Quadric bundles and hyperbolic equivalence

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = ( L , [ . , . ] , ♯ ) be a holomorphic Lie algebroid over X of rank one and degree ( L ) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes

Matrices of linear forms of constant rank from vector bundles on projective spaces

The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space X.A homomorphism is defined between them analogous to the generalized Hasse invariant.Upon evaluation, the Witt group is seen to be KO(X), the graded Brauer group 1+H1(X;ZZ) + H2(X; Z2) with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: I + m^ + h^.Introduction.The Hasse invariant of a quadratic form on a vector space over a field k is an element of the Brauer group of k.It can be generalized to a homomorphism from the Witt group of classes of quadratic forms to the graded Brauer group of classes of graded ^-algebras [3], [16].Curiously enough, in this guise it is quite analogous to the homomorphism given by the first two Stiefel-Whitney classes which maps real vector bundles over a space X to the group 1 + H\X; Z2) + H2(X;Z2), where multiplication is by truncated cup product.The purpose of this paper is to display this analogy.Delzant has defined Stiefel-Whitney classes of quadratic forms [4], [13] using the discriminant and Hasse invariant.It should add insight into his definition to see that the first two topological Stiefel-Whitney classes can also be defined in an algebraic way.A vector bundle L is known to be orientable for real L-theory if its first two Stiefel-Whitney classes vanish.In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].The connection between algebra and topology comes from considering bundles over X whose fibers are vector spaces with quadratic forms or graded algebras and subjecting them to the same analysis the fiber alone is accorded in algebra.This is described in some detail in 1.11 is devoted to the algebraic computations and 111 to a study of the topological problems.

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions.

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring with . The problem of the construction of a `normal' diagonal form of a quadratic form over a ring faces obstacles in the case of indices greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for (an extension of the law of inertia for real quadratic forms) and in Theorem 2.2 for containing an invertible non-square. Under the same conditions on a ring with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free -module of rank is explicitly calculated (Proposition 3.2). Up to projectivities, the list of forms is presented for the projective plane over and also (Theorem 3.3) over the local ring with non-principal maximal ideal, where is a field with an invertible non-square in and . In the latter case the number of classes of non-diagonalizable quadratic forms of rank 0 depends on one's choice of the field and is not even always finite; all the other forms make up 21 classes.

According to Horrocks (Quart. J. Math. Oxford (2) 17 (1966), 14–18), a vector bundle E on the projective n-space extends stably to the projective N-space, N > n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N > n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coandă and Trautmann (Commun. Algebra 34 (2006), 2485–2488). Combining this result with a theorem of Mohan Kumar et al. (Manuscripta Math. 112 (2003), 183–189), we get a new effective version of the Babylonian tower theorem for vector bundles on projective spaces.

The Kobayashi indicatrix (infinitesimal unit ball) of a domain in IE n is known to be a biholomorphic invariant. In particular, if a domain is biholomorphic to a ball, then the indicatrix is the ball. Until the recent deep results of Lempert [4], it was not known to what extent the indicatrix characterizes the domain. Sibony had shown earlier that the indicatrix of any pseudoconvex circular domain is the domain itself [10]; hence the indicatrix determines such domains up to biholomorphism. This is not true in general. Lempert has shown that, for smoothly bounded strictly convex domains, three invariants determine the domain up to biholomorphism and a linear change of variables at a base point. These invariants consist of not only the Kobayashi indicatrix but also a quadratic form and a hermitian form on a certain vector bundle. In two dimensions, there is an open set in a Frechet space, corresponding to domains not biholomorphic to the ball but with indicatrix the ball. The result of this paper is that, for ellipsoids, if the Kobayashi indicatrix is the ball then the ellipsoid is biholomorphic to the ball. This is proved in the following equivalent form: if an ellipsoid symmetric about 0 is not biholomorphic to the ball, then its infinitesimal Kobayashi metric at 0 is not hermitian. In this language, our result answers a question posed by Reiffen in [7] about the differential-geometric nature of the Carath6odory metric for ellipsoids. By two results of Lempert ([3] and [2]), for strictly convex domains the Carathe6dory and Kobayashi metrics coincide and determine a Finsler metric. One now sees that, for ellipsoids, the metric is not hermitian and hence, by a theorem of Reiffen [7], not Riemannian. We prove that the infinitesimal Kobayashi metric at 0 is not hermitian by proving that the parallelogram law fails to hold. An ellipsoid ~ may be written (via a linear change of coordinates) as )-], ] zj] 2 ..~ ,~j Re (z 2) < , 0 < Aj < 1. By a

We study the parameters of codes spanned by quadratic and Hermitian forms in projective space. The resulting codes have high minimum distance and few distinct weights. An example is a new ternary [91.9,54] code.

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ( H 1 {H^1} in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.

The study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some of the leading researchers in the field to explore further these interactions. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles. Topics covered include those linking gauge theory and geometric invariant theory such as augmented bundles and coherent systems; Donaldson invariants of algebraic surfaces; Floer homology and quantum cohomology; conformal field theory and the moduli spaces of bundles on curves; the Horrocks–Mumford bundle and codimension 2 subvarieties in P4 and P5; exceptional bundles and stable sheaves on projective space.

We study the probability that an $(n - m)$-dimensional linear subspace in $\mathbb{P}^n$ or a collection of points spanning such a linear subspace is contained in an m-dimensional variety $Y \subset \mathbb{P}^n$. This involves a strategy used by Galkin–Shinder to connect properties of a cubic hypersurface to its Fano variety of lines via cut-and-paste relations in the Grothendieck ring of varieties. Generalizing this idea to varieties of higher codimension and degree, we can measure growth rates of weighted probabilities of k-planes contained in a sequence of varieties with varying initial parameters over a finite field. In the course of doing this, we move an identity motivated by rationality problems involving cubic hypersurfaces to a motivic statistics setting associated with cohomological stability.

We produce counterexamples to the birational Torelli theorem for Calabi–Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non-birational pairs of Calabi–Yau ( n 2 − 1 ) (n^2-1) -folds which, for n ≥ 2 n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an L \mathbb {L} -equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi–Yau pairs, namely all those which are realized as pushforwards of a general ( 1 , 1 ) (1,1) -section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.