##
**Algebraic cycles on degenerate fibers.**
*(English)*
Zbl 0955.14007

Catanese, Fabrizio (ed.), Arithmetic geometry. Proceedings of a symposium, Cortona, Arezzo, Italy, October 16-21, 1994. Cambridge: Cambridge University Press. Symp. Math. 37, 45-69 (1997).

From the introduction: In this paper we extend part of the theory of degeneration of Hodge structures to algebraic cycles.

The theory of limiting mixed Hodge structures leads to a proof of the local invariant cycle theorem for proper families of complex manifolds over the unit disk with semi-stable degeneration at the origin. In the proof of this result, a key role is played by the spectral sequence coming from the weight filtration on the complex of nearby cycles. One can compute the \(E_1\) term and the first differential \(d_1\) of this spectral sequence. One finds that \((E_1,d_1)\) is isomorphic to a complex \((K,d)\) where \(K\) is a sum of cohomology groups of the different strata of the special fiber, and the differential \(d\) is defined by means of the corestriction and Gysin morphisms relating these cohomology groups. Since the weight spectral sequence degenerates from \(E_2\) on, one can prove properties of the limit Hodge structure on the cohomology of the general fiber from this computation of \((E_1,d_1)\) and the fact that each stratum is a compact Kähler manifold.

Our basic remark is that the complex \((K,d)\) makes sense in a more general set up, for most cohomology theories and also for algebraic cycles modulo any adequate equivalence relation, given any principal reduced Cartier divisor on a scheme instead of a family of complex manifolds, even though, in general, the abutment of the weight spectral sequence needs not be defined. Furthermore, one can reproduce the proof of the local invariant cycle theorem in this more general setting, as soon as each stratum of the special fibre satisfies both the hard Lefschetz theorem and the Hodge index theorem.

The main motivation for our study came from our attempt to develop a non-archimedean analog of Arakelov theory [cf. S. Bloch, H. Gillet and C. Soulé, J. Algebr. Geom. 4, No. 3, 427-485 (1995; Zbl 0866.14011)]. If \({\mathcal X}\) is a regular proper scheme over a discrete valuation ring with special fiber \(Y\) a reduced divisor with normal crossings, we may consider Chow homology groups \(\text{CH}_p(Y)\) and Fulton’s Chow cohomology groups \(\text{CH}^p(Y)\), \(p\geq 0\). The inclusion \(i:Y \to X\) induces a group morphism \(i^*i_*\) from \(\text{CH}_p(Y)\) to \(\text{CH}^{ \dim ({\mathcal X})-p}(Y)\), and it was shown in the cited paper that, when resolution of singularities holds, both the kernel and the cokernel of \(i^*i_*\) depend only on the generic fiber \({\mathcal X}-Y\) and not on the model \({\mathcal X}\). We then raised the question of whether \(\text{ker} (i^*i_*)\) and \(\text{coker} (i^*i_*)\) could be isomorphic in appropriate degrees, the same way \(\partial\overline \partial\)-cohomology groups of Kähler manifolds coincide with the usual cohomology. Here, we prove that it is indeed the case when all strata satisfy the hard Lefschetz and Hodge index theorem (theorem 5).

The paper is organized as follows. In section one, under general assumptions, we define a bigraded group \(K''\) with two differentials \(d'\) and \(d''\) and a “monodromy” operator \(N\), which is zero or the identity map on direct summands of \(K''\). In section two, we define using \(K''\) several cohomology groups. In section 3, we assume that all strata satisfy the hard Lefschetz and Hodge index theorems, and, we deduce that the cohomology of \((K'',d)\) has the structure of a bigraded polarized Hodge-Lefschetz module in the sense of Deligne and Saito (theorem 1). This implies analogs of the fact that weight and monodromy filtrations coincide (theorem 2), of the local invariant cycle theorem (theorem 3) and of the Clemens-Schmidt exact sequence (theorem 4). As a consequence, we prove in theorem 5 and 6 that the kernel and cokernel of \(i^*i_*\) coincide and enjoy properties similar to the cohomology of compact Kähler manifolds. Finally, we discuss in section 5 possible relations of our work with motivic cohomology.

Note that H. Gillet and C. Soulé [J. Reine Angew. Math. 478, 127-176 (1996; Zbl 0863.19002)] found recently another manifestation of the motivic nature of the weight spectral sequence: For any variety \(X\) over a field of characteristic zero, the \(E_1\)-term of the weight spectral sequence converging to the cohomology with compact supports can be computed from a complex of pure Chow motives canonically attached to \(X\) (up to homotopy). It would be of interest to express the results in the present paper in terms of these complexes of motives.

For the entire collection see [Zbl 0864.00054].

The theory of limiting mixed Hodge structures leads to a proof of the local invariant cycle theorem for proper families of complex manifolds over the unit disk with semi-stable degeneration at the origin. In the proof of this result, a key role is played by the spectral sequence coming from the weight filtration on the complex of nearby cycles. One can compute the \(E_1\) term and the first differential \(d_1\) of this spectral sequence. One finds that \((E_1,d_1)\) is isomorphic to a complex \((K,d)\) where \(K\) is a sum of cohomology groups of the different strata of the special fiber, and the differential \(d\) is defined by means of the corestriction and Gysin morphisms relating these cohomology groups. Since the weight spectral sequence degenerates from \(E_2\) on, one can prove properties of the limit Hodge structure on the cohomology of the general fiber from this computation of \((E_1,d_1)\) and the fact that each stratum is a compact Kähler manifold.

Our basic remark is that the complex \((K,d)\) makes sense in a more general set up, for most cohomology theories and also for algebraic cycles modulo any adequate equivalence relation, given any principal reduced Cartier divisor on a scheme instead of a family of complex manifolds, even though, in general, the abutment of the weight spectral sequence needs not be defined. Furthermore, one can reproduce the proof of the local invariant cycle theorem in this more general setting, as soon as each stratum of the special fibre satisfies both the hard Lefschetz theorem and the Hodge index theorem.

The main motivation for our study came from our attempt to develop a non-archimedean analog of Arakelov theory [cf. S. Bloch, H. Gillet and C. Soulé, J. Algebr. Geom. 4, No. 3, 427-485 (1995; Zbl 0866.14011)]. If \({\mathcal X}\) is a regular proper scheme over a discrete valuation ring with special fiber \(Y\) a reduced divisor with normal crossings, we may consider Chow homology groups \(\text{CH}_p(Y)\) and Fulton’s Chow cohomology groups \(\text{CH}^p(Y)\), \(p\geq 0\). The inclusion \(i:Y \to X\) induces a group morphism \(i^*i_*\) from \(\text{CH}_p(Y)\) to \(\text{CH}^{ \dim ({\mathcal X})-p}(Y)\), and it was shown in the cited paper that, when resolution of singularities holds, both the kernel and the cokernel of \(i^*i_*\) depend only on the generic fiber \({\mathcal X}-Y\) and not on the model \({\mathcal X}\). We then raised the question of whether \(\text{ker} (i^*i_*)\) and \(\text{coker} (i^*i_*)\) could be isomorphic in appropriate degrees, the same way \(\partial\overline \partial\)-cohomology groups of Kähler manifolds coincide with the usual cohomology. Here, we prove that it is indeed the case when all strata satisfy the hard Lefschetz and Hodge index theorem (theorem 5).

The paper is organized as follows. In section one, under general assumptions, we define a bigraded group \(K''\) with two differentials \(d'\) and \(d''\) and a “monodromy” operator \(N\), which is zero or the identity map on direct summands of \(K''\). In section two, we define using \(K''\) several cohomology groups. In section 3, we assume that all strata satisfy the hard Lefschetz and Hodge index theorems, and, we deduce that the cohomology of \((K'',d)\) has the structure of a bigraded polarized Hodge-Lefschetz module in the sense of Deligne and Saito (theorem 1). This implies analogs of the fact that weight and monodromy filtrations coincide (theorem 2), of the local invariant cycle theorem (theorem 3) and of the Clemens-Schmidt exact sequence (theorem 4). As a consequence, we prove in theorem 5 and 6 that the kernel and cokernel of \(i^*i_*\) coincide and enjoy properties similar to the cohomology of compact Kähler manifolds. Finally, we discuss in section 5 possible relations of our work with motivic cohomology.

Note that H. Gillet and C. Soulé [J. Reine Angew. Math. 478, 127-176 (1996; Zbl 0863.19002)] found recently another manifestation of the motivic nature of the weight spectral sequence: For any variety \(X\) over a field of characteristic zero, the \(E_1\)-term of the weight spectral sequence converging to the cohomology with compact supports can be computed from a complex of pure Chow motives canonically attached to \(X\) (up to homotopy). It would be of interest to express the results in the present paper in terms of these complexes of motives.

For the entire collection see [Zbl 0864.00054].

### MSC:

14C25 | Algebraic cycles |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

14F42 | Motivic cohomology; motivic homotopy theory |

14C15 | (Equivariant) Chow groups and rings; motives |