## Boundaries of cycle spaces and degenerating Hodge structures.(English)Zbl 1310.32017

The aim of the work is to interpret the boundary points of the Satake compactifications $$\Gamma \setminus \mathcal{H}_S$$ and the toroidal compactifications $$\Gamma \setminus \mathcal{H} _{\Sigma}$$ of arithmetic quotients $$\Gamma \setminus \mathcal{H}$$ of the Siegel space $$\mathcal{H} = \text{Sp} (n, {\mathbb R}) / \text{U}(n)$$ by limiting mixed Hodge structures. This is accomplished by extending the natural projection $$D = \text{Sp} (n, {\mathbb R}) / \prod _j \text{U}(n_j) \rightarrow \mathcal{H}$$, $$\sum _j n_j = n$$, of an odd weight period domain $$D$$ to a map $$\Gamma \setminus D_{\Sigma} \rightarrow \Gamma \setminus \mathcal{H}_S$$ from Kato-Usui’s partial compactification $$\Gamma \setminus D_{\Sigma}$$, associated with an appropriate fan $$\Sigma$$ or to a map $$\Gamma \setminus D_{\Sigma} \rightarrow \overline{\Gamma \setminus \mathcal{H}_S}$$ to the complex conjugate $$\overline{\Gamma \setminus \mathcal{H}_S}$$ of $$\Gamma \setminus \mathcal{H}_S$$. The maps in question are factored through the toroidal compactifications $$\Gamma \setminus \mathcal{H}_{\Sigma}$$ and their complex conjugates $$\overline{\Gamma \setminus \mathcal{H}_{\Sigma}}$$, respectively. Bearing in mind that the Satake and the toroidal compactifications of $$\Gamma \setminus \mathcal{H}$$ are studied extensively, the constructions are expected to shed a light on the geometry of Kato-Usui’s compactifications $$\Gamma \setminus D_{\Sigma}$$. In the case of $$D = \text{Sp} (2, {\mathbb R}) / \text{U}(1) \times \text{U}(1)$$, the discussed correspondence is shown to be continuous.
In order to sketch the construction, recall that $$D$$ classifies the polarized Hodge structures with Hodge numbers $$n_j$$ on the complexification $$H_{\mathbb C}$$ of a lattice $$H_{\mathbb Z}$$. Let us identify the points of $$D$$ with their corresponding Hodge filtrations $$F$$ of $$H_{\mathbb C}$$, shift the weight to $$-1$$ and restrict to mixed Hodge structures on the points of the compact dual $$\check{D}$$, which arise from a two-step nilpotent element $$N \in \mathfrak{g}_{\mathbb C}$$. The restriction $$N^2=0$$ is quite natural for the set up of the problem, as far as the Satake and the toroidal compactifications of $$\mathcal{H}$$ and $$\Gamma \setminus \mathcal{H}$$, $$\Gamma \leq G_{\mathbb Z} \simeq \text{Sp} (n, {\mathbb Z})$$, are related to the parabolic subgroups $$P$$ of $$G$$ and the unipotent radicals of $$P$$ are two-step nilpotent. The assumption $$N^2 =0$$ provides a monodromy weight filtration $$W(N) _0 = H_{\mathbb R} \supseteq W(N) _{-1} = \text{Ker} N \supseteq W(N) _{-2} = \text{Im} N$$. Under appropriate restrictions, the pair $$(W(N), F)$$ is a limiting mixed Hodge structure and admits the Deligne decomposition $$H_{\mathbb C} = \bigoplus _{p+q = 0,-1,-2} I^{p,q}$$. For any $$z \in {\mathbb C}$$ with $$\text{Im} (z) >0$$, the author shows the existence of a decomposition $$H^{p, -p-1} = I^{p,-p-1} \oplus \exp (zN) I^{p,-p} \oplus \exp ( \overline{z} N) I^{p+1,-p-1}$$. Denote by
${\mathbf n} _{-} = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right), \;\;{\mathbf h} = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right), \;\;{\mathbf n} _{+} = \left( \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right)$
the standard generators of the Lie algebra $$\text{sl} (2, {\mathbb C})$$. According to the $$\text{SL}(2)$$-orbit theorem, for any $$N \in \mathfrak{g}_{\mathbb C}$$ with $$N^2 = 0$$ there is a Lie group homomorphism $$\rho : \text{SL} (2, {\mathbb C}) \rightarrow G_{\mathbb C} = \text{Sp} (n, {\mathbb C})$$ defined over $${\mathbb R}$$ and a $$\rho$$-equivariant holomorphic map $$\phi : {\mathbb P} ^1 ( {\mathbb C}) \rightarrow \check{D}$$ such that $$(W(N), \phi (0) = \widehat{F})$$ is an $${\mathbb R}$$-split limiting mixed Hodge structure, $$\rho _* : \text{sl}(2, {\mathbb C}) \rightarrow \mathfrak{g}_{\mathbb C}$$ is a $$(0,0)$$-morphism, $$\rho _* ( {\mathbf n} _{-}) = N$$ and $$\rho _* ( {\mathbf h}) v = (p+q+1)v$$ for all $$v \in I^{p,q}$$.
The article establishes that $$X := \frac{1}{2} ( i \rho _* ( {\mathbf n} _{-} ) - \rho _* ( {\mathbf h}) + i \rho _* ( {\mathbf n} _{+} ) ) \in \mathfrak{g} ^{-1,1} _{\mathbb C}$$ is of weight $$(-1,1)$$ with respect to the reference point $$F_0 = \exp (iN) \widehat{F}$$ of $$D$$ and $$X \exp (iN) v = - \text{exp} ( - iN) v$$ for all $$v \in I^{p,-p}$$. If $$K_0 \simeq U(n)$$ is the maximal compact subgroup of $$G = \text{Sp} (n, {\mathbb R})$$ containing the stabilizer of $$F_0$$, then the $$K_0$$-orbit $$C_0$$ of $$F_0$$ is a compact submanifold of $$D$$. The cycle space $$\mathcal{M}_{\check{D}}$$ of the compact dual $$\check{D}$$ of $$D$$ consists of the translates $$gC_0$$ of $$C_0$$ by all $$g \in G_{\mathbb C}$$. The cycle space $$\mathcal{M}_D$$ of the period domain $$D$$ contains only those $$gC_0 \in \mathcal{M}_{\check{D}}$$ which lie in $$D$$. The space $$\mathcal{M}_D$$ is known to be isomorphic to $$\mathcal{B} \times \overline{\mathcal{B}}$$, where $$\mathcal{B}$$ stands for the Siegel space $$\mathcal{H} = \text{Sp} (n, {\mathbb R}) / U(n)$$, viewed as a bounded symmetric domain. The article shows that $$\exp (X) \in G_{\mathbb C}$$ maps the reference point $$C_0 \in \mathcal{M}_D$$ into the point $$\exp (X) C_0 \in \mathcal{M}_{\check{D}}$$ from the topological closure $$\mathcal{M}_D ^{\text{cl}} \simeq \mathcal{B}^{\text{cl}} \times \overline{{\mathcal{B}}^{\text{cl}}}$$ of $$\mathcal{M}_D$$ in $$\mathcal{M}_{\check{D}}$$. This relates the boundary of the cycle space $$\mathcal{M}_D$$ with the limiting mixed Hodge structure $$(W(N), \widehat{F})$$ associated with $$N$$ and $$X$$.
A nilpotent cone $$\sigma$$ in $$\mathfrak{g}$$ is a finitely generated rational polyhedral cone $$\sigma = \sum _{j=1} ^n {\mathbb R}_{\geq 0} N_j \subset \mathfrak{g}_{\mathbb R}$$, whose elements are nilpotent and commute with each other. For $$F \in \check{D}$$ the pair $$( \sigma, \exp ( \sigma _{\mathbb C}) F)$$ is a $$\sigma$$-nilpotent orbit if $$N F^p \subset F^{p-1}$$ for all $$N \in \sigma$$ and $$\text{exp} (zN) F \in D$$ for $$z \in {\mathbb C}$$ with sufficiently large $$\text{Im} z >0$$ and all $$N$$ from the relative interior $$\sigma ^{o}$$ of $$\sigma$$. The monodromy weight filtration $$W(N)$$ on $$F$$ is independent of the choice of $$N \in \sigma$$ and denoted by $$W( \sigma )$$. A nilpotent orbit $$( \sigma, \exp ( \sigma _{\mathbb C}) F)$$ is of even type (respectively, of odd type) if $$N^2 =0$$ for all $$N \in \sigma ^{o}$$ and $$I^{p,-p} =0$$ for all odd integers (respectively, $$I^{p,-p} =0$$ for all even integers $$p$$) with respect to the limiting mixed Hodge structure $$(W( \sigma), F)$$. A nilpotent cone $$\sigma$$ is of even type (respectively, of odd type) if all $$\sigma$$-nilpotent orbits are of even type (respectively, of odd type). A fan $$\Sigma$$ is of even type (respectively, of odd type) if any face of $$\Sigma$$ is an even type (respectively, an odd type) nilpotent cone. Let $$( \sigma, \exp ( \sigma _{\mathbb C}) F)$$ be a nilpotent orbit and $$C_0$$ be the orbit of $$F$$ under the maximal compact subgroup $$K_0$$ of $$G = \text{Sp} (n, {\mathbb R})$$ containing the stabilizer of $$F$$. Any $$N \in \sigma^{o}$$ is associated with $$X(N) \in \mathfrak{g} ^{-1,1}$$ by means of the $$\text{SL}(2)$$-orbit theorem. The article establishes that a basic feature of the nilpotent orbits $$(\sigma, \exp ( \sigma _{\mathbb C} ) F)$$ of even type and odd type is $$\exp (X(N)) C_0 \subset \mathcal{B}^{\text{cl}}$$ and $$\exp (X(N)) C_0 \subset \overline{\mathcal{B}}^{\text{cl}}$$, respectively, for all $$N \in \sigma ^{o}$$. Moreover, $$\exp (X(N)) C_0$$ is shown to be independent of $$N \in \sigma ^{o}$$ and $$F \in \exp (\sigma _{\mathbb C})F$$. Let $${\mathbf B}_S( \sigma)$$ be the Satake boundary component of $$\mathcal{B} \simeq \text{Sp} (n, {\mathbb R}) / \text{U}(n)$$ corresponding to the real totally isotropic subspace $$W( \sigma) _{-2} = \text{Im} N$$, $$N \in \sigma ^{o}$$, and let $${\mathbf B} ( \sigma)$$ be the set of all $$\sigma$$-nilpotent orbits for a nilpotent cone $$\sigma$$ of even type or of odd type. The aforementioned properties of $$\exp (X(N)) C_0$$ imply the presence of well defined maps $${\mathbf B} ( \sigma) \rightarrow {\mathbf B} _S( \sigma)$$ and $${\mathbf B} ( \sigma) \rightarrow \overline{{\mathbf B} _S ( \sigma )}$$, respectively. Thus, for any fan $$\Sigma$$ of even type or odd type, there arise correspondences $$p^{\text{ev}} : D_{\Sigma} := \coprod _{\sigma \in \Sigma} {\mathbf B} ( \sigma) \rightarrow \mathcal{H} _S$$ and $$p^{\text{od}}: D_{\Sigma} \rightarrow \overline{\mathcal{H}_S}$$, respectively, in the Satake compactification $$\mathcal{H}_S$$ of $$\mathcal{H} = \text{Sp} (n, {\mathbb R}) / U(n)$$ or in its complex conjugate $$\overline{\mathcal{H}_S}$$. A subgroup $$\Gamma$$ of $$G_{\mathbb Z} = \text{Sp} (n, {\mathbb Z})$$ is compatible with a fan $$\Sigma$$ of nilpotent cones $$\sigma \subset \mathfrak{g}_{\mathbb R}$$ if $$\Sigma$$ is invariant under the adjoint action of $$\Gamma$$. If $$\Gamma$$ is compatible with a fan $$\Sigma$$ of even type or of odd type, then there are maps $$p^{\text{ev}} : \Gamma \setminus D_{\Sigma} \rightarrow \Gamma \setminus \mathcal{H} _S$$ and $$p^{\text{od}} : \Gamma \setminus D_{\Sigma} \rightarrow \overline{\Gamma \setminus \mathcal{H}_S}$$, respectively, in the Satake compactification $$\Gamma \setminus \mathcal{H}_S$$ of $$\Gamma \setminus \mathcal{H}$$ or in its complex conjugate $$\overline{\Gamma \setminus \mathcal{H}_S}$$.
Let $$( \sigma , \exp ( \sigma _{\mathbb C}) \widehat{F})$$ be an $${\mathbb R}$$-split nilpotent orbit and $$H_{\mathbb C} = \bigoplus _{p +q = 0, -1,-2} I^{p,q}$$ be the Deligne decomposition with respect to the limiting mixed Hodge structure $$(W( \sigma), \widehat{F})$$. If $$( \sigma , \exp ( \sigma _{\mathbb C}) \widehat{F})$$ is of even type, consider the point $$\widetilde{F} ^{o} = \left( \bigoplus _{ p: \, \text{even} } I^{p, -p-1} \right) \oplus \left( \bigoplus _p I^{p,-p} \right)$$ of the compact dual $$\check{\mathcal{H}}$$ of $$\mathcal{H}$$. For $$(\sigma , \exp ( \sigma _{\mathbb C} ) \widehat{F})$$ of odd type put $$\widetilde{F}^{o} = \left( \bigoplus _{ p: \, \text{odd}} I^{p, -p-1} \right) \oplus \left( \bigoplus _p I^{p,-p} \right) \in \check{\mathcal{H}}$$. After showing that $$p^{\text{ev}} (\exp (zN) \widehat{F}) = \exp (zN) \widetilde{F}$$ and $$p^{\text{od}} ( \exp ( \overline{z} N) \widehat{F}) = \exp ( \overline{z} N) \widetilde{F}$$, respectively, for all $$z \in {\mathbb C}$$ with $$\text{Im} z >0$$ and all $$N \in \sigma ^{o}$$, the author establishes that $$p^{\text{ev}} : \Gamma \setminus D_{\Sigma} \rightarrow \Gamma \setminus \mathcal{H} _S$$ and $$p^{\text{od}} : \Gamma \setminus D_{\Sigma} \rightarrow \overline{\Gamma \setminus \mathcal{H}_S}$$, respectively, factor through maps $$\widetilde{p} ^{\text{ev}} : \Gamma \setminus D_{\Sigma} \rightarrow \Gamma \setminus \mathcal{H}_{\Sigma}$$ and $$\widetilde{p} ^{\text{od}} : \Gamma \setminus D_{\Sigma} \rightarrow \overline{\Gamma \setminus \mathcal{H}_{\Sigma}}$$, respectively, in the toroidal compactification $$\Gamma \setminus \mathcal{H}_{\Sigma}$$ of $$\Gamma \setminus \mathcal{H}$$ associated with $$\Sigma$$ and in its complex conjugate $$\overline{\Gamma \setminus \mathcal{H}_{\Sigma}}$$, respectively.
In the case of $$D = \text{Sp} (2, {\mathbb R}) / \text{U}(1) \times \text{U}(1)$$, let us fix a basis $$e_1, \dots , e_4$$ of $$H_{\mathbb Z}$$ with $( \langle e_j, e_k \rangle )_{j,k} = \left( \begin{matrix} 0 & - I_2 \\ I_2 & 0 \end{matrix} \right),$ where $$I_2$$ stands for the unit $$2 \times 2$$-matrix. If $$\sigma = {\mathbb R}_{\geq 0} N$$ for some $$N \in G_{\mathbb Z} = \text{Sp} (2, {\mathbb Z})$$ with $$N(e_3) = e_1$$ and $$N(e_j) =1$$ for $$j = 1,2,4$$, then $$\Sigma _{\text{ev}} := \{ \text{Ad} (g) \sigma \;\;| \;\;g \in G_{\mathbb Z} \}$$ is the fan of all nilpotent cones of even type. The article shows that for any subgroup $$\Gamma < G_{\mathbb Z}$$ containing the generators of the cones from $$\Sigma _{\text{ev}}$$, the maps $$\widetilde{p} ^{\text{ev}} : \Gamma \setminus D_{\Sigma _{\text{ev}}} \rightarrow \Gamma \setminus \mathcal{H}_{\Sigma _{\text{ev}}}$$ and $$p^{\text{ev}} : \Gamma \setminus D_{\Sigma _{\text{ev}}} \rightarrow \Gamma \setminus \mathcal{H}_S$$ are continuous. This is done by explicit descriptions of neighborhoods of the boundary points $$(\sigma, \text{exp} ( \sigma _{\mathbb C}) \widetilde{F}) \in \Gamma \setminus \mathcal{H} _{\Sigma _{\text{ev}}}$$ and $$(\sigma, \text{exp} ( \sigma _{\mathbb C}) F) \in \Gamma \setminus D_{\Sigma _{\text{ev}}}$$. Similarly, the fan $$\Sigma _{\text{od}}$$ of all nilpotent cones of odd type consists of $$Ad (g) \sigma _m$$, where $$g \in G_{\mathbb Z}$$ and $$\sigma _m = {\mathbb R} _{\geq 0} N_m$$ for some square-free positive integer $$m$$, $$N_m \in \mathfrak{g}_{\mathbb C}$$, $$N_m ^2 =0$$, $$N_m (e_3) = -e_1$$, $$N_m ( e_4) = - m e_2$$, $$N_m (e_1) = N_m (e_2) =0$$. If the generators of all the cones from $$\Sigma _{\text{od}}$$ belong to a subgroup $$\Gamma$$ of $$G_{\mathbb Z}$$, then $$\widetilde{p} ^{\text{od}} : \Gamma \setminus D_{\Sigma _{\text{od}}} \rightarrow \overline{\Gamma \setminus \mathcal{H}_{\Sigma _{\text{od}}}}$$ and $$p^{\text{od}} : \Gamma \setminus D_{\Sigma _{\text{od}}} \rightarrow \overline{\Gamma \setminus \mathcal{H}_S}$$ are shown to be continuous.

### MSC:

 32G20 Period matrices, variation of Hodge structure; degenerations 14D07 Variation of Hodge structures (algebro-geometric aspects)
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