Line Bundle Research Articles

The RO(ΠB)-graded C2-equivariant ordinary cohomology of [formula omitted

We calculate the ordinary C2-cohomology, with Burnside ring coefficients, of CPC2∞=BC2U(1), the complex projective space, a model for the classifying space for C2-equivariant complex line bundles. The RO(C2)-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the RO(C2)-graded theory.

Half-integrality of line bundles on partial flag schemes of classical Lie groups

In this paper, we develop a theory of Galois descent for equivariant line bundles on partial flag schemes. In particular, we study computational aspects of the classification of descent data of equivariant line bundles attached to characters of parabolic subgroups. As an application, we classify equivariant line bundles on partial flag schemes of the standard Z[1/2]-forms of classical Lie groups.

Lower bounds for Seshadri constants via successive minima of line bundles

Given a nef and big line bundle L L on a projective variety X X of dimension d ≥ 2 d \geq 2 , we prove that the Seshadri constant of L L at a very general point is larger than ( d + 1 ) 1 d − 1 (d+1)^{\frac {1}{d}-1} . This slightly improves the lower bound 1 / d 1/d established by Ein, Küchle and Lazarsfeld [J. Differential Geom. 42 (1995), pp. 193–219]. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito [Adv. Math. 365 (2020), 38 pp.].

Intermedialities as Sociopolitical Assemblages in Contemporary Art

This article is an introductory essay to the Special Issue “A Comparative Study of Media in Contemporary Visual Art”. It starts with a short overview of the terminological discussion about intermediality as a concept and its relationship with medialities with other prefixes—such as mixed, intra-, multi-, and transmedialities. So far, intermediality has been discussed less by art historians than by literary scholars. This introductory essay argues that critical analysis of intermediality in contemporary artworks may offer additional insights for investigation of the issues addressed in these artworks. The case studies in this Special Issue underscore this view. As a kind of kick-off, the second part of this essay includes a short case study that focuses on two artworks by the Lebanese artist Rabih Mroué in order to provide insight into how intermedial relations can act as metaphors for the sociopolitical relations addressed in his artworks. Applying philosopher Manuel DeLanda’s “assemblage theory”, philosopher Edward S. Casey’s concept of “absorptive mapping”, and anthropologist Tim Ingold’s view of living beings as consisting of a bundle of lines facilitates the highlighting of the sociopolitical aspects of intermediality in Mroué’s artworks.

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential −log⁡u such that u is C∞-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M,g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.

Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued (p, q)-forms on Kähler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, ℙ3 and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications.

The modular class of a singular foliation

The modular class of a regular foliation is a cohomological obstruction to the existence of a volume form transverse to the leaves which is invariant under the flow of the vector fields of the foliation. It has been used in dynamical systems and operator theory, from which the name modular comes from. By drawing on the relationship between Lie algebroids and regular foliations, this paper extends the notion of modular class to the realm of singular foliations. The singularities are dealt with by replacing the singular foliations by any of their universal Lie ∞-algebroids, and by picking up the modular class of the latter. The geometric meaning of the modular class of a singular foliation F is not as transparent as for regular foliations: it is an obstruction to the existence of a universal Lie ∞-algebroid of F whose Berezinian line bundle is a trivial F-module. This paper illustrates the relevance of using universal Lie ∞-algebroids to extend mathematical notions from regular foliations to singular ones, thus paving the road to defining other characteristic classes in this way.

Ein–Lazarsfeld–Mustopa conjecture for the blow-up of a projective space

We solve the Ein–Lazarsfeld–Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let X be the blow up of {mathbb {P}}^n at a linear subspace and let L be any ample line bundle on X. We show that the syzygy bundle M_{L} defined as the kernel of the evalution map H^{0}(X,L)otimes {mathcal {O}}_{X}rightarrow L is L-stable. In the last part of this note we focus on the rigidness of M_{L} to study the local shape of the moduli space around the point [M_{L}].

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.

Integral Points on Varieties With Infinite Étale Fundamental Group

Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

A compactification of the moduli space of multiple-spin curves

We construct a smooth Deligne–Mumford compactification for the moduli space of curves with an m-tuple of spin structures using line bundles on quasi-stable curves as limiting objects, as opposed to line bundles on stacky curves. For all m, we give a combinatorial description of the local structure of the corresponding coarse moduli spaces. We also classify all irreducible and connected components of the resulting moduli spaces of multiple-spin curves.

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are$\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

Stratified Hilbert Modules on Bounded Symmetric Domains

We analyze the “eigenbundle” (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank r, giving rise to complex-analytic fibre spaces which are stratified of length r+1. The fibres are described in terms of Kähler geometry as line bundle sections over flag manifolds, and the metric embedding is determined by taking derivatives of reproducing kernel functions. Important examples are the determinantal ideals defined by vanishing conditions along the various strata of the stratification.

Ample line bundles and generation time

AbstractWe prove that ifXis a regular quasi-projective variety of dimensiond, the set of line bundles{𝒪X⁢(n)}n∈ℤ{\{\mathcal{O}_{X}(n)\}_{n\in{\mathbb{Z}}}}generates the bounded derived category ofXindsteps. This proves new cases of a conjecture of Orlov as well as a conjecture of Elagin and Lunts.

Stability of line bundle mean curvature flow

Let ( X , ω ) (X,\omega ) be a compact Kähler manifold of complex dimension n n and ( L , h ) (L,h) be a holomorphic line bundle over X X . The line bundle mean curvature flow was introduced by Jacob-Yau in order to find deformed Hermitian-Yang-Mills metrics on L L . In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric h ^ \hat h on L L . We prove that the line bundle mean curvature flow converges to h ^ \hat h exponentially in C ∞ C^\infty sense as long as the initial metric is close to h ^ \hat h in C 2 C^2 -norm.

Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus–Toma manifolds

Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus–Toma manifolds

Ginzburg-Landau equations on non-compact Riemann surfaces

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

The Demailly system for a direct sum of ample line bundles on Riemann surfaces

The Demailly system for a direct sum of ample line bundles on Riemann surfaces

Moduli spaces of vector bundles on a curve and opers

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ be a holomorphic line bundle on X with $$\xi ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix a theta characteristic $${\mathbb {L}}$$ on X. Let $${\mathcal M}_X(r,\xi )$$ be the moduli space of stable vector bundles E on X of rank r such that $$\bigwedge ^r E\,=\, \xi $$ and $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ . Consider the quotient of $${\mathcal M}_X(r,\xi )$$ by the involution given by $$E\, \longmapsto \, E^*$$ . We construct an algebraic morphism from this quotient to the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers on X. Since $$\dim {\mathcal M}_X(r,\xi )$$ coincides with the dimension of the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers, it is natural to ask about the injectivity and surjectivity of this map.

The Verlinde traces for SU_{X}(2,xi ) and blow-ups

Given a compact Riemann surface X of genus at least 2 with automorphism group G, we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of G-linearized line bundles defined on certain blow-ups of projective spaces along the curve X. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular, we can compute the traces of automorphisms of X on the Verlinde spaces corresponding to the moduli space SU_{X}(2,xi ) when xi is a line bundle G-linearized of suitable degree.