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A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories. [ABSTRACT FROM AUTHOR]

We give a geometric characterization of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line arrangements. [ABSTRACT FROM AUTHOR]

We generalize to vector bundles the techniques introduced for line bundles in Liu (TAMS 374:367–405, 2021). We then use this method to prove the injectivity of the Petri map for vector bundles and the surjectivity of a map related to deformation theory of Poincaré sheaves. [ABSTRACT FROM AUTHOR]

We consider vector bundles of rank 2 with trivial generic fiber on the projective line over ℤ . For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to superscript 풪 2 in the fiber over ℚ and are isomorphic to superscript 풪 2 or direct-sum 풪 1 풪 1 over each closed point of Spec ℤ , we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle. [ABSTRACT FROM AUTHOR]

Purpose – The aims of this paper is to prove that every semisimple Jordan algebra bundle is locally trivial and establish the decomposition theorem for locally trivial Jordan algebra bundles using the decomposition theorem of Lie algebra bundles. Design/methodology/approach – Using the decomposition theorem of Lie algebra bundles, this paper proves the decomposition theorem for locally trivial Jordan algebra bundles. Findings – Findings of this paper establish the decomposition theorem for locally trivial Jordan algebra bundles. Originality/value – To the best of the author’s knowledge, all the results are new and interesting to the field of Mathematics and Theoretical Physics community.

We define cohomology of module bundles over associative algebra bundles. We establish a one to one correspondence between the first cohomology classes and the extensions of module bundles. Using this correspondence we give sufficient condition in terms of cohomology for an algebra bundle to be semisimple. [ABSTRACT FROM AUTHOR]

We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles. [ABSTRACT FROM AUTHOR]

A new approach to finite elasto-plasticity of crystalline materials, with a differential geometry point of view toward the material description, is proposed. In order to define the plastic and elastic distortions, traditionally it is accepted the existence of an intermediate configuration, which is endowed with a differential manifold structure. A more restrictive but physically and mathematically well-motivated structure is assigned to the intermediate configuration, namely, a vector bundle structure. Two approaches to multiplicative decomposition of the deformation gradient can be rebuilt if on the same total space (intermediate configuration) the vector bundle structures are considered, but either the reference configuration, B (i.e., plastic assumption) or the deformed configuration, B t (i.e., elastic assumption) stands for the base space. As the base spaces are homeomorphic, a pullback vector bundle over one base space is provided by the other one through the motion diffeomorphism. Thus, the total space is endowed with different differential manifold structures. The plastic distortion and the inverse elastic distortion, respectively, are defined for any material points as linear, invertible, and non-integrable maps from the tangent vector space at their material points, with the value on the attached vector fiber. The multiplicative decompositions into their non-integrable components are derived, based on the associate inverse elastic distortion and plastic distortion, respectively. When the plastic and elastic assumptions are simultaneously accepted, then a three-term multiplicative decomposition of the deformation gradient is achieved. The transition functions, which characterize the compatibility of the overlapping charts which belong to these different structures, namely of the vector bundle and of the pullback vector bundle over the same configuration, say B define the non-uniqueness of the two-term multiplicative decomposition. The material symmetry transformation is defined for elastic-type materials. The compatibility of the models and examples of the bundle vector structures are also discussed. [ABSTRACT FROM AUTHOR]

We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the variety? We exhibit some conditions under which this question has an affirmative answer. We primarily consider ample bundles on projective spaces and Hirzebruch surfaces. We also show that Seshadri constants of ample vector bundles can be arbitrarily small. [ABSTRACT FROM AUTHOR]

In analogy with the theory of cohomology of Lie algebras, we develop the concept of module bundle and Lie algebra bundle extensions. [ABSTRACT FROM AUTHOR]

Trudinger–Moser inequalities are important tools in partial differential equations and geometric analysis. Although there have been many results in this regard, there are few studies on vector valued function spaces. Years ago, joined with Y. Li and P. Liu, the second named author established a Trudinger–Moser inequality on vector bundle over a closed Riemann surface. In that article (Calc Var 28:59–87, 2007), we do not settle down the existence of extremal bundle sections. In the present paper, we have two purposes, one is to prove the existence of extremal bundle sections, and the other is to generalize such an inequality to a stronger one with remainder terms. The proof is based on a much more delicate analysis and a very careful construction of a sequence of bundle sections. [ABSTRACT FROM AUTHOR]

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory, (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case. [ABSTRACT FROM AUTHOR]

In this paper, we define algebraic Lie algebra bundles, discuss some results on algebraic Lie algebra bundles and derivations of Lie algebra bundles. Some results involving inner derivations and central derivations of Lie algebra bundles are obtained. [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and let n ≥ 3 be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to H 1 (X , Z / (n)) ⋊ (Out (SL (n , C)) × Aut (X)) . Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of SL (n , C) on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of H 1 (X , Z / (n)) , H 1 (X , Z / (n)) ⋊ Out (SL (n , C)) , and Out (SL (n , C)) × Aut (X) on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of Out (SL (n , C)) × Aut (X) , the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution σ X of X. Finally, it is discussed how the description of fixed points for the action of the elements of is covered by the descriptions above. [ABSTRACT FROM AUTHOR]

In this paper, we consider spaces whose geometry is generated by a homogeneous function of degree m ≥ 2, which is invariant under the action of some subgroup of the linear group of the given space. A general method is proposed and examples of realization of such spaces on linear algebras are given. [ABSTRACT FROM AUTHOR]

This paper classifies rank two vector bundles on a del Pezzo threefold X of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on X , and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result gives a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one. [ABSTRACT FROM AUTHOR]

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G_2-variety, including its quantum version. We also consider the Spin_7-variety, which deserves a different treatment. [ABSTRACT FROM AUTHOR]

We show that the radical bundle of an algebra bundle is a characteristic ideal bundle. Further we prove an algebra bundle is semisimple if and only if its derivation algebra bundle is either semisimple or zero. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate harmonic functions on vector bundle manifolds endowed with the class of spherically symmetric metrics. We will study three classes of harmonic functions on total spaces of vector bundles. We attempt to establish the link between harmonic function theories on the base and total space. [ABSTRACT FROM AUTHOR]

We use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization. [ABSTRACT FROM AUTHOR]

Given a vector bundle E on a tree of smooth rational curves C, we give necessary and sufficient conditions for a vector bundle E ′ on P 1 to specialize to E on C, generalizing the rank 2 case, due to Coskun. [ABSTRACT FROM AUTHOR]

In this article, we completely classify locally free sheaves E of rank 2 such that h 1 (E (t)) = 0 for t ∈ Z on the double embedding of a smooth quadric surface P 1 × P 1 , and prove the existence of them. Geometric properties of the sets of points corresponding to E by Serre correspondence are obtained. [ABSTRACT FROM AUTHOR]

A K-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple K-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple K-equivalent maps, and classify such maps in several cases, including the case of dimension at most 8. [ABSTRACT FROM AUTHOR]

We complete the classification of globally generated vector bundles with c1≤5$c_1 \le 5$ on projective spaces by treating the case c1=5$c_1 = 5$ on Pn$\mathbb {P}^n$, n≥4$n \ge 4$. It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on P5$\mathbb {P}^5$, and its restriction to P4$\mathbb {P}^4$. We recall, in an appendix, from one of our previous papers, the main results allowing the classification of globally generated vector bundles with c1=5$c_1 = 5$ on P3$\mathbb {P}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to P4$\mathbb {P}^4$ as globally generated vector bundles. [ABSTRACT FROM AUTHOR]

In this paper, we construct a correspondence between the complex vector bundles of finite rank on smooth manifolds and the representation of the generated vector space by a basis of the vector bundles and derive its non-triviality via the representation of the generated space. We also provide some applications of this correspondence. [ABSTRACT FROM AUTHOR]

We classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion. [ABSTRACT FROM AUTHOR]

We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter k R -spaces and locally convex spaces E such that E × E is a k R -space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a complete hypertetrahedral arrangement A in P n . We address two basic problems. First, we describe the local freeness of A in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustaţă–Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of A . [ABSTRACT FROM AUTHOR]

In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle M V , L : = ker (V ⊗ O C → L) . In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of M L . The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7. [ABSTRACT FROM AUTHOR]

This paper treats the strict semi-stability of the symmetric powers S k E of a stable vector bundle E of rank 2 with even degree on a smooth projective curve C of genus g ≥ 2. The strict semi-stability of S 2 E is equivalent to the orthogonality of E or the existence of a bisection on the ruled surface ℙ C (E) whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of E with strictly semi-stable S 3 E. Moreover, it is shown that when S 2 E is stable, every symmetric power S k E is stable for all but a finite number of E in the moduli of stable vector bundles of rank  2 with fixed determinant of even degree on C. [ABSTRACT FROM AUTHOR]

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results. [ABSTRACT FROM AUTHOR]

We describe completely the so called jet like functors of a vector bundle E over an m-dimensional manifold M, i.e. bundles FE over M canonically depending on E such that F(E1 ×M E2) = FE1 ×M FE2 for any vector bundles E1 and E2 over M. Then we study how a linear vector field on E can induce canonically a vector field on FE. [ABSTRACT FROM AUTHOR]

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian G and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective complex curve of genus g ≥ 2 and let MX(2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦k ⊂ MX(2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦k in the Picard group of MX(2, Λ). [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P 3. [ABSTRACT FROM AUTHOR]

Let (X , L) be a complex polarized n -fold with the structure of a classical scroll over a smooth projective threefold Y. The Hilbert curve of such a pair (X , L) is a complex affine plane curve of degree n , consisting of n − 3 evenly spaced parallel lines plus a cubic. This paper is devoted to a detailed study of this cubic. In particular, existence of triple points, behavior with respect to the line at infinity, and non-reducedness, are analyzed in connection with the structure of (X , L). Special attention is reserved to the case n = 4 , where various examples are presented and the possibility that the cubic is itself the Hilbert curve of the base threefold Y for a suitable polarization is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms. [ABSTRACT FROM AUTHOR]

A method of the Riemann–Hilbert problem is employed for Zhang's conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang's conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry. [ABSTRACT FROM AUTHOR]