We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

We study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles” over matrix algebras in the literature of theoretical high-energy physics.

Modular forms for a discrete subgroup Γ of SL(2,ℝ) can be identified with holomorphic sections of line bundles over the modular curve U corresponding to Γ, and quasimodular forms generalize modular forms. We construct vector bundles over U whose sections can be identified with quasimodular forms for Γ.

We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles.

We prove the Bertram–Feinberg–Mukai conjecture for a generic curve C of genus g and a semistable vector bundle E of rank two and determinant K on C, namely we prove the injectivity of the Petri-canonical map S2(H0(E))→H0(S2(E)).

Let $\mathcal{E}$ be a rank-2 bundle over a smooth complex projective surface X. Whenever the subscheme of zeros of a global section of $\mathcal{E}$ is zero-dimensional, it gives a geometric realization of the second Chern class $c_2(\mathcal{E})$. Taking the incidence correspondence

\[\xymatrix{{\mathcal{Z} =\{(x,[e]) \in X\times \bf{P} (H^0(\mathcal{E}))\mid e(x)= 0\}} \ar[r]^>>>>>{p_2}& {\bf{P} (H^0(\mathcal{E}))}}\]

and considering the Zariski open subset $U_f \subset \bf{P} (H^0(\mathcal{E}))$ over which the morphism p2 is finite, we have

\[\mathcal{Z}_f = p^{-1}_2(U_f)\longrightarrow U_f,\]

a family of zero-dimensional subschemes of X of length $\deg (c_2(\mathcal{E}))$. This can be viewed as a distinguished geometric representative of $c_2(\mathcal{E})$.

We define a new invariant of $\mathcal{E}$ which can be viewed as a ‘lifting’ of $c_2(\mathcal{E})$ to Uf. This invariant is a sequence of sections of some coherent sheaves on Uf. These sheaves are built from the following cohomology cup-products:

\begin{align}\gamma_1 &: H^1 (\mathcal{E}^{\ast}) \otimes H^0 (\mathcal{E}) \longrightarrow H^1 (\mathcal{O}_X),\\ \gamma_2&: S^2 H^1 (\mathcal{E}^{\ast})\longrightarrow H^2 (\mathcal{O}_X (\det(\mathcal{E}^{\ast})) ).\end{align}

The main property of our invariant is as follows: either it determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$, or the vector bundle $\mathcal{E}$ is ‘special’. The speciality is expressed in terms of the special geometry of zero-loci of global sections of $\mathcal{E}$ and the special geometry of X.

The sequence of sections entering the definition of our invariant is obtained by starting with the one defined by the cup-product $\gamma_2$ and deriving others inductively by using a geometric interpretation of a part of the cup-product $\gamma_1$ together with the Grothendieck residue map. So one can view our invariant as $\gamma_2$ together with some kind of higher-order cohomology cup-products.

The emergence of these higher-order cohomology cup-products is explained conceptually as higher-order derivatives of a natural deformation of $\gamma_2$ associated to a certain ‘natural’ deformation of the complex structure on $\mathcal{E}$. The variety of these natural deformations of $\mathcal{E}$ has all the features of the classical Jacobian of curves: it carries a distinguished divisor which either determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$ or ‘sees’ that $\mathcal{E}$ is special in the aforementioned sense. An essentially new feature of this Jacobian of $\mathcal{E}$ is that it also carries a variation of Hodge-like structures which arises naturally from our invariant.

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Let $\mathbb{C}$ be a complete non-archimedean-valued algebraically closed field of characteristic p > 0 and consider the punctured unit disc $\dot{D} \subset \mathbb{C}$. Let q be a power of p and consider the arithmetic Frobenius automorphism $\sigma_{\dot{D}}: x \mapsto x^{q^{-1}}$. A $\sigma$-bundle is a vector bundle $\mathcal{F}$ on $\dot{D}$ together with an isomorphism $\tau_\mathcal{F}: \sigma^\ast_{\dot{D}} \mathcal{F} \stackrel{\sim}{\longrightarrow} \mathcal{F}$. The aim of this article is to develop the basic theory of these objects and to classify them. It is shown that every $\sigma$-bundle is isomorphic to a direct sum of indecomposable $\sigma$-bundles $\mathcal{F}_{d,r}$ which depend only on rational numbers d/r. This result has close analogies with the classification of rational Dieudonné modules and of vector bundles on the projective line or on an elliptic curve. It has interesting consequences concerning the uniformizability of Anderson's t-motives that will be treated in a future paper.

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles coincides with the Waldhausen K-theory of perfect complexes. Examples show that, on non-separated schemes, usually many coherent sheaves are not quotients of vector bundles.

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

On P$^3$, we show that mathematical instantons in characteristic two are unobstructed. We produce upper bounds for the dimension of the moduli space of stable rank two bundles on P$^3$ in characteristic two. In cases where there is a phenomenon of good reduction modulo two, these give similar results in characteristic zero. We also give an example of a nonreduced component of the moduli space in characteristic two.

The cohomology ring of the moduli space $M(n,d)$ of semistable bundles of coprime rank $n$ and degree $d$ over a Riemann surface $M$of genus $g \geqslant 2$ has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when $n>2$. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these ‘dual’ Mumford relations form a complete set for the rational cohomology ring of $M(3,d)$ and show (Sect. 5) that the Pontryagin ring vanishes in degree$12g-8$ and above.