Geometric model for vector bundles via infinite marked strips

We introduce a notion of quasiconvexity for continuous functions f defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold (M,g) and \mathbb{R}^{m} , naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional F(u,\Omega) = \int_{\Omega}f(du)\,d\mu with respect to the weak ^{*} topology of W^{1,\infty}(\Omega,\mathbb{R}^{m}) , for every bounded open subset \Omega\subseteq M .

We characterize the existence of an Ulrich vector bundle on a variety X ⊂ P N X \subset \mathbb {P}^N in terms of the existence of a subvariety satisfying some precise conditions. Then we use this fact to prove that a complete intersection of dimension n ≥ 4 n \ge 4 , which if n = 4 n=4 is very general and not of type ( 2 , 2 ) (2,2) , does not carry any Ulrich bundles of rank r ≤ 3 r \le 3 unless n = 4 , r = 2 n=4, r=2 and X X is a quadric.

A vector bundle approach to Nash equilibria

Z-critical equations for holomorphic vector bundles on Kähler surfaces

Abstract A finite CW-complex X is called P $\mathcal{P}$ -trivial if, for any real vector bundle ξ over X , the total Pontrjagin class p ( ξ ) = 1. In this article, we completely determine when each of the following spaces are P $\mathcal{P}$ -trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces. Furthermore, we show that the 3-fold suspension of a compact orientable surface of genus ≥ 1 $\ge \text{ }1$ is not P $\mathcal{P}$ -trivial.

Abstract In parallel to the characterization of global hypoellipticity for G -invariant operators on homogeneous vector bundles obtained by Cardona and the author [J. Pseudo-Differ. Oper. Appl. 16, 23 (2025)], in this paper we obtain necessary and sufficient conditions for an arbitrary system of left-invariant operators on a compact Lie group to be globally hypoelliptic, via a proof which avoids the homogeneous vector bundle structure of that paper. We then prove alternative sufficient conditions for globally hypoellipticity for a large class of systems making use of lower bounds for the smallest singular value of complex matrices.

In this paper, we study the positivity of an Ulrich vector bundle defined with respect to a globally generated ample line bundle. First, we prove a generalization of a Lopez theorem on the first Chern class and the bigness of an Ulrich bundle. Then, under some additional assumptions on the polarization, we give a description of its augmented base locus, which consequently leads to a characterization of the V-bigness and of the ampleness of an Ulrich bundle in this setting.

We introduce a geometric approach to the construction of moment maps in finite and infinite-dimensional complex geometry. We apply this to two settings: Kähler manifolds and holomorphic vector bundles. We first give a new, geometric proof of Donaldson–Fujiki’s moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern characters of the manifold—namely to a central charge—we further introduce a geometric PDE determining a Z -critical Kähler metric, and show that these general equations also satisfy moment map properties. For holomorphic vector bundles, we similarly give a geometric proof that the PDE determining a Z -critical connection can be viewed as a moment map. Our main assertion is that this is the canonical way of producing moment maps in complex geometry, and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.

We review the construction (due to Brenner–Schröer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for \mathbb{N} -graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the \mathbb{N} -graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner–Schröer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism.

We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel–Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.

A bstract We develop the field space geometry of scalar-fermion effective field theories as a vector bundle supermanifold. We further establish a Fermi normal coordinate system on the bundle that clarifies the geometric content in scattering amplitudes, particularly the imprints of field space non-analyticities. Specializing to the Standard Model Higgs sector, we examine the geometric consequences of custodial symmetry violation, including implications for the physical Higgs field as a distinguished scalar axis and deformations in the fermionic sector. Our results enable a systematic and realistic geometric interpretation of Higgs sector phenomenology.

We obtain an explicit solution of the Riemann–Hilbert problem on an elliptic curve for the two-dimensional commutative monodromy representations. By an arbitrary set of points together with a representation of the fundamental group of the curve punctured at these points, we construct a semistable holomorphic vector bundle of degree zero with a logarithmic connection possessing the required singularities and monodromy.

Abstract Griffiths’ conjecture asserts that a holomorphic vector bundle is ample if and only if it admits a Hermitian metric with positive curvature. In this paper, we present a new proof of this conjecture on compact Riemann surfaces using a system of PDEs introduced by Demailly. Our argument combines techniques developed by Uhlenbeck–Yau for Hermitian–Einstein metrics with Pingali’s reduction of the problem to an a priori estimate.

Abstract Given a number field F with ring of integers $\mathcal {O}_{F}$ , one can associate to any torsion free subgroup of $\operatorname {SL}(2,\mathcal {O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp ends. For natural choices of flat vector bundles on such a manifold, we show that analytic torsion is identified with the Reidemeister torsion of the Borel-Serre compactification. This is used to obtain exponential growth of torsion in the cohomology for sequences of congruence subgroups.

This paper presents a simplified geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem, which states that the closure of a singular Riemannian foliation on a complete Riemannian manifold is itself a smooth singular Riemannian foliation. Our approach circumvents several technical and analytical tools employed in the previous proof of the theorem, resulting in a more direct geometric demonstration. We first establish conditions for a projectable foliation to be Riemannian, focusing on compatible connections. We then apply these results to linear foliations on vector bundles and their lifts to frame bundles. Finally, we use these findings for the linearization of singular Riemannian foliations around leaf closures. This method allows us to prove the smoothness of the closure directly for the linear semi-local model, bypassing the need for intermediate results on orbit-like foliations.

This paper considers vector bundles on smooth quadric hypersurfaces defined over an algebraically closed field in characteristic zero. Let p be an integer with p ≥ 2 . We show that there exists a uniform but nonhomogeneous vector bundle E of rank 3 N + 2 on a smooth quadric hypersurface Q N ( N ≥ 6 ) . We also construct indecomposable uniform but nonhomogeneous vector bundles of rank 3 N + p − 9 over Q N ( N ≥ 4 ) , which gives an up bound 3 N − 8 for the uniform-homogeneous threshold of Q N ( N ≥ 4 ) .

Abstract Let ℰ be a rank-2 vector bundle over an elliptic curve E , decomposable as a sum of line bundles of degrees d ' > d ≥ 2, and ℒ the determinant of ℰ. The subspace L (ℰ) ⊂ ℙ n −1 ≅ ℙExt 1 (ℒ, 𝒪 E ) consisting of classes of extensions with middle term isomorphic to ℰ is one of the symplectic leaves of a remarkable Poisson structure on ℙ n −1 defined by Feigin–Odesskii and Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes L (ℰ) as the base space of a principal Aut(ℰ)-fibration. Here, we embed L (ℰ) into a larger, projective base space L ˜ ( E ) $\widetilde{L}(\mathcal{E})$ of a principal Aut(ℰ)-fibration whose total space parametrizes sections of ℰ. The embedding realizes L ( E ) ⊂ L ˜ ( E ) $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ as a complement of an anticanonical divisor Y (one of the main results), and we give an explicit description of the normalization of Y as a projective-space bundle over a projective space. For d = 2 , L ˜ ( E ) $d=2,\, \widetilde{L}(\mathcal{E})$ is one of the three Hirzebruch surfaces Σ i , i = 0, 1, 2; we determine which occurs when and hence also the cases when L (ℰ) is affine. Separately, we prove that for d < n /2 the singular locus of the secant slice Sec d , z ( E ) ⊂ ℙ n −1 , the portion of the d th secant variety of E consisting of points lying on spans of d -tuples with sum z ∈ E , is precisely Sec d −2 . This strengthens the result that L (ℰ) is smooth, appearing in prior joint work with R. Kanda and S. P. Smith.

Monge-Ampère Type Equation for the Nakano Positive Curvature Tensor of Holomorphic Vector Bundles

Abstract A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original variety. We study the relative geometry of this map. We give a polyhedral criterion for the map to have the structure of a toric variety bundle, and prove that this structure always exists over the interior of the codomain. We give examples demonstrating that this is the strongest statement one can hope for in general. In addition, we provide a combinatorial recipe for constructing the toric variety bundle as a fibrewise GIT quotient of an explicit split vector bundle. Our proofs make systematic use of Artin fans as a language for globalising local toric models.