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nLab > Latest Changes: stable vector bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2012

    started stable vector bundle, but still vague

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 7th 2012
    • (edited Nov 7th 2012)

    We already have the entry stable bundle, which I created in Spring 2010. It seems it is essentially about the same notion.

    Note also the entry stability where one of the points is

    in geometry, in the construction of moduli spaces there is a notion of stable and semistable objects (e.g. Mumford stability in geometric invariant theory); they can be taken with respect to some stability data; more recently those are important in physical applications and abstract versions appeared in the context of triangulated categories, most notably Bridgeland stability conditions.

    I had an ambitious plan to create and integrate a circle of entries on that kind of stability (e.g. some pointers like Kontsevich’s stable curves and the related moduli space, used e.g. in CFT, Mumford stability, Bridgeland stability, stable bundle etc.). I hope you are in phase with my thoughts here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 7th 2012

    I hope you are in phase with my thoughts here.

    I am currently focusing on aspects of the geometric quantization of CS theory and the appearance of stable vector bundles there.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeNov 7th 2012

    this is provided by the Narasimhan–Seshadri theorem which establishes that the moduli space of flat connections on a Riemann surface is naturally a complex manifolds.

    My understanding is that this is an application of the variant of the general picture of stable objecs from Mumford’s geometric invariant theory, which he designed precisely for moduli spaces). So yes, it fits with the general picture I wanted to (eventually become able to) spell out.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2018
    • (edited Sep 29th 2018)

    Supposedly there is a generalization to stable equivariant vector bundles? If so, then in the special case over the point there should be an answer to: When is a G-representation stable, in this sense?

    Is there any literature on this?

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorraghu
    • CommentTimeSep 30th 2018
    • (edited Sep 30th 2018)

    Supposedly there is a generalization to stable equivariant vector bundles? … Is there any literature on this?

    Here are a couple of references:

    • C. S. Seshadri, Moduli of π-vector bundles over an algebraic curve. 1970. Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) pp. 139–260. Edizioni Cremonese, Rome. LaTeX version. Constructs the moduli space of (semi)stable π\pi-vector bundles over a compact Riemann surface XX, where π\pi is a finite group acting on XX.

    • O. García-Prada, Invariant connections and vortices. Comm. Math. Phys. 156 (1993), no. 3, 527–546. Considers stability of GG-vector bundles on a compact K"ahler manifold XX, where GG is a compact Lie group acting on XX.

    If so, then in the special case over the point there should be an answer to: When is a G-representation stable, in this sense?

    I guess that, according to any of the usual definitions of GG-stability, every GG-vector bundle on a point ** is semistable. It is stable if and only if it is irreducible as a representation of GG. For instance, in the notation of [Huybrechts and Lehn, Definition 1.2.3], p(E,m)=1p(E,m)=1 for every non-zero sheaf EE on **.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2018

    Thanks a million! I’ll have a look.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2018
    • (edited Sep 30th 2018)

    Allow me to ask regarding your pointer to Huybrechts-Lehn 96:

    Their Def. 1.2.3 seems to to be stated a in the non-equivariant context, no? They seem to introduce group actions not before section 4.2.

    Are you saying that (semi-)stability of equivariant coherent sheaves should be just that of their underlying plain coherent sheaves? Why would that be so?

    (I am not doubting it, just trying to understand.)

    • CommentRowNumber9.
    • CommentAuthorraghu
    • CommentTimeSep 30th 2018
    • (edited Sep 30th 2018)

    regarding your pointer to Huybrechts-Lehn 96: Their Def. 1.2.3 seems to to be stated a in the non-equivariant context, no?

    The GG-semistability of a GG-sheaf is usually defined to be just the semistability of the underlying sheaf. Thus, a GG-sheaf EE is defined to be GG-semistable if F⪯EF \preceq E (in the chosen preorder) for every non-zero proper subsheaf FF of EE. See:

    • Seshadri, page 32: “We say that a π-vector bundle V on X is π-semi-stable (or semi-stable π-bundle) if the underlying vector bundle of V is semi- stable.”

    On the other hand, the GG-stability of a GG-sheaf EE is defined by requiring that F≺EF \prec E (in the chosen preorder) for every non-zero proper GG-subsheaf FF of EE. See:

    • Seshadri, page 32: “A π-vector bundle V on X is said to be π-stable if V is π-semi-stable and for every proper π-sub-bundle W of V , we have µ(W ) < µ(V ).” (The condition of π\pi-semistability, that is semistability of the underlying sheaf, need not be assumed, because it follows from the other condition, see Garcia-Prada, Proposition 7.)

    • Garcia-Prada, page 533: “Definition: The bundle EE is GG-invarίantly stable with respect to ω\omega if for every GG-ίnvarίant coherent subsheaf FF with 0<rk(E)<rk(F)0 &lt; \rk(E) &lt; \rk(F), we have μ(F)<μ(E)\mu(F) &lt; \mu(E).”

    Are you saying that (semi-)stability of equivariant coherent sheaves should be just that of their underlying plain coherent sheaves?

    As per the above definitions, the action of GG does not affect the definition of the GG-semistability of a GG-sheaf. It does enter the definition of GG-stability of a GG-sheaf through the condition that the strict preorder relation holds only for GG-subsheaves and not necssarily for all subsheaves. Also, the action of GG does not affect the definition of the preorder in the situations that I’ve seen; e.g., in both the above references the slope μ\mu is the slope of the underlying sheaf, and does not take into account the action of GG.

    Why would that be so?

    I don’t know. However, it works, where I can specify the meaning of “works” to some extent.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2018

    Okay, thanks.

    Could it be that there are, in addition, other possible choices of stability conditions on GG-representations, possibly with other stable objects?

    I am thinking of the following, but please set me straight if this seems wrong:

    For G⊂SU(2)G \subset SU(2) a finite subgroup of SU(2)SU(2), the K-theoretic McKay correspondence (GSV83) gives an isomorphism

    R ℂ(G)≃K G(*)→≃K(X˜) R_{\mathbb{C}}(G) \simeq K_G(\ast) \overset{\simeq}{\to} K(\tilde X)

    from the representation ring to the KK-theory over the resolution X˜\tilde X of the corresponding du Val singularity.

    Now there is an interesting space of stability conditions over X˜\tilde X, due to Bridgeland 05, hence of stability functions

    K(X˜)⟶Zℂ K(\tilde X) \overset{Z}{\longrightarrow} \mathbb{C}

    But these pull back along the above isomorphism to stability functions

    R ℂ(G)⟶Zℂ R_{\mathbb{C}}(G) \overset{Z}{\longrightarrow} \mathbb{C}

    on the representation ring, hence on the K-theory of equivariant vector bundles over the point.

    Maybe the class of stable GG-representations can change as we move around in that space of stability conditions?

    Possibly the “standard” stability condition that you point out, with the irreps being precisely the stable objects, is “just” one point in this space of stability conditions?

    I suppose page 2 in Bridgeland 05 is sort of about this. But I am hoping that after pullback to R ℂ(G)R_{\mathbb{C}}(G) the stability functions ZZ can be given a simple explicit description in terms of characters.

    • CommentRowNumber11.
    • CommentAuthorraghu
    • CommentTimeSep 30th 2018
    • (edited Sep 30th 2018)

    Could it be that there are, in addition, other possible choices of stability conditions on GG-representations, possibly with other stable objects?

    I regret that I have not read about Bridgeland stability conditions or about the McKay correspondence, so I cannot say anything about most of what you have written. However, even from my limited perspective, I think that the above should be true. For example, if GG is finite, A=C[G]A = \mathbf{C}[G], and 𝒜=A-mod\mathcal{A} = A\text{-}\mathbf{mod}, then every homomorphism θ:K 0(𝒜)→Z\theta: K_0(\mathcal{A}) \to \mathbf{Z} defines a stability structure on the abelian category 𝒜\mathcal{A}, as in Definition 1.1 of

    • A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515–530.

    The moduli space M(θ)M(\theta) of θ\theta-semistable objects of 𝒜\mathcal{A} may vary as θ\theta varies. In the cases that I’ve seen, where instead of C[G]\mathbf{C}[G], I take AA to be the path algebra of a finite quiver, and consider θ\theta real-valued instead of integer-valued, then the space of all θ\theta is a finite-dimensional real affine space, with a distinguished hyperplane arrangement ℋ\mathcal{H}. As θ\theta varies within a facet of ℋ\mathcal{H}, the moduli spaces M(θ)M(\theta) lie in the same isomorphism class. When θ\theta crosses a wall of ℋ\mathcal{H}, the isomorphsim class of M(θ)M(\theta) may change. (This kind of phenomenon is described even as early as in Mehta and Seshadri’s work on parabolic vector bundles over a curve. The wall-crossing phenomenon is explicitly described in some situations by Thaddeus and others in terms of flips.)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2018

    Excellent, that’s just what I was looking for. I’ll have a look at that article by King, thanks again.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2018
    • (edited Sep 30th 2018)

    Okay, if the simple statement of Def. 1.1 with Thm. 4.1 in King 94 gives stability conditions on GRep ℂG Rep_{\mathbb{C}} compatible with the “modern” meaning as used by Bridgeland etc., then that’s exactly the kind of simplification I was hoping for, in the case of a hands-on abelian category like GRepG Rep.

    So, as you said above, a stability condition on GRep ℂG Rep_{\mathbb{C}} should just be a linear function

    R ℂ(G)=K(GRep ℂ)⟶AAθAAℤ R_{\mathbb{C}}(G) = K(G Rep_{\mathbb{C}}) \overset{ \phantom{AA} \theta \phantom{AA} }{\longrightarrow} \mathbb{Z}

    from the represention ring to the integers,

    and an element V∈GRep ℂV \in G Rep_{\mathbb{C}} is to be called stable with respect to θ\theta if

    1. θ(V)=0\theta(V) = 0

    2. W⊂VW \subset V with W≠VW \neq V implies θ(W)<θ(V)\theta(W) \lt \theta(V).

    (I am just repeating this for the record, and because I am taking the liberty of switching from King’s “>\gt” to Bridgeland’s “<\lt”.)

    So let me explore that a bit.

    In these terms, what is that “standard” stability condition which we talked about, the one coming from slope-stability of equivariant vector bundles, when restricting to equivariant vector bundles over the point. I suppose that standard stability condition should in fact be the one corresponding to θ\theta vanishing identically

    θ=0, \theta = 0 \,,

    because that indeed makes the irreps and only the irreps be stable.

    Does that sound right?

    (Of course this is elementary, but I need to warm up to this stability business here.)

    Okay, so what else can we get? In general, we can identify

    θ=(θ i)∈ℤ Irreps(G) \theta = (\theta^i) \in \mathbb{Z}^{ Irreps(G) }

    given by sending a rep

    V=⨁i∈Irreps(G)⟨V,V i⟩⏟∈ℤV i V \;=\; \underset{i \in Irreps(G)}{\bigoplus} \underset{ \in \mathbb{Z} }{\underbrace{\langle V,V_i\rangle}} \, V_i

    to

    θ(V)≔∑iθ i⟨V,V i⟩. \theta(V) \;\coloneqq\; \underset{i}{\sum} \theta^i \langle V,V_i\rangle \,.

    Hm, so which other choice of θ\theta would be interesting?

    Hm. Is there any non-vanishing choice of θ\theta that would have a non-trivial space of stable objects here?

    Looks like as soon as θ i\theta^i is non-vanishing, then V iV_i may not appear in a direct summand of a stable VV. Because if θ i\theta^i is positive, then V iV_i may not be a subobject of a stable object, while if it is negative, then it can only appear as a summand together with another object that contributes positively, but then that object may not appear as a subobject.

    So then the only thing that can happen is that the stable objects are a subset of all irreps. That would be a bit sad.

    Hm. I am missing something here.(?)

    [edit: I think the issue is that King’s θ\theta plays the role of the slope, but the slope itself is not meant to be additive, while King asks θ\theta to be additive. That seems to be an overly strong condition.]

    • CommentRowNumber14.
    • CommentAuthorraghu
    • CommentTimeOct 2nd 2018

    Your analysis is correct, and I was hasty when I said in #11 that I expect that varying the homomorphism θ\theta can produce a different set of θ\theta-stable representations.

    When GG is finite, the ring A=C[G]A = \mathbf{C}[G] is semisimple, and this implies that for any homomorphism θ:K 0(A-mod)→Z\theta: K_0(A\text{-}\mathbf{mod}) \to \Z, every AA-module EE is θ\theta-polystable (direct sum of θ\theta-stables); it is θ\theta-stable if and only if it is simple, that is, irreducible as a representation of GG. So the stability structures of King don’t give anything different for AA from I’d mentioned in #6.

    My apologies in case my comment in #11 led you astray.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018

    Thanks, your comment were very helpful. It was good for me to look into some basics here.

    Luckily, today I realized that for the application that I have in mind I may not need stability conditions after all.

    • CommentRowNumber16.
    • CommentAuthorraghu
    • CommentTimeOct 3rd 2018
    • (edited Oct 3rd 2018)

    A correction to my attempted correction in #14: I’d written

    When GG is finite, the ring A=C[G]A = \mathbf{C}[G] is semisimple, and this implies that for any homomorphism θ:K 0(A-mod)→Z\theta: K_0(A\text{-}\mathbf{mod}) \to \Z, every AA-module EE is θ\theta-polystable (direct sum of θ\theta-stables); it is θ\theta-stable if and only if it is simple, that is, irreducible as a representation of GG.

    What I meant was that an AA-module EE is θ\theta-semistable if and only if it is θ\theta-polystable, that is, isomorphic to a direct sum of θ\theta-stable AA-modules; it is θ\theta-stable if and only if θ(E)=0\theta(E) = 0 and EE is simple, that is, irreducible as a representation of GG.

    Incidentally, the following fundamental work about stability structures on abelian categories may be of interest here:

    • A. Rudakov, Stability for an abelian category, J. Algebra 197 (1997), 231–245.

    Rudakov wrote some sequels to this article, but I haven’t read them.

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