Vector Bundles Research Articles

A quick journey on stable sheaves and vector bundles

Abstract In this article, we collect main results obtained from our collaboration with Edoardo Ballico. The topics cover various aspects on the theory of vector bundles and stable sheaves related to Castelnuovo–Mumford regularity, cohomological splitting criteria, globally generated vector bundles, arithmetically Cohen–Macaulay sheaves, logarithmic vector bundles, stable sheaves of rank zero and co-Higgs sheaves.

Galilei covariance of the theory of Thouless pumps

Abstract The Thouless theory of quantum pumps establishes the conditions for quantized particle transport per cycle, and determines its value. When describing the pump from a moving reference frame, transported and existing charges transform, though not independently. This transformation is inherent to Galilean space and time, but it is underpinned by a transformation of vector bundles. Different formalisms can be used to describe this transformation, including one based on Bloch theory. Depending on the chosen formalism, the two types of charges will be realized as indices of either the same or different kinds. Finally, we apply the bulk-edge correspondence principle, so as to implement the transformation law within Büttiker’s scattering theory of quantum pumps.

On the existence of Parseval frames for vector bundles

Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and important, since they provide a simple reconstruction scheme and are maximally robust against certain types of noise. In this paper we describe a theory of frames on arbitrary vector bundles—this is the natural setting for signals which are realized as parameterized families of vectors rather than as single vectors—and discuss the existence of Parseval frames in this setting. Our approach is phrased in the language of G G -bundles, which allows us to use many tools from classical algebraic topology. In particular, we show that orientable vector bundles always admit Parseval frames of sufficiently large size and provide an upper bound on the necessary size. We also give sufficient conditions for the existence of Parseval frames of smaller size for tangent bundles of several families of manifolds, and provide some numerical evidence that Parseval frames on vector bundles share the desirable reconstruction properties of classical Parseval frames.

ON THE DEGREE OF IRRATIONALITY OF LOW GENUS $K3$ SURFACES

AbstractGiven a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$ , we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$ , and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.

MUKAI’S PROGRAM FOR NONPRIMITIVE CURVES ON K3 SURFACES

AbstractMukai’s program in [16] seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier–Mukai partner to a Brill–Noether locus of vector bundles on the curve. In the case X has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh in [11, 13] for $g\geq 11$ and $g\neq 12$ . More recently, Feyzbakhsh has shown in [12] that certain moduli spaces of stable bundles on X are isomorphic to the Brill–Noether locus of curves in $|H|$ if g is sufficiently large. In this paper, we work with irreducible curves in a nonprimitive ample linear system $|mH|$ and prove that Mukai’s program is valid for any irreducible curve when $g\neq 2$ , $mg\geq 11$ and $mg\neq 12$ . Furthermore, we introduce the destabilising regions to improve Feyzbakhsh’s analysis in [12]. We show that there are hyper-Kähler varieties as Brill–Noether loci of curves in every dimension.

Nef or pseudoeffective cycles on projective bundles

Let E be a vector bundle of rank r over a smooth projective variety X, and let π:Y=P(E)→X be the associated projective bundle of one-dimensional quotients. We show that if the relative anticanonical divisor -KY/X is nef (or equivalently E is universally semistable in the sense of Fulger and Langer (J Algebra 609:657–687, 2022), then the rth self intersection (KY/X)r is numerically trivial. Using this, we show that if -KY/X is nef, then nef (resp. pseudoeffective) cycles of codimension c on Y are precisely those of the form ∑i=0r-1(-KY/X)i·π∗αiwhere αi are nef (resp. pseudoeffective) cycles of codimension c-i on X. We also describe the nef/pseudoeffective cycles on Y when E is certain extension of a universally semistable bundle. Our results generalize the key propositions of Fulger (Math Z 269:449–459, 2011) about pseudoeffective cycles on projective bundles over curves. Even when specialized to nef or pseudoeffective divisors, our results are cleaner and more general than some recent computation of Misra et al. (Osaka J Math 59:639–651, 2022; On pseudoeffective cones of projective bundles and volume function, preprint https://arxiv.org/abs/2203.07007).

Genus-zero permutation-twisted conformal blocks for tensor product vertex operator algebras: The tensor-factorizable case

Let [Formula: see text] be a vertex operator algebra (VOA), let [Formula: see text] be a finite set, and let [Formula: see text] be a subgroup of the permutation group [Formula: see text] which acts on [Formula: see text] in a natural way. For each [Formula: see text], the [Formula: see text]-twisted [Formula: see text]-modules were first constructed and characterized in [K. Barron, C. Dong and G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227(2) (2002) 349–384] when [Formula: see text] has only one orbit, i.e. [Formula: see text] for some [Formula: see text]. In general, if [Formula: see text] is a disjoint union of several [Formula: see text]-orbits [Formula: see text], and if for each orbit [Formula: see text] one chooses a [Formula: see text]-twisted [Formula: see text]-module [Formula: see text], then [Formula: see text] is a [Formula: see text]-twisted [Formula: see text]-module. A direct sum of such [Formula: see text] is called a ⊗-factorizable [Formula: see text]-twisted [Formula: see text]-module. It is known that all [Formula: see text]-twisted modules are ⊗-factorizable if [Formula: see text] is rational [K. Barron, C. Dong and G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227(2) (2002) 349–384]. In this paper, we use the main result of [B. Gui, Sewing and propagation of conformal blocks, New York J. Math. 30 (2024) 187–230] to construct an explicit isomorphism from the space of genus-[Formula: see text] conformal blocks associated to the [Formula: see text]-twisted [Formula: see text]-modules (i.e. [Formula: see text]-twisted [Formula: see text]-modules for some [Formula: see text]) that are ⊗-factorizable to the space of conformal blocks associated to the untwisted [Formula: see text]-modules and a branched covering [Formula: see text] of the Riemann sphere [Formula: see text]. When [Formula: see text] is CFT-type, [Formula: see text]-cofinite, and rational, we use the above result, the (untwisted) factorization property [C. Damiolini, A. Gibney and N. Tarasca, On factorization and vector bundles of conformal blocks from vertex algebras, Ann. Sci. École Norm. Sup. (2022)], and the Riemann–Hurwitz formula to completely determine the fusion rules among [Formula: see text]-twisted [Formula: see text]-modules. Furthermore, assuming [Formula: see text] is as above, we prove that the sewing/factorization of genus-[Formula: see text] [Formula: see text]-twisted [Formula: see text]-conformal blocks holds, and corresponds to the sewing/factorization of untwisted [Formula: see text]-conformal blocks associated to the branched coverings of [Formula: see text]. This proves, in particular, the operator product expansion (i.e. associativity) of [Formula: see text]-twisted [Formula: see text]-intertwining operators (a key ingredient of the [Formula: see text]-crossed braided tensor category [Formula: see text] of the [Formula: see text]-twisted [Formula: see text]-modules) without assuming that the fixed point subalgebra [Formula: see text] is [Formula: see text]-cofinite (and rational), a condition known so far only when [Formula: see text] is solvable and remains a conjecture in the general case. More importantly, this result implies that besides the fusion rules, the associativity isomorphism of [Formula: see text] is also characterized by the higher genus data of untwisted [Formula: see text]-conformal blocks, which gives a new insight into the category [Formula: see text]. We also discuss the applications to conformal nets, which are indeed the original motivations for the author to study the subject of this paper.

Holomorphic factorization of vector bundle automorphisms

Holomorphic factorization of vector bundle automorphisms

On rigidity of ALE vector bundles

On rigidity of ALE vector bundles

Vanishing and Residue of Pontryagin Classes on Singular Foliations

In this paper, we study the vanishing and residue of the Pontryagin classes on singular foliations on smooth manifolds. Specifically, we extend the Bott Vanishing Theorem to singular foliations that admit resolutions by vector bundles, which can be represented by -algebroids, and subsequently prove the Residue Existence Theorem for this type of singular foliations. Our results provide a way of computing the characteristic classes, particularly the Pontryagin classes on smooth manifolds using cohesive modules developed by J. Block. This approach potentially offers a new path in studying the differential geometry and topology of singular foliations beyond the traditional operator algebraic approach.

Basic Concept of Isomorphism and its Use in Group Theory and Physics

Isomorphism is a central concept in mathematics and other sciences for defining structures and their equivalencies. The aim of this paper would be to describe the historical commencement of group theory and introduce the basics of concepts in isomorphisms. This paper defines some of the fundamental theorems related to isomorphisms, such as three isomorphism theorems and the epimorphism lemma, whereby the notion of how mathematical objects could have maps onto each other in a structure-preserving way can be defined. It also summarizes the practical applications of isomorphisms in mathematics and physics. Isomorphisms play a huge role in K-theory-a branch of mathematics that deals, in essence, with vector bundles and modules-and in quantum mechanics to further help in understanding symmetries and transformations in physical systems. The importance and implication of isomorphism are very far-reaching in unifying the concepts across scientific fields. This work underscores the importance of isomorphism and related concepts in group theory and other subjects.

Interpolation in higher codimension

Abstract Following the suggestions contained in [5], I will discuss constructions that generalizes the interpolation problems for divisors to the case of varieties of higher codimension, with emphasis on the case of curves arising as 0-loci of vector bundles of rank 2 in $$\mathbb {P}^3$$ P 3 .

EXTENSION OF COHOMOLOGY CLASSES FOR VECTOR BUNDLES WITH SINGULAR HERMITIAN METRICS

Abstract In this paper, we present an extension theorem which is equivalent to an injectivity theorem on the cohomology groups of vector bundles equipped with singular Hermitian metrics over holomorphically convex Kähler manifolds.

Global hypoellipticity of G-invariant operators on homogeneous vector bundles

Global hypoellipticity of G-invariant operators on homogeneous vector bundles

Infinitesimally equivariant bundles on complex manifolds

AbstractWe study holomorphic vector bundles equipped with a compatible action of vector field by Lie derivatives. We will show that the dependence of the Lie derivative on a vector field is almost ‐linear. More precisely, after an algebraic reformulation, we show that any continuous ‐linear Lie algebra splitting of the symbol map from the Atiyah algebra of a vector bundle on a complex manifold is given by a differential operator, which is further of order at most the rank of the bundle plus one. The proof is quite elementary. When the differential operator we obtain has order 0 we have simply a vector bundle with flat connection, so in a sense, our theorem says that we are always a uniformly bounded order away from this simplest case.

A p-adic Simpson correspondence for smooth proper rigid varieties

For any smooth proper rigid analytic space X over a complete algebraically closed extension of Qp, we construct a p-adic Simpson correspondence: an equivalence of categories between vector bundles on Scholze’s pro-étale site of X and Higgs bundles on X. This generalises a result of Faltings from smooth projective curves to any higher dimension, and further to the rigid analytic setup. The strategy is new, and is based on the study of rigid analytic moduli spaces of pro-étale invertible sheaves on spectral varieties.

A vector bundle valued mixed hard Lefschetz theorem

A vector bundle valued mixed hard Lefschetz theorem

Derived categories of symmetric products and moduli spaces of vector bundles on a curve

Derived categories of symmetric products and moduli spaces of vector bundles on a curve

Drinfel’d doubles, twists and all that. in stringy geometry and M theory

Drinfel’d doubles of Lie bialgebroids play an important role in T-duality of string theories. In the presence of H and R fluxes, Lie bialgebroids should be extended to proto Lie bialgebroids. For both cases, the pair is given by two dual vector bundles, and the Drinfel’d double yields a Courant algebroid. However for U-duality, more complicated direct sum decompositions that are not described by dual vector bundles appear. In a previous work, we extended the notion of a Lie bialgebroid for vector bundles that are not necessarily dual. We achieved this by introducing a framework of calculus on algebroids and examining compatibility conditions for various algebroid properties in this framework. Here our aim is two-fold: extending our work on bialgebroids to include both H- and R-twists, and generalizing proto Lie bialgebroids to pairs of arbitrary vector bundles. To this end, we analyze various algebroid axioms and derive twisted compatibility conditions in the presence of twists. We introduce the notion of proto bialgebroids and their Drinfel’d doubles, where the former generalizes both bialgebroids and proto Lie bialgebroids. We also examine the most general form of vector bundle automorphisms of the double, related to twist matrices, that generate a new bracket from a given one. We analyze various examples from both physics and mathematics literatures in our framework.

On the spectrum of a stable rank 2 vector bundle on ℙ3

The spectrum of a stable rank 2 vector bundle E with c 1 = 0 on the projective 3-space is a finite sequence of positive integers s ( 0 ) ,…, s ( m ) characterizing the Hilbert function of the graded H 1 -module of E in negative degrees. Hartshorne [Invent. Math. 66 (1982), 165–190] showed that if s ( i ) = 1 for some i > 0 then s ( i + 1 ) = 1 ,…, s ( m ) = 1 . We show that if s ( 0 ) = 1 then E ( 1 ) has a global section whose zero scheme is a double structure on a space curve. We deduce, then, the existence of sequences satisfying Hartshorne’s condition that cannot be the spectrum of any stable 2-bundle. This provides a negative answer to a question of Hartshorne and Rao [J. Math. Kyoto Univ. 31 (1991), 789–806].