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We introduce a theory of multigraded Cayley-Chow forms associated to subvarieties of products of projective spaces. Two new phenomena arise: first, the construction turns out to require certain inequalities on the dimensions of projections; and second, in positive characteristic the multigraded Cayley-Chow forms can have higher multiplicities. The theory also provides a natural framework for understanding multifocal tensors in computer vision.
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In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.
We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation proceeds along the lines of the classical computation of the integral cohomology of ${rm BO}(n)$ with local coefficients, as done by Cadek. The computations of Chow-Witt rings of classifying spaces of ${rm GL}_n$ are then used to compute the Chow-Witt rings of the finite Grassmannians. As before, the formulas are close parallels of the formulas describing integral cohomology rings of real Grassmannians.
In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayleys trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.
Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.
We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a motivic decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X.The result is a Chow-theoretic analogue of Borho-MacPhersons observation concerning the cohomology of the fibers and their relation to the relevant strata for f. Under suitable assumptions on the stratification, we prove an explicit version of the motivic decomposition theorem. The assumptions are fulfilled in many cases of interest, e.g. in connection with resolutions of orbifolds and of some configuration spaces. We compute the Chow motives and groups in some of these cases, e.g. the nested Hilbert schemes of points of a surface. In an appendix with T. Mochizuki, we do the same for the parabolic Hilbert scheme of points on a surface. The results above hold for mixed Hodge structures and explain, in some cases, the equality between orbifold Betti/Hodge numbers and ordinary Betti/Hodge numbers for the crepant semismall resolutions in terms of the existence of a natural map of mixed Hodge structures. Most results hold over an algebraically closed field and in the Kaehler context.
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