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.

For any integer $m\geq 4$ we construct twistor spaces of algebraic dimension a = 2 associated to some self-dual structures on mP2, the connected sum of m complex projective planes P2. Together with previously known results this implies that for any $m\geq 5$ all the possible values for algebraic dimension $0\leq a \leq 3$ are attained for some such twistor spaces. Our method is to construct such twistor spaces by small deformations of Joyce twistor spaces.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group $GL_r(\mathbb{C}[z]/z^n)$. We prove that they carry complete hyper-Kähler metrics.

We introduce a new approach to the study of a system of algebraic equations in $(\mathbb{C}^\times)^n$ whose Newton polytopes have sufficiently general relative positions. Our method is based on the theory of Parshin's residues and tame symbols on toroidal varieties. It provides a uniform algebraic explanation of the recent result of Khovanskii on the product of the roots of such systems and the Gel'fond–Khovanskii result on the sum of the values of a Laurent polynomial over the roots of such systems, and extends them to the case of an algebraically closed field of arbitrary characteristic.

In this paper, we construct automorphic forms on the five-dimensional complex ball which give the inverse of the period map for cyclic 4-ple coverings of the complex projective line branching at eight points. We use theta constants associated to the Prym varieties of these coverings.

Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the étale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $\ne 2$, where genus(C) = $g \ge 3$, Pic$^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$, we prove that ‘Riemann's singularity theorem holds at $\mathcal L$’, i.e. mult$_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$, if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$, and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$. This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$, hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$.

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.

We prove the vanishing of the equivariant holomorphic torsion forms associated to the holomorphic de Rham complex. We also consider corresponding torsion forms associated to complexes in hypercohomology, for which we give a local formula. Rigidity of certain genera explains part of these formulas. The Lefschetz operators in Hodge theory play a prominent role in the proofs.

Let G be a real special or general linear group and $\sigma_{0}$ be the transpose-inverse involution. We characterize the image of $f \mapsto {\rm tr}(\pi(f) \pi(\sigma_{0}) )$ for irreducible representations $\pi$ of $G \rtimes \langle \sigma_{0}\rangle$, and K-finite $f \in C_{c}^{\infty}(G)$.

In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.

We use the $A_{\infty}$-formalism to study variation in cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of some singular points in twisted Brill–Noether loci in the moduli spaces of vector bundles on a curve. Another application is a computation of the Fourier–Mukai transform of some natural line bundles on symmetric powers of a curve.

In this paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result is the restriction on k: it can equal only 3, 4, or 5. The most interesting class of nets is formed by 3-nets that relate to finite geometries, latin squares, loops, etc. All known examples of 3-nets in $\mathbb{P}^2$ realize finite Abelian groups. We study the question of which groups can be realized in this way. Our main result is that, except for groups with all invariant factors below 10, the realizable groups are isomorphic to subgroups of a 2-torus. This follows from the ‘algebraization’ result asserting that, in the dual plane, the points dual to lines of a net lie on a plane cubic.

Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of $\widetilde{P}$-polynomials. A ‘quantum Schubert calculus’ is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov--Witten invariants. As an application, we show that the table of three-point, genus zero Gromov–Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.

Let $f:X\to Y$ be a Cohen–Macaulay map of finite type between Noetherian schemes, and $g:Y'\to Y$ a map, with Y′ Noetherian. Let $f':X'\to Y'$ be the base change of f under g and $g':X'\to X$ the base change of g under f. We show that there is a canonical isomorphism $\theta_g^f: {g'}^*\omega_f \simeq \omega_{f'}$, where $\omega_f$ and $\omega_{f'}$ are the relative dualizing sheaves. The map $\theta_g^f$ is easily described when f is proper, and has a more subtle description when f is not. If f is smooth we show that $\theta_g^f$ corresponds to the canonical identification $g'^*\Omega_f^r= \Omega_{f'}^r$ of differential forms, where r is the relative dimension of f. This work is closely related to B. Conrad's work on base change. However, our approach to the problems and our viewpoint are very different from Conrad's: dualizing complexes and their Cousin versions, residual complexes, do not appear in this paper.

We describe the Zariski closure of the geometric realization of Coxeter groups. When the Coxeter system is irreducible and the Tits form B non-positive and non-degenerate, this Zariski closure is equal to the orthogonal group of B.

In this paper, we present a conjecture on a relation between the Zelevinski involutions for reductive groups over a p-adic field and the conjectural A-packets. As evidence for this conjecture, we prove that the Zelevinski involutions, which are regarded as operators on the space of virtual characters, commute with the endoscopic transfers under the assumption of the fundamental lemma for groups and for Lie algebras.

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.

On a real algebraic variety there may exist an algebraic cycle that is algebraically equivalent to zero and whose cohomology class is non-zero. The group of such cohomology classes can be highly non-trivial. It is interesting since it allows one to detect cohomology classes, in complementary dimension, which cannot be represented by algebraic cycles.

In this paper, we give a formal algebraic notion of exponents for linear differential systems at any singularity as eigenvalues of the residue of a regular connection on a maximal lattice (that we call ‘Levelt's lattice’). This allows us to establish upper and lower bounds for the sum of these exponents for differential systems on ${\mathbb P}^{1}(\mathbb{C})$.

A skeletal structure (M, U) in ${\mathbb R}^{n+1}$ is a special type of n-dimensional Whitney stratified set M on which is defined a multivalued ‘radial vector field’ U. This is an extension of the notion of the Blum medial axis of a region in ${\mathbb R}^{n+1}$ with generic smooth boundary. For such a skeletal structure an ‘associated boundary’ $\mathcal{B}$ is defined. In part I of this paper, we introduced radial and edge shape operators, which are geometric invariants of the radial vector field U on M, and a ‘radial flow’ from M to $\mathcal{B}$. In this paper, in the partial Blum case we derive formulas for the differential geometric shape operator of the boundary (and hence all curvature invariants) in terms of the shape operators on the medial axis. We further derive the effects of a diffeomorphism of the skeletal structure on the radial and edge shape operators using a distortion operator which is computed from the second derivative of the diffeomorphism evaluated on the unit radial vector field. This allows one to compute the geometry of the boundary associated to a deformed skeletal structure purely in terms of operators defined on the original skeletal set.