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Register a new userOn the Chow theory of projectivizations
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Authors
Qingyuan Jiang
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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.
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We prove a formula for Chow groups of $Quot$-schemes which resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This result provides a unified way to understand known formulae for various geometric situations such as blowups, Cayleys trick, projectivizations, Grassmannian bundles, flops from Springer type resolutions, as well as provide new phenomena such as formulae for Grassmannain type flips/flops and virtual flips. We also give applications to blowups of determinantal ideals, moduli spaces of linear series on curves, and Hilbert schemes of points on surfaces.
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.
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.
We prove results describing the structure of a Chow ring associated to a product of graphs, which arises from the Gross-Schoen desingularization of a product of regular proper semi-stable curves over discrete valuation rings. By the works of Johannes Kolb and Shou-Wu Zhang, this ring controls the behavior of the non-Archimedean height pairing on products of smooth proper curves over non-Archimedean fields. We provide a complete description of the degree map, and prove vanishing results affirming a conjecture of Kolb, which, combined with his work, leads to an analytic formula for the arithmetic intersection number of adelic metrized line bundles on products of curves over complete discretely valued fields.
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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