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Hodge theory of degenerations. I: Consequences of the decomposition theorem. (English) Zbl 1478.14024

The reviewed article is the first of a series of papers where the authors study the relationship between the asymptotic Hodge theory of a degeneration and the mixed Hodge theory of its singular fiber. The motivation to study this relationship comes from studying of compactifications of moduli spaces. The period map has been used to study moduli spaces of several examples, including abelian varieties, \(K3\) surfaces, hyper-Kähler manifolds, and cubic threefolds and fourfolds. In these examples, the period map embeds each moduli space as an open subset of a locally symmetric variety, which facilitates the comparison between Hodge-theoretic and geometric compactifications.
The authors are interested in extending the use of period maps in studying moduli of other examples, including surfaces of general type and Calabi-Yau threefolds, and establishing a similar strong connection between the compactifications. A key challenge is to compute the limiting mixed Hodge structures in the geometric boundary from the geometry of the fibers over this boundary.
With this goal in mind, the authors generalize the Clemens-Schmid sequence [C. H. Clemens, Duke Math. J. 44, 215–290 (1977; Zbl 0353.14005)] in various ways. The first of these is in the case of a projective family of varieties over a disk.
{Theorem 1.} Let \(f:\mathcal{X}\to\Delta\) be a flat projective family of varieties over the disk, which is the restriction of an algebraic family over a curve, such that \(f\) is smooth over \(\Delta*\). If \(\mathcal{X}\) is smooth, then we have exact sequences of mixed Hodge structures \[ 0 \to H^{k-2}_{\lim}(X_t)_T(-1) \to H_{2n-k+2}(X_0)(-n-1) \to H^k(X_0) \to H^k_{\lim}(X_t)^T\to 0\] for every \(k\in\mathbb{Z}\), where the outer terms are the coinvariants and the invariants of the monododromy operator \(T\) on the limiting mixed Hodge structure.
The authors obtain more precise information when considering the Hodge numbers \(h^{p,q}\) with \(pq = 0\), namely, that these numbers are preserved under such degenerations. The first isomorphism of the following theorem extends a result from Steenbrink [J. H. M. Steenbrink, Compos. Math. 42, 315–320 (1981; Zbl 0428.32017)], who proved it in the case when \(X_0\) has Du Bois singularities (using the result in [J. Kollár and S. Kovács, J. Am. Math. Soc. 23, No. 3, 791–813 (2010; Zbl 1202.14003 )].
{Theorem 2.} Let \(f:\mathcal{X}\to\Delta\) be a flat projective family of varieties over the disk, which is the restriction of an algebraic family over a curve, such that \(f\) is smooth over \(\Delta*\). Suppose that \(\mathcal{X}\) is normal and \(\mathbb{Q}\)-Gorenstein, and that the special fiber \(X_0\) is reduced.
1.
If \(X_0\) is semi-log-canonical, then \[ Gr^0_FH^k(X_0) \cong Gr^0_FH^k_{\lim}(X_t)\cong Gr^0_FH^k_{\lim}(X_t)^{T^{ss}}\] for all \(k\in\mathbb{Z}\), where \(T=T^{n}T^{ss}\) is the Jordan decomposition of the monodromy into the unipotent and (finite) semisimple parts.
2.
If \(X_0\) is log-terminal, then additionally \[W_{k-1}Gr^0_FH^k_{\lim}(X_t) = \{0\}\] for all \(k\in\mathbb{Z}\).

The authors obtain a similar result one step higher in the Hodge filtration under stronger assumptions.
{Theorem 3.} Let \(f:\mathcal{X}\to\Delta\) be as in Theorem 2. Assume that the total space \(\mathcal{X}\) is smooth and the special fiber \(X_0\) is log-terminal (or more generally, has rational singularities). Then \[Gr^1_FH^k(X_0) \cong Gr^1_F(H^k(X_t))^{T^{ss}}.\]
The main technique used in the reviewed article is the theory of mixed Hodge modules from M. Saito [M. Saito, Publ. Res. Inst. Math. Sci. 26, No. 2, 221–333 (1990; Zbl 0727.14004)], and especially the Decomposition Theory in this setting. The first part of the article reviews the result, with an emphasis on the Decomposition Theorem over a curve. In addition, several concrete geometric examples are discussed in Section 6.
The article ends with an Appendix by the third author, which discusses the Decomposition Theorem over a curve in the context of analytic spaces and includes the following general equivalence result.
{Theorem A.} Let \(f:X\to C\) be a proper surjective morphism of a connected complex manifold \(X\) to a connected non-compact curve \(C\). The Decomposition Theorem for \(\mathbf{R} f_*\mathbb{Q}_X\) is equivalent to the Clemens-Schmid exact sequence (or the local invariant cycle theorem) for every singular fiber of \(f\).
A condition for the Decomposition Theorem to hold in the context of Theorem 4 is described in terms of a resolution of singularities of \(X\) (Corollary A).

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
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