Abstract bundles

An abstract bundle on an abstract variety $X$ is determined by its rank and its Chern character (or, equivalently, its total Chern class). Abstract bundles support the standard operations on vector bundles in algebraic geometry: direct sum, tensor product, duals, determinant bundles, exterior and symmetric powers, as well as pullback and pushforward along abstract variety maps. They also carry the usual characteristic classes: Chern classes, Segre classes, Todd class, and Pontryagin classes.

The arithmetic operations +, -, * on abstract bundles correspond to direct sum, formal difference, and tensor product, respectively. In particular, multiplying a bundle by an integer n gives the direct sum of n copies.

Grothendieck ring and virtual bundles

Abstract bundles live in the Grothendieck ring $\mathrm{K}^0(X)$ of vector bundles on $X$. In this ring, every element can be written as a formal difference $[E] - [F]$ of genuine bundles. The Chern character $\mathrm{ch}\colon\mathrm{K}^0(X) \to \mathrm{N}^*(X)_{\mathbb Q}$ is a ring homomorphism that identifies the Grothendieck ring (after tensoring with $\mathbb Q$) with the Chow ring.

In OSCAR, an AbstractBundle is stored as a pair (rank, Chern character), so virtual bundles with zero (and negative) rank are fully supported:

julia> P2 = abstract_projective_space(2);

julia> T = tangent_bundle(P2);

julia> F = T - 2*OO(P2); # a virtual bundle of rank 0

julia> rank(F)
0

julia> chern_character(F)
3//2*h^2 + 3*h

Segre classes

The total Segre class $s(E) = c(E)^{-1}$ is the formal inverse of the total Chern class and arises naturally as the fundamental class of a projective bundle. For a bundle $E$ of rank $r$ the individual Segre classes satisfy $s_k(E) = (-1)^k c_k(E^\vee)$ when $k \le r$, but the relation $s(E)\cdot c(E) = 1$ also determines the higher Segre classes uniquely.

julia> P3 = abstract_projective_space(3);

julia> T = tangent_bundle(P3);

julia> total_segre_class(T)
-20*h^3 + 10*h^2 - 4*h + 1

julia> total_segre_class(T) * total_chern_class(T) # must be 1
1

Types

The OSCAR type for abstract vector bundles is AbstractBundle.

Constructors

abstract_bundle — Method

abstract_bundle(X::AbstractVariety, ch::Union{MPolyDecRingElem, MPolyQuoRingElem})
abstract_bundle(X::AbstractVariety, r::RingElement, c::Union{MPolyDecRingElem, MPolyQuoRingElem})

Return an abstract vector bundle on X by specifying its Chern character ch. Equivalently, specify its rank r and total Chern class c.

Examples

We show two ways of constructing the Horrocks-Mumford bundle F [HM73]. First, we create F as the cohomology bundle of its Beilinson monad (see Equation (2.1) in [HM73]):

\[0 \rightarrow \mathcal O_{\mathbb P^4}^5(2)\rightarrow \bigwedge^2\mathrm{T}^*_{\mathbb P^4}(5) \rightarrow \mathcal O_{\mathbb P^4}^5(3)\rightarrow 0.\]

Then, we show the constructor above at work.

julia> P4 = abstract_projective_space(4)
AbstractVariety of dim 4

julia> A = 5*OO(P4, 2)
AbstractBundle of rank 5 on AbstractVariety of dim 4

julia> B = 2*exterior_power(cotangent_bundle(P4), 2)*OO(P4, 5)
AbstractBundle of rank 12 on AbstractVariety of dim 4

julia> C = 5*OO(P4, 3)
AbstractBundle of rank 5 on AbstractVariety of dim 4

julia> F = B - A - C
AbstractBundle of rank 2 on AbstractVariety of dim 4

julia> total_chern_class(F)
10*h^2 + 5*h + 1

julia> h = gens(P4)[1]
h

julia> F == abstract_bundle(P4, 2, 10*h^2 + 5*h + 1)
true

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.