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Roitman's Theorem for Singular Projective Varieties Compositio Mathematica 119: 213–237, 1999. 213 © 1999 Kluwer Academic Publishers. Printed in the Netherlands. Roitman’s Theorem for Singular Projective Varieties J. BISWAS and V. SRINIVAS School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India. e-mail: [email protected], [email protected] (Received: 9 January 1998; accepted in revised form: 26 August 1998) Abstract. We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties. Mathematics Subject Classification (1991): 14C025. Key words: 0-cycle, Roitman theorem, Lefschetz theorem. 1. Introduction If X is a smooth complete variety of dimension n over the complex field C,there n is a natural map CH (X)deg 0 → Alb (X) from the Chow group of 0-cycles of degree 0 (modulo rational equivalence) to the Albanese variety of X. It is known 0 i 6= > that this is surjective, and has a ‘large’ kernel if H (X, X/C) 0forsomei 2. n A famous result of Roitman [R] asserts that CH (X)deg 0 → Alb (X) is always an isomorphism on torsion subgroups. Our aim in this paper is to generalize this theorem to reduced projective varieties with arbitrary singularities. We begin with some background. Let X be any reduced projective variety of dimension n over C.LetXsing (the singular locus) denote the set of points x ∈ X such that the module of Kähler differentials 1 is not a free module of rank n; Ox,X/C thus Xsing includes points x through which X has a component of dimension <n. For Y ⊂ X a closed subvariety containing Xsing, Levine and Weibel [LW] have defined the relative Chow (cohomology) group of 0-cycles CHn(X, Y ) to be the quotient of the free Abelian group on points of X − Y modulo a suitable notion of rational equivalence (given by the subgroup of cycles of the form (f )C where C is a ‘Cartier curve’ on (X, Y ) and f a rational function on C which is a unit at points of C ∩ Y ). If Y has codimension > 2, the relations may also be given by sums of cycles (f )C,whereC is any curve which does not meet Y .IfX is integral and Y has codimension > 2, Levine [L3] and Collino [Co] have shown that Roitman’s theorem holds for CHn(X, Y ),ifwetakeAlbX to be the Albanese variety of any resolution of singularities of X. Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 02:00:48, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001793226084 214 J. BISWAS AND V. SRINIVAS n n In this paper, we will take CH (X) to be the group CH (X, Xsing) as defined in [LW] (see also [L2]). In a recent paper [BPW], the authors show that if X is a projective surface, then the Albanese variety of X,nowdefined to be the group H 3(X, C) J 2(X) := , F 2H 3(X, C) + im H 3(X, Z) is in fact a semi-Abelian variety (here F 2 denotes the subspace for the Hodge filtration of the mixed Hodge structure). They construct a natural homomorphism CH2(X)→→Z⊕t ,whereX has t irreducible components of dimension 2; the kernel 2 2 is denoted by CH (X)deg 0. They then construct a homomorphism CH (X)deg 0 → J 2(X), using simplicial techniques and Chern classes with values in Deligne co- homology. The main theorem of [BPW] is that this map is an isomorphism on torsion subgroups. Their proof is technically quite complicated, and uses partic- ular features of the two-dimensional case, like the relationship with K2,which implies (via excision, double-relative K-groups, etc.) a formula for the Chow group in terms of that of its normalization, and a quotient of SK1 groups of possibly nonreduced curves. This motivates our main result. THEOREM 1.1. Let X be a reduced projective variety over C of dimension n. (a) Define H 2n−1(X, C) J n(X) := . F nH 2n−1(X, C) + imH 2n−1(X, Z) Then J n(X) is the group of closed points of a semi-Abelian variety over C (which we again denote by J n(X)). The association X 7→ J n(X) is a contrav- ariant functor from n-dimensional projective varieties over C to semi-Abelian varieties over C. n → Z⊕t (b) ThereisadegreemapdegX: CH (X) , where X has t irreducible n components of dimension n.LetCH (X)deg 0 denote the kernel of degX.Then n n → n = there is a surjective Abel–Jacobi map AJX: CH (X)deg 0 J (X). If U X0 − Xsing,whereX0 is any n-dimensional component of X, and x0 ∈ U is a 7→ n [ ]−[ ] base point, then the map x AJX( x x0 ), defined on closed points of U, is induced by a morphism U → J n(X). (c) If Y ⊂ X is a reduced subscheme of dimension m which fits into a chain of subschemes Y = Ym ⊂ Ym+1 ⊂ ··· ⊂ Yn = X, such that Yj is a reduced Cartier divisor on Yj+1 for each j<n, then there is a commutative diagram m - n CH (Y )deg 0 CH (X)deg 0 AJ m AJ n Y ? X ? J m(Y ) - J n(X) Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 02:00:48, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001793226084 ROITMAN’S THEOREM FOR SINGULAR PROJECTIVE VARIETIES 215 (where CHm(Y ) → CHn(X) is induced by the obvious map on 0-cycles de- termined by the inclusion Y ⊂ X). The map J m(Y ) → J n(X) is independent of the particular chain {Ym,Ym+1,...} chosen. (d) (Lefschetz theorem) If in (c) above, Y is a general complete intersection of very ample divisors in X,thenJ m(Y ) → J n(X) is an isomorphism if dim Y = m > 2, and is surjective on N-torsion for each N,ifm = 1 (in particular, J 1(Y )→→J n(X) in this case). n (e) (Roitman theorem) The map AJX is an isomorphism on torsion subgroups. Here part (d) is obtained as a corollary of the following version of the Lef- schetz Hyperplane Theorem, valid for singular projective varieties, which may be of independent interest (see Section 4.1 below). Part (a) (‘without base points’), formulated for cohomology with C-coefficients, has been obtained earlier [GNPP]; we thank the referee for providing us this reference. The rather technical statement made in (b) is needed in the proof of the Roitman Theorem (part (e) of the Main Theorem). Following the referee’s suggestions, the two separate ‘Lefschetz Theor- ems’ stated in an earlier version of our paper are now combined. LEFSCHETZ THEOREM (a) Let X be a reduced projective variety over C of dimension n and let Y be a general hyperplane section. Then the the Gysin map H i(Y, Z(j)) → H i+2(X, Z(j + 1)) is an isomorphism for i<dim Y and surjective for i = dim Y . (b) Let X be as above, π: Xe → X the normalization, and A ⊂ X a closed subvariety such that if U = X − A and V = Xe − π −1(A), the following are satisfied: (i) V is nonsingular of dimension n, (ii) V → U is the normalization of U, −1 (iii) if W = π (Using), then Using and W are nonsingular of dimension n − 1 and W → Using is an analytic covering space. Let C ⊂ X be a reduced curve such that (i0) C is a local complete intersection in X. 0 (ii ) C ∩ Xsing is reduced, and supported at smooth points of Xsing. 0 (iii ) C ∩ (X − Xsing) has only plane curve singularities. (iv0) C ∩ A =∅. Let Y be a general hypersurface section of X of sufficiently large degree, and which contains C. Then the Gysin map H i(Y, Z(j)) → H i+2(X, Z(j + 1)) is an isomorphism for i<dim Y and surjective for i = dim Y . Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 02:00:48, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001793226084 216 J. BISWAS AND V. SRINIVAS Thus, there are 3 main results in the paper: the construction of the Abel–Jacobi map on rational equivalence classes for arbitrary dimensional varieties, and the proofs of the Lefschetz theorem (with Z-coefficients, and with base conditions) and the Roitman theorem. We comment further on the proofs. The Abel–Jacobi map is constructed on the level of 0-cycles using Deligne’s 1-motives [D] (or equivalently, extension classes for mixed Hodge structures [C]). To show that rational equivalence is preserved, we use a moving lemma for Cartier curves, together with functoriality properties of mixed Hodge structures with re- spect to point blow-ups, pullbacks and Gysin maps. The discussion of the functori- ality for the Gysin maps uses M. Saito’s theory [MS] of Mixed Hodge Modules, to give a clean exposition, though ad hoc (messy) arguments of a more elementary type are possible in the cases at hand.
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