorBundles package by René Birkner, Nathan Ilten and Lars Petersen.
To check nefness and ampleness,
PositivityToricBundles uses a result of Milena Hering, Mircea Mustaţă and Sam Payne, namely, that it is sufficient to check this for the restriction of the bundle to the torus invariant curves. The central method for this is
restrictToInvCurves; the methods
isNef and
isAmple are based on it.
For global generation and very ampleness,
PositivityToricBundles uses results of Sandra Di Rocco, Kelly Jabbusch and Gregory Smith, who describe these properties in terms of the so-called parliament of polytopes of a toric vector bundle. From the parliament of polytopes one can extract the information up to which order jets are separated by the vector bundle. Globally generated or very ample toric vector bundles are those that separate 0-jets or 1-jets, respectively. Here, the central method is
separatesJets; built on it are
isGloballyGenerated and
isVeryAmple.
For the mathematical background see
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[K] Alexander Klyachko, Equivariant bundles over toral varieties, Izv. Akad. Nauk SSSR Ser. Mat., 53, 1989.
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[P] Sam Payne, Moduli of toric vector bundles, Compos. Math, 144, 2008.
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[HMP] Milena Hering, Mircea Mustaţă, Sam Payne, Positivity properties of toric vector bundles, Ann. Inst. Fourier (Grenoble), 60, 2010
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[RJS] Sandra Di Rocco, Kelly Jabbusch, Gregory Smith, Toric vector bundles and parliaments of polytopes, Trans. AMS, 370, 2018.
The following example computes the positivity for the tangent sheaf of \mathbb P^2:
i1 : E = tangentBundle projectiveSpaceFan 2
o1 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
|
i2 : isNef E
o2 = true
|
i3 : isAmple E
o3 = true
|
i4 : isVeryAmple E
o4 = true
|
i5 : isGloballyGenerated E
o5 = true
|
i6 : separatesJets E
o6 = 1
|
The toric Chern character can be computed:
i7 : toricChernCharacter E
o7 = HashTable{| -1 0 | => {| 1 |, | 1 |}}
| -1 1 | | 0 | | -1 |
| 1 -1 | => {| -1 |, | 0 |}
| 0 -1 | | 1 | | 1 |
| 1 0 | => {| -1 |, | 0 |}
| 0 1 | | 0 | | -1 |
o7 : HashTable
|
which associates to each maximal cone (its rays put into matrices) the corresponding components.
The restrictions of the bundle to the torus invariant curves can be computed:
i8 : restrictToInvCurves E
o8 = HashTable{| -1 | => {2, 1}}
| -1 |
| 0 | => {2, 1}
| 1 |
| 1 | => {1, 2}
| 0 |
o8 : HashTable
|
Here, in all three cases, the restriction splits into $\mathcal{O}_{\mathbb P^1}(2) \oplus \mathcal{O}_{\mathbb P^1}(1)$.
Most methods of
PositivityToricBundles support the option
Verbosity. So by adding
Verbosity => n with n a positive integer to the arguments of a method, hopefully useful insight about the course of the calculation is provided.
Caveat
The description of a toric variety and a toric vector bundle by filtrations involves the choice of signs.
PositivityToricBundles follows the same choice of signs as
ToricVectorBundles, which are
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the fan associated to a polytope will be generated by inner normals,
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the filtrations for describing a toric vector bundle are increasing.
Unfortunately, the above cited articles use decreasing filtrations and, moreover, [HMP], [P] and [RJS] use outer normals.
Another warning concerns the toric variety: the methods of
PositivityToricBundles implicitly assume that the variety is complete (to apply the results of [HMP] and [P]) and in addition smooth (for [RJS]). For non-complete or singular toric varieties, methods might break or results might become meaningless.