Chow Theory Research Articles

On the Chow theory of Quot schemes of locally free quotients

We prove a formula for the Chow groups of Quot schemes that resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This formula simultaneously generalizes the formulas for projective bundles, Grassmannian bundles, blowups, Cayley's trick, projectivizations, flops from Springer-type resolutions, and Grassmannian-type flips and flops. We also apply the formula to study the Chow groups of (i) blowups of determinantal ideals; (ii) moduli spaces of linear series on curves; and (iii) (nested) Hilbert schemes of points on surfaces.

Additive operations between connective K-theory and Chow theory

We determine all additive operations, stable and unstable, between connective K-theory and Chow theory modulo a prime integer p. It is proved the module of all stable operations is free of rank p−1 over the reduced Steenrod algebra.

Virtual intersection theories

We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.

Operational $K$-theory

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational K -theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothen­dieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is {A}^1 -homotopy invariant. Furthermore, we show that the operational K -theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the K -theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational K -theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.

Splitting vector bundles outside the stable range and 𝔸¹-homotopy sheaves of punctured affine spaces

We discuss the relationship between theA1{\mathbb A}^1-homotopy sheaves ofAn∖0{\mathbb A}^n {\setminus } 0and the problem of splitting off a trivial rank11summand from a ranknnvector bundle. We begin by computingπ3A1(A3∖0)\boldsymbol {\pi }_3^{{\mathbb A}^1}({\mathbb A}^3 {\setminus } 0)and providing a host of related computations of “non-stable”A1{\mathbb A}^1-homotopy sheaves. We then use our computation to deduce that a rank33vector bundle on a smooth affine44-fold over an algebraically closed field having characteristic unequal to22splits off a trivial rank11summand if and only if its third Chern class (in Chow theory) is trivial. This result provides a positive answer to a case of a conjecture of M.P. Murthy.

Around rationality of cycles

Abstract We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

Variations on a theme of rationality of cycles

Abstract We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

Around rationality of integral cycles

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for Z/2Z-coefficients. Those results have already been proved by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠2.

Modified theory of secondary electron emission from spherical particles and its effect on dust charging in complex plasma

The authors have modified Chow's theory of secondary electron emission (SEE) to take account of the fact that the path length of a primary electron in a spherical particle varies between zero to the diameter or xm the penetration depth depending on the distance of the path from the centre of the particle. Further by including this modified expression for SEE efficiency, the charging kinetics of spherical grains in a Maxwellian plasma has been developed; it is based on charge balance over dust particles and number balance of electrons and ionic species. It is seen that this effect is more pronounced for smaller particles and higher plasma temperatures. Desirable experimental work has also been discussed.

Bivariant Chow groups and the topology of envelopes

Let K be an algebraically closed field and let SchK denote the category of schemes over K. Our objective is to extend the operational Chow theory to presheaves (of sets) on SchK. More generally, we introduce bivariant Chow groups CHp(f:F→G), p∈Z for a representable morphism of presheaves on SchK. These are then studied with the help of the topology of envelopes on SchK.

Gazes, targets, (en)visions: reading Fatima Mernissi through Rey Chow

The primary focus of this article will be the implications of the relations between the work of Rey Chow and that of Fatima Mernissi, two very different but eminently “relatable” thinkers, who have not yet been read through and in terms of each other's thought. The themes of colonialism, orientalism, ethnicity, sexism and visuality are key issues in both scholars' thinking, whose works, although faithful to the practice of deconstruction as a method of analysis, diverge on the ground of language, in terms of their writing style. In Chow and Mernissi's work, representation, vision, visibility and its politics entangle in thick power discourses, but Chow's theory unravels them, focusing on the axis between power and knowledge in discursive formations.

Thermal expansion behavior of composites based on axisymmetric ellipsoidal particles

A new model for calculating the coefficients of thermal expansion, CTE, in all three coordinate directions is developed for composites containing aligned, axisymmetric elliptical particles, i.e., characterized by a single aspect ratio, that in the limit approximate the shapes of spheres, fibers and discs. This model is based on Elshelby's method but employs a somewhat different formulation than used in prior papers; a main advantage of the current approach is that it can be readily extended to composites based on ellipsoidal particles with no axes of symmetry, i.e., all three major axes are different, as recently demonstrated for modulus. CTE predictions for the simple case of axisymmetric particles are illustrated by calculations for glass particles in the shape of spheres, fibers and discs in an epoxy resin and are compared to those from the popular Chow theory. For spherical-shaped particles, the CTEs in all directions are the same and decrease modestly as the volume fraction of filler particles increases. As the particle aspect ratio increases from unity, the thermal expansion becomes anisotropic. The coefficient of longitudinal linear thermal expansion always decreases with increasing aspect ratio and filler loading due to the mechanical constraint of the filler. For aligned axisymmetric particles, the coefficients of linear thermal expansion are always the same in two directions. The values in the transverse direction may be higher or lower than that of the matrix depending on the values of aspect ratio and filler loading; these regions are mapped out for this particular set of matrix and filler properties. The two-dimensional constraints on matrix expansion caused by discs versus the one-dimensional effects of fibers cause quantitative differences in behavior for the two shapes.

Steenrod operations in Chow theory

An action of the Steenrod algebra is constructed on the mod p p Chow theory of varieties over a field of characteristic different from p p , answering a question posed in Fulton’s Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.

The Error-Reject Tradeoff

We investigate the error versus reject tradeoff for classifiers. Our analysis is motivated by the remarkable similarity in error-reject tradeoff curves for widely differing algorithms classifying handwritten characters. We present the data in a new scaled version that makes this universal character particularly evident. Based on Chow's theory of the error-reject tradeoff and its underlying Bayesian analysis we argue that such universality is in fact to be expected for general classification problems. Furthermore, we extend Chow's theory to classifiers working from finite samples on a broad, albeit limited, class of problems. The problems we consider are effectively binary, i.e., classification problems for which almost all inputs involve a choice between the right classification and at most one predominant alternative. We show that for such problems at most half of the initially rejected inputs would have been erroneously classified. We show further that such problems arise naturally as small perturbations of the PAC model for large training sets. The perturbed model leads us to conclude that the dominant source of error comes from pairwise overlapping categories. For infinite training sets, the overlap is due to noise and/or poor preprocessing. For finite training sets there is an additional contribution from the inevitable displacement of the decision boundaries due to finiteness of the sample. In either case, a rejection mechanism which rejects inputs in a shell surrounding the decision boundaries leads to a universal form for the error-reject tradeoff. Finally, we analyze a specific reject mechanism based on the extent of consensus among an ensemble of classifiers. For the ensemble reject mechanism we find an analytic expression for the error-reject tradeoff based on a maximum entropy estimate of the problem difficulty distribution.

Chow motif and higher Chow theory ofG/P

Chow motif and higher Chow theory ofG/P