Chow cohomology groups of algebraic surfaces

Abstract

For an algeraic surface with isolated singularities we consider its higher Chow group and Chow cohomology (the latter defined by the author). We study the canonical map from the Chow cohomology to the higher Chow group by relating it to the canonical map from the higher Chow group to Chow cohomology of the exceptional divisor of a desingularization of the surface. Introduction. For a quasi-projective variety S over a field, S. Bloch defined its higher Chow groups CH(S, n) as the homology of a certain complex Z(S, ·) called the cycle complex, [Bl 1], [Bl 2], [Bl 3]. One may view this as a Borel-Moore homology theory; for example it is covariantly functorial for proper maps, and contravariantly functorial for open immersions (more generally for flat maps). For a quasi-projective variety S over a field of characteristic zero, using resolution of singularities and Bloch’s cycle complexes, we defined the Chow cohomology groups CHC(S, n) (see [Ha 2] for details). To briefly recall it, we take a cubical hyperresolution X• → S, which gives a strict truncated simplicial scheme; it consists of smooth varieties Xa for 0 ≤ a ≤ N with some N , and the face maps di : Xn → Xn−1, i = 0, . . . , n, satisfying the usual identities. (It differs from a simplicial scheme in that there are only face maps and no degeneracies, and there are only finite many terms.) We then form the double complex Z(X0, ·) d∗ −−−→Z(X1, ·) d∗ −−−→· · · d ∗ −−−→Z(XN , ·) where the a-th column is the cycle complex of Xa, and the horizontal differentials d ∗ : Z(Xa, ·) → Z(Xa+1, ·) are the alternating sums of the pull-backs di by the face maps. (Strictly speaking one must take appropriate quasi-isomorphic subcomplexes for d∗ be defined, see §1.) The total complex of this double complex is denoted Z(X•, ·)∗, and called the cohomological cycle complex of S. Then CHC(S, n) is by definition the (−n)-th cohomology of Z(X•, ·)∗. It is proven in [Ha 2] that CHC(S, n) is well-defined up to canonical isomorphism, independent of the choice of a hyperresolution, and that the association S 7→ CHC(S, n) is contravariantly functorial for all maps. The condition the characteristic being zero is unnecessary if dimS ≤ 2, since its desingularization (and thus its cubical hyperresolution) exist. In that case we exhibit the cubical hyperresolution and the cohomological cycle complex of S explicitly in §§1 and 2. (The relationship of our Chow cohomology to the motivic cohomology of [FV] will be discussed in a separate paper.) 2010Mathematics Subject Classification. Primary 14C25; Secondary 14C15, 14C35.