##
**Degeneration of Hodge structures.**
*(English)*
Zbl 0617.14005

The aim of the paper is to describe the behaviour of Hodge structures for singular varieties. A variation of Hodge structure of weight k, as it arises from the geometric situation, consists of a flat complex vector bundle \(H\to M\) over a complex variety M, equipped with a flat \({\mathbb{Q}}\)-structure, and a decomposition \(H=\oplus_{p}H^{p,k-p}\) by \(C^{\infty}\)-subbundles, such that \(H^{p,k-p}\) is complex conjugate to \(H^{k-p,p}\). By assumption, the Hodge flags \(F^ p=\oplus_{j\geq p}H^{j,k-j}\) are holomorphic and satisfy the ”transversability relation” \(\nabla F^ p\subseteq F^{p-1}\). Moreover, the existence of a polarisation is required. (Both conditions hold whenever the variation arises from a family of projective varieties.) By a suitable resolution of singularities, the base space M lies as a Zariski-open subset in some completion \(\bar M\) such that \(\bar M-M\) is smooth except for normal crossings; then they shall characterize the possible singularities of the bundles \(F^ p\) along the subvariety \(\bar M-M\). The ”nilpotent orbit theorem” by W. Schmid [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] provides a first reduction of the problem: near any \(x\in \bar M-M\), the bundle \(F^ p\) have at most logarithmic singularities, when expressed in terms of a flat, multiple valued frame of H. Both H and its subbundles \(F^ p\) can be realized as the pullback of universal bundles over a complex homogeneous space D which classifies the Hodge structures on the typical fibre of H; the pullback is induced by Griffiths’ period map \(M\to \Gamma \setminus D\), which takes values in the quotient of the classifying space by \(\Gamma\), the image of \(\pi_ 1(M)\) via the monodromy action. The nilpotent orbit theorem approximates the period mapping near \(p\in \bar M-M\) by the projection into \(\Gamma\setminus D\) of an orbit of a nilpotent group in D. For variations depending depending on a single parameter, the approximating nilpotent orbits have been analyzed in the above cited paper of W. Schmid. According to the ”\(SL_ 2\)-orbit theorem”, these orbits are in turn asymptotic to equivariantly embedded copies of the upper half plane in D. Still in the one parameter case, the combination of the two orbit theorems gives a fairly explicit description of the singularities of the Hodge bundles. In particular, the singularities encode certain limiting mixed Hodge structures, whose existence had been predicted by Deligne (they have geometric significance in the projective case).

The principal result of the paper is an extension of the \(SL_ 2\)-orbit theorem to variations of Hodge structure in any number of variables. The starting point is a result conjectured by Deligne and proved by E. Cattani and E. Kaplan in Inventiones Math. 67, 101-115 (1982; Zbl 0516.14005) when \(p\in \bar M-M\) is approached along any direction transverse to the components of \(\bar M-M\), one obtains the same limiting mixed Hodge structure. After choosing an ordering of the variables, the given n-dimensional nilpotent orbit is viewed as an \((n-1)\)-parameter pencil of one dimensional nilpotent orbits; the constancy of the limiting mixed Hodge structure ensures that the approximating upper half-planes (equivalently unit discs) vary smoothly with the (n-1)-parameters. The origins in the resulting pencil of discs trace out a nilpotent orbit of dimension n-1. An inductive procedure enables the authors to construct an equivariantly embedded polydisc which approximates the nilpotent orbit on a suitable sector. The various possible orderings of the variables give rise, in general, to different polydisc embeddings approximating the orbit on each of n! overlapping sectors. The appearance of more than one polydisc reflects certain properties of the local monodromy.

In carrying out the proof of the several variables \(SL_ 2\)-orbit theorem, the authors need more detailed information on the one variable version that appears in the afore mentioned paper of Schmid. A constructruction of P. Deligne [appendix to the paper cited below] assigns to any mixed Hodge structure another one of a very special type, i.e. split over \({\mathbb{R}}\). As was shown by E. Cattani and A. Kaplan [”On the \(SL_ 2\)-orbits in Hodge theory”, Inst. Hautes Étud. Sci. (pre-publication, Oct. 1982)], the \(SL_ 2\)-orbit and various related data can be constructed algebraically from the limiting Hodge structure and its \({\mathbb{R}}\)-split relative. Here the authors systematically develop this approach to the one-variable \(SL_ 2\)-orbit theorem, which clarifies the dependence on parameters. Like the original proof, the arguments make use of a certain nonlinear system of differential equations.

The general setting of the problem is described in section 1, which also contains a sharper version of the nilpotent orbit theorem. Section 2 is devoted to the linear algebra of mixed Hodge structures, Deligne’s construction, and its interplay with \(sl_ 2\)-actions. The one-variable \(SL_ 2\)-orbit theorem is recalled in section 3, in the form needed for its generalization; the more technical arguments are postponed until section 6. The multi-variable \(SL_ 2\)-orbit theorem is stated and proved in section 4. Section 5 contains a particular application: the authors give estimates for the Hodge norms - or the size of cycles, in the geometric situation - and they show that the Chern forms of a variation of Hodge structure define currents on the completion \(\bar M\) of the parameter space M. Another application will be given by the authors [Invent. Math. 87, 217-252 (1987; Zbl 0611.14006)] proving that the \(L_ 2\) and intersection cohomologies of M with values in H are isomorphic.

The principal result of the paper is an extension of the \(SL_ 2\)-orbit theorem to variations of Hodge structure in any number of variables. The starting point is a result conjectured by Deligne and proved by E. Cattani and E. Kaplan in Inventiones Math. 67, 101-115 (1982; Zbl 0516.14005) when \(p\in \bar M-M\) is approached along any direction transverse to the components of \(\bar M-M\), one obtains the same limiting mixed Hodge structure. After choosing an ordering of the variables, the given n-dimensional nilpotent orbit is viewed as an \((n-1)\)-parameter pencil of one dimensional nilpotent orbits; the constancy of the limiting mixed Hodge structure ensures that the approximating upper half-planes (equivalently unit discs) vary smoothly with the (n-1)-parameters. The origins in the resulting pencil of discs trace out a nilpotent orbit of dimension n-1. An inductive procedure enables the authors to construct an equivariantly embedded polydisc which approximates the nilpotent orbit on a suitable sector. The various possible orderings of the variables give rise, in general, to different polydisc embeddings approximating the orbit on each of n! overlapping sectors. The appearance of more than one polydisc reflects certain properties of the local monodromy.

In carrying out the proof of the several variables \(SL_ 2\)-orbit theorem, the authors need more detailed information on the one variable version that appears in the afore mentioned paper of Schmid. A constructruction of P. Deligne [appendix to the paper cited below] assigns to any mixed Hodge structure another one of a very special type, i.e. split over \({\mathbb{R}}\). As was shown by E. Cattani and A. Kaplan [”On the \(SL_ 2\)-orbits in Hodge theory”, Inst. Hautes Étud. Sci. (pre-publication, Oct. 1982)], the \(SL_ 2\)-orbit and various related data can be constructed algebraically from the limiting Hodge structure and its \({\mathbb{R}}\)-split relative. Here the authors systematically develop this approach to the one-variable \(SL_ 2\)-orbit theorem, which clarifies the dependence on parameters. Like the original proof, the arguments make use of a certain nonlinear system of differential equations.

The general setting of the problem is described in section 1, which also contains a sharper version of the nilpotent orbit theorem. Section 2 is devoted to the linear algebra of mixed Hodge structures, Deligne’s construction, and its interplay with \(sl_ 2\)-actions. The one-variable \(SL_ 2\)-orbit theorem is recalled in section 3, in the form needed for its generalization; the more technical arguments are postponed until section 6. The multi-variable \(SL_ 2\)-orbit theorem is stated and proved in section 4. Section 5 contains a particular application: the authors give estimates for the Hodge norms - or the size of cycles, in the geometric situation - and they show that the Chern forms of a variation of Hodge structure define currents on the completion \(\bar M\) of the parameter space M. Another application will be given by the authors [Invent. Math. 87, 217-252 (1987; Zbl 0611.14006)] proving that the \(L_ 2\) and intersection cohomologies of M with values in H are isomorphic.

Reviewer: A.Brezuleanu

### MSC:

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14B05 | Singularities in algebraic geometry |