Equivariant Vector Bundles Research Articles

Equivariant orientation of vector bundles over disconnected base spaces

AbstractIn this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group , a precise condition under which an ‐orientation of a ‐equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous” ‐equivariant bundle with respect to an ‐ring spectrum . As an application, we show that the 2‐fold direct sum of any homogeneous bundle is ‐orientable, where is the Burnside Mackey functor. We notice that ‐orientability is equivalent to ‐orientability when the order of is odd. When the order of is even, we show that a ‐equivariant analog of the tautological line bundle over is ‐orientable but not ‐orientable.

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Dual exceptional collections on Lagrangian Grassmannians

We construct graded left dual exceptional collections to the exceptional collections generating the Kuznetsov-Polishchuk blocks on Lagrangian Grassmannians. As an application, we find explicit resolutions for some natural irreducible equivariant vector bundles. Bibliography: 13 titles.

Visible actions of compact Lie groups on complex spherical varieties

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.

Seshadri constants of equivariant vector bundles on toric varieties

We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.

Equivariant fundamental classes in RO(C2)–graded cohomology with ℤ∕2–coefficients

Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this paper, we develop a theory of fundamental classes for equivariant submanifolds in $RO(C_2)$-graded cohomology in constant $\mathbb{Z}/2$ coefficients. We show the cohomology of any $C_2$-surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.

\'Etale triviality of finite equivariant vector bundles

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite \'etale covering of X is trivial as an H-equivariant bundle.

The pro-étale cohomology of Drinfeld's upper half space

We determine the geometric pro-étale cohomology of Drinfeld's upper half space over a p -adic field K . The strategy is different from the one given by [ P. Colmez et al., Invent. Math. 219, No. 3, 873–985 (2020; Zbl 1475.11110)] and is based on the approach developed in [ S. Orlik , Invent. Math. 172, No. 3, 585–656 (2008; Zbl 1136.22009)] describing global sections of equivariant vector bundles.

Erratum to: "Stability of equivariant vector bundles over toric varieties"

We correct the proof of [the authors, ibid. 25, 1787–1833 (2020; Zbl 1445.14071), Proposition 3.1.1].

ON THE BOUNDED DERIVED CATEGORY OF IGr(3, 7)

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\mathsf{IGr}(3,7)$. Moreover, we show that $\mathsf{IGr}(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.

Locally analytic representations in the étale coverings of the Lubin-Tate moduli space

The Lubin-Tate moduli space X0rig is a p-adic analytic open unit polydisc which parametrizes deformations of a formal group H0 of finite height defined over an algebraically closed field of characteristic p. It is known that the natural action of the automorphism group Aut(H0) on X0rig gives rise to locally analytic representations on the topological duals of the spaces H0(X0rig , (ℳ0 )rig) of global sections of certain equivariant vector bundles (ℳ0 )rig over X0rig . In this article, we show that this result holds in greater generality. On the one hand, we work in the setting of deformations of formal modules over the valuation ring of a finite extension of ℚp. On the other hand, we also treat the case of representations arising from the vector bundles (ℳ )rig over the deformation spaces Xrig with Drinfeld level-m-structures. Finally, we determine the space of locally finite vectors in H0(Xrig , (ℳ )rig). Essentially, all locally finite vectors arise from the global sections of invertible sheaves over the projective space via pullback along the Gross-Hopkins period map.

Arithmetically rigid schemes via deformation theory of equivariant vector bundles

We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $$p \ne 0$$ . This gives the same conclusion for Totaro’s examples of Fano varieties violating Kodaira vanishing. We also provide an alternative purely geometric proof of non-liftability mod $$p^2$$ and to characteristic zero of the Frobenius homomorphism of a reductive group of non-exceptional type. In the appendix, written jointly with Piotr Achinger, we provide examples of non-liftable Calabi–Yau varieties in every characteristic $$p \geqslant 5$$ .

The symmetric tensor product on the Drinfeld centre of a symmetric fusion category

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.

Descent of vector bundles under wildly ramified extensions

Given an irreducible normal Noetherian scheme and a finite Galois extension of the field of rational functions, we discuss the comparison of the categories of vector bundles on the scheme and equivariant vector bundles on the integral closure in the extension. This is well understood in the tame case (geometric stabilizer groups of order invertible in the local rings), so we focus on the wild (non-tame) case, which may be reduced to the case of cyclic extensions of prime order. In this case, under an additional flatness hypothesis, we give a characterization of the equivariant vector bundles that arise by base change from vector bundles on the scheme.

Spinor modules for Hamiltonian loop group spaces

Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.

Stability of Equivariant Vector Bundles over Toric Varieties

We give a complete answer to the question of (semi)stability of tangent bundles on any nonsingular complex projective toric variety with Picard number 2 by using combinatorial criterion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundles on all toric Fano 4-folds with Picard number \leq 3 which are classified by Batyrev [1]. We construct a collection of equivariant indecomposable rank 2 vector bundles on Bott towers and pseudo-symmetric toric Fano varieties. Furthermore, in case of Bott towers, we show the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.

On Gromov–Witten Theory of Projective Bundles

Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.

On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let A be a -algebra, M a left A-module, H a Hopf -algebra, an algebra coaction, and let denote with the right A-module structure induced by δ. The usual definitions of equivariant vector bundle naturally lead, in the context of -algebras, to an -module homomorphismthat fulfills some appropriate conditions. On the other hand, sometimes an (A, H)-Hopf module is considered instead, for the same purpose. When Θ is invertible, as is always the case when H is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra H for which there exists such a Θ that is not invertible and a left-right (A, H)-Hopf module whose corresponding homomorphism is not an isomorphism.

Grothendieck Ring of Varieties with Actions of Finite Groups

AbstractWe define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

Equivariant Vector Bundles Over Quantum Projective Spaces

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.