[Paper Review] The Chow motive of semismall resolutions

Mark Andrea A. de Cataldo, Luca Migliorini|ArXiv.org|Apr 5, 2002

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper establishes a motivic decomposition theorem for semismall algebraic maps from complex manifolds, showing that the Chow motive of the total space decomposes into a direct sum of twisted motives of relevant strata, indexed by monodromy-invariant components of fibers. The key result is an explicit isomorphism of Chow motives, Chow groups, and mixed Hodge structures via a correspondence built from strata and monodromy data, with applications to Hilbert schemes and parabolic Hilbert schemes of points on surfaces.

ABSTRACT

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-MacPherson's 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.

Motivation & Objective

• To extend the Decomposition Theorem from cohomology to Chow motives and algebraic cycles for semismall maps.
• To provide a Chow-theoretic analogue of Borho-MacPherson's cohomological decomposition for semismall resolutions.
• To compute Chow motives and groups for specific cases, including nested Hilbert schemes and parabolic Hilbert schemes of points on surfaces.
• To explain the equality of orbifold and ordinary Betti/Hodge numbers in crepant resolutions via natural maps of mixed Hodge structures.

Proposed method

• Use the Decomposition Theorem for the direct image complex $ Rf_*\mathbb{Q}_X[\dim X] $ to construct orthogonal projectors in the Chow ring of $ X \times X $, indexed by relevant strata.
• Construct a correspondence $ \overline{\Gamma} $ from monodromy-invariant components of fibers over relevant strata, using refined Gysin formalism and intersection theory.
• Define twisted motives $ [X_\chi](t_\chi) $, where $ t_\chi = n - \sum_{v_* = 0} \chi(v) $, to account for dimension shifts in the decomposition.
• Prove that the correspondence $ \overline{\Gamma}_* $ induces an isomorphism between the direct sum of twisted motives of strata and the motive of the total space.
• Apply the theory to Hilbert schemes and parabolic Hilbert schemes by analyzing stratifications of the Hilbert-Chow morphism.
• Use generating functions to express the motivic, Betti, and Hodge polynomial generating series of the Hilbert schemes in terms of symmetric products of the base surface and divisor.

Experimental results

Research questions

• RQ1 How can the Decomposition Theorem be lifted from cohomology to the category of Chow motives for semismall maps?
• RQ2 What is the explicit structure of the Chow motive of a semismall resolution in terms of strata and monodromy?
• RQ3 Why do orbifold and ordinary Hodge numbers coincide for crepant semismall resolutions, and how is this reflected in Hodge structures?
• RQ4 What is the Chow motive of the parabolic Hilbert scheme of points on a surface, and how does it decompose?
• RQ5 Can generating functions for motives, Betti, and Hodge numbers of Hilbert schemes be expressed in terms of symmetric products of the base surface and divisor?

Key findings

• The Chow motive of a proper semismall map $ f: X \to Y $ decomposes as a direct sum of twisted motives of relevant strata, with the sum of projectors equaling the diagonal of $ X $.
• An explicit isomorphism of Chow groups $ \overline{\Gamma}_*: \bigoplus_{\chi \in \overline{\mathcal{S}}} A_*(X_\chi) \to A_*(Hilb(X,D;n,h,l_*)) $ holds, induced by a correspondence built from strata and monodromy.
• For the parabolic Hilbert scheme, the motive decomposes as $ \bigoplus_{\chi \in \overline{\mathcal{S}}} [X_\chi](t_\chi) \cong [Hilb(X,D;n,h,l_*)] $, with $ t_\chi $ encoding dimension shifts.
• The generating function for Chow motives is expressed as a product over symmetric powers of $ X $ and $ D $, with twists by $ t^m $ and $ s_\alpha^m $, reflecting the stratification.
• The generating functions for Betti and Hodge polynomials are given explicitly, with rational functions involving $ t $, $ s_\alpha $, and Betti/Hodge numbers of $ X $ and $ D $.
• The equality of orbifold and ordinary Hodge numbers in crepant resolutions is explained by the existence of a natural map of mixed Hodge structures arising from the motivic decomposition.