##
**Compactifications of smooth families and of moduli spaces of polarized manifolds.**
*(English)*
Zbl 1238.14009

Let \(M_{h}\) be the moduli scheme of canonically polarized manifolds with Hilbert polynomial \(h\). One constructs in this paper for \(\nu\geq2\) with \(h(\nu)>0\) a projective compactification \(\overline{M}_{h}\) of the reduced moduli scheme \((M_{h})_{\mathrm{red}}\) such that the ample invertible sheaf \(\lambda_{\nu}\), corresponding to \(\det(f_{*}\omega_{X_{0}/Y_{0}}^{\nu})\) on the moduli stack, has a natural extension \(\overline{\lambda}_{\nu}\in\text{Pic}(\overline{M}_{h})_{\mathbb{Q}}\). A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero.

The useful technical statements are:

Theorem 1. Let \(f_{0}:X_{0}\rightarrow Y_{0}\) be a smooth projective morphism of quasi-projective reduced schemes such that \(\omega_{F}\) is semiample for all fibers \(F\) of \(f_{0}\). Let \(I\) be a finite set of positive integers. Then there exists a projective compactification \(Y\) of \(Y_{0}\), a finite covering \(\phi:W\rightarrow Y\) with a splitting trace map, and for \(\nu\in I\) a locally free sheaf \(\mathcal{F}_{W}^{(\nu)}\) on W with: (i) For \(W_{0}=\phi^{-1}(Y_{0})\) and \(\phi_{0}=\phi|_{W_{0}}\) one has \(\phi_{0}^{*}f_{0*}\omega^{\nu}_{X_{0}/Y_{0}}=\mathcal{F}_{W}^{(\nu)}|_{W_{0}}\). (ii) Let \(\xi:\widehat{Y}\rightarrow W\) be a morphism from a nonsingular projective variety \(\widehat{Y}\) with \(\widehat{Y}_{0}=\xi^{-1}(W_{0})\) dense in \(\widehat{Y}\). Assume either that \(\widehat{Y}\) is a curve, or that \(\widehat{Y}\rightarrow W\) is dominant. For some \(r\geq1\) let \(X^{(r)}\) be a nonsingular projective model of the \(r\)-fold product family \(\widehat{X}_{0}^{r}=(X_{0}\times_{Y_{0}}\times \dots \times_{Y_{0}}X_{0})\times_{Y_{0}}\widehat{Y}_{0}\), which admits a morphism \(f^{(r)}:X^{(r)}\rightarrow\widehat{Y}\). Then \(f_{*}^{(r)}\omega^{\nu}_{X^{(r)}/\widehat{Y}}=\bigotimes^{r}\xi^{*}\mathcal{F}_{W}^{(\nu)}\). The second result is

Theorem 2. Conditions (i) and (ii) in Theorem 1 imply: (iii) The sheaf \(\mathcal{F}_{W}^{(\nu)}\) is nef. (iv) Assume that for some \(\eta_{1}, \dots , \eta_{s}\in I\) and for some \(a_{1}, \dots, a_{s}\in\mathbb{N}\) the sheaf \(\bigotimes_{i=1}^{s}\det(\mathcal{F}_{W}^{(\eta_{i})})^{a_{i}}\) is ample with respect to \(W_{0}\). Then, if \(\nu\geq2\) and if \(\mathcal{F}_{W}^{(\nu)}\) is nonzero, it is ample with respect to \(W_{0}\).

The formulation of Theorem 1 is motivated by what is needed to prove positivity properties of direct image sheaves. The author used two main ingredients which allow the construction of W and \(\mathcal{F}_{W}^{(\nu)}\). The first one is the Weak Semistable Reduction Theorem see [D. Abramovich and K. Karu, “Weak semistable reduction in characteristic 0”, Invent. Math. 139, No. 2, 241–273 (2000; Zbl 0958.14006)]. The second ingredient is Gabber’s Extension Theorem (see §5.1 in [E. Viehweg, Quasi-projective moduli for polarized manifolds. Berlin: Springer-Verlag (1995; Zbl 0844.14004)]). Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools used: a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves. The construction of the moduli scheme \(M_{h}\) is done by using geometric invariant theory and stability criteria. As pointed out by the author, suitable variants of Theorems 1 and 2 could allow to get another proof of the quasi-projectivity of \(M_{h}\), replacing the GIT-approach by the analytic methods presented in the second part of the paper [G. Schumacher and H. Tsuji, “Quasi-projectivity of moduli spaces of polarized varieties”, Ann. Math. (2) 159, No. 2, 597–639 (2004; Zbl 1068.32011)].

By using a variant of Theorems 1 and 2 one obtains for the coarse moduli schemes the following result:

Theorem 5. Let \(M_{h}\) be the coarse moduli scheme of canonically polarized manifolds with Hilbert polynomial h. Given a finite set I of integers \(\nu\geq2\) with \(h(\nu)>0\), one finds a projective compactification \(\overline{M}_{h}\) of \((M_{h})_{\mathrm{red}}\) and for \(\nu\in I\) and some \(p>0\) invertible sheaves \(\lambda_{\nu}^{(p)}\) on \(\overline{M}_{h}\) with: (1) \(\lambda_{\nu}^{(p)}\) is nef, and it is ample with respect to \((M_{h})_{\mathrm{red}}\). (2) The restrictions of \(\lambda_{\nu}^{(p)}\) and \(\lambda_{0,\nu}^{(p)}\) to \((M_{h})_{\mathrm{red}}\) coincide. (3) Let \(\zeta:C\rightarrow\overline{M}_{h}\) be a morphism from a nonsingular curve with \(C_{0}=\zeta^{-1}(M_{h})\) dense in C and such that \(C_{0}\rightarrow M_{h}\) is induced by a family \(h_{0}:S_{0}\rightarrow C_{0}\). If \(h_{0}\) extends to a semistable family \(h:S\rightarrow C\), then \(\zeta^{*}\lambda_{\nu}^{(p)}=\det(h_{*}\omega^{\nu}_{S/C})^{p}\).

For the moduli functor \(\mathcal{M}_{h}\) of polarized minimal manifolds \((F, \mathcal{H})\) of Kodaira dimension zero and with Hilbert polynomial h one obtains:

Theorem 6. Let \(M_{h}\) be the coarse moduli scheme of polarized manifolds \((F, \mathcal{H})\) with \(\omega_{F}^{\nu}=\mathcal{X}(\mathcal{H}^{\mu})\). Then there exists a projective compactification \(\overline{M}_{h}\) of \((M_{h})_{\mathrm{red}}\) and for some \(p>0\) an invertible sheaf \(\lambda_{\nu}^{(p)}\) on \(\overline{M}_{h}\) with: (1) \(\lambda_{\nu}^{(p)}\) is nef and ample with respect to \((M_{h})_{\mathrm{red}}\). (2) Let \(Y_{0}\) be reduced and \(\varphi:Y_{0}\rightarrow M_{h}\) induced by a family \(f_{0}:X_{0}\rightarrow Y_{0}\) in \(\mathcal{M}_{h}(Y_{0})\). Then \(\varphi^{*}\lambda_{\nu}^{(p)}=f_{0*}\omega^{p\nu}_{X_{0}/Y_{0}}\). (3) Let \(\zeta:C\rightarrow\overline{M}_{h}\) be a morphism from a nonsingular curve C, with \(C_{0}=\zeta^{-1}(M_{h})\) dense in C and such that \(C_{0}\rightarrow M_{h}\) is induced by a family \(h_{0}:S_{0}\rightarrow C_{0}\). If \(h_{0}\) extends to a semistable family \(h:S\rightarrow C\), then \(\zeta^{*}\lambda_{\nu}^{(p)}=h_{*}\omega^{p\nu}_{S/C}\).

By using Theorem 5, one obtains the uniform boundedness of families of canonically polarized manifolds in: [S. J. Kovács and M. Lieblich, “Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”, Ann. Math. (2) 172, No. 3, 1719–1748 (2010; Zbl 1223.14040), republished in “Erratum: ‘Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”, ibid. 173, No. 1, 585–617 (2011; Zbl 1234.14029)].

The useful technical statements are:

Theorem 1. Let \(f_{0}:X_{0}\rightarrow Y_{0}\) be a smooth projective morphism of quasi-projective reduced schemes such that \(\omega_{F}\) is semiample for all fibers \(F\) of \(f_{0}\). Let \(I\) be a finite set of positive integers. Then there exists a projective compactification \(Y\) of \(Y_{0}\), a finite covering \(\phi:W\rightarrow Y\) with a splitting trace map, and for \(\nu\in I\) a locally free sheaf \(\mathcal{F}_{W}^{(\nu)}\) on W with: (i) For \(W_{0}=\phi^{-1}(Y_{0})\) and \(\phi_{0}=\phi|_{W_{0}}\) one has \(\phi_{0}^{*}f_{0*}\omega^{\nu}_{X_{0}/Y_{0}}=\mathcal{F}_{W}^{(\nu)}|_{W_{0}}\). (ii) Let \(\xi:\widehat{Y}\rightarrow W\) be a morphism from a nonsingular projective variety \(\widehat{Y}\) with \(\widehat{Y}_{0}=\xi^{-1}(W_{0})\) dense in \(\widehat{Y}\). Assume either that \(\widehat{Y}\) is a curve, or that \(\widehat{Y}\rightarrow W\) is dominant. For some \(r\geq1\) let \(X^{(r)}\) be a nonsingular projective model of the \(r\)-fold product family \(\widehat{X}_{0}^{r}=(X_{0}\times_{Y_{0}}\times \dots \times_{Y_{0}}X_{0})\times_{Y_{0}}\widehat{Y}_{0}\), which admits a morphism \(f^{(r)}:X^{(r)}\rightarrow\widehat{Y}\). Then \(f_{*}^{(r)}\omega^{\nu}_{X^{(r)}/\widehat{Y}}=\bigotimes^{r}\xi^{*}\mathcal{F}_{W}^{(\nu)}\). The second result is

Theorem 2. Conditions (i) and (ii) in Theorem 1 imply: (iii) The sheaf \(\mathcal{F}_{W}^{(\nu)}\) is nef. (iv) Assume that for some \(\eta_{1}, \dots , \eta_{s}\in I\) and for some \(a_{1}, \dots, a_{s}\in\mathbb{N}\) the sheaf \(\bigotimes_{i=1}^{s}\det(\mathcal{F}_{W}^{(\eta_{i})})^{a_{i}}\) is ample with respect to \(W_{0}\). Then, if \(\nu\geq2\) and if \(\mathcal{F}_{W}^{(\nu)}\) is nonzero, it is ample with respect to \(W_{0}\).

The formulation of Theorem 1 is motivated by what is needed to prove positivity properties of direct image sheaves. The author used two main ingredients which allow the construction of W and \(\mathcal{F}_{W}^{(\nu)}\). The first one is the Weak Semistable Reduction Theorem see [D. Abramovich and K. Karu, “Weak semistable reduction in characteristic 0”, Invent. Math. 139, No. 2, 241–273 (2000; Zbl 0958.14006)]. The second ingredient is Gabber’s Extension Theorem (see §5.1 in [E. Viehweg, Quasi-projective moduli for polarized manifolds. Berlin: Springer-Verlag (1995; Zbl 0844.14004)]). Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools used: a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves. The construction of the moduli scheme \(M_{h}\) is done by using geometric invariant theory and stability criteria. As pointed out by the author, suitable variants of Theorems 1 and 2 could allow to get another proof of the quasi-projectivity of \(M_{h}\), replacing the GIT-approach by the analytic methods presented in the second part of the paper [G. Schumacher and H. Tsuji, “Quasi-projectivity of moduli spaces of polarized varieties”, Ann. Math. (2) 159, No. 2, 597–639 (2004; Zbl 1068.32011)].

By using a variant of Theorems 1 and 2 one obtains for the coarse moduli schemes the following result:

Theorem 5. Let \(M_{h}\) be the coarse moduli scheme of canonically polarized manifolds with Hilbert polynomial h. Given a finite set I of integers \(\nu\geq2\) with \(h(\nu)>0\), one finds a projective compactification \(\overline{M}_{h}\) of \((M_{h})_{\mathrm{red}}\) and for \(\nu\in I\) and some \(p>0\) invertible sheaves \(\lambda_{\nu}^{(p)}\) on \(\overline{M}_{h}\) with: (1) \(\lambda_{\nu}^{(p)}\) is nef, and it is ample with respect to \((M_{h})_{\mathrm{red}}\). (2) The restrictions of \(\lambda_{\nu}^{(p)}\) and \(\lambda_{0,\nu}^{(p)}\) to \((M_{h})_{\mathrm{red}}\) coincide. (3) Let \(\zeta:C\rightarrow\overline{M}_{h}\) be a morphism from a nonsingular curve with \(C_{0}=\zeta^{-1}(M_{h})\) dense in C and such that \(C_{0}\rightarrow M_{h}\) is induced by a family \(h_{0}:S_{0}\rightarrow C_{0}\). If \(h_{0}\) extends to a semistable family \(h:S\rightarrow C\), then \(\zeta^{*}\lambda_{\nu}^{(p)}=\det(h_{*}\omega^{\nu}_{S/C})^{p}\).

For the moduli functor \(\mathcal{M}_{h}\) of polarized minimal manifolds \((F, \mathcal{H})\) of Kodaira dimension zero and with Hilbert polynomial h one obtains:

Theorem 6. Let \(M_{h}\) be the coarse moduli scheme of polarized manifolds \((F, \mathcal{H})\) with \(\omega_{F}^{\nu}=\mathcal{X}(\mathcal{H}^{\mu})\). Then there exists a projective compactification \(\overline{M}_{h}\) of \((M_{h})_{\mathrm{red}}\) and for some \(p>0\) an invertible sheaf \(\lambda_{\nu}^{(p)}\) on \(\overline{M}_{h}\) with: (1) \(\lambda_{\nu}^{(p)}\) is nef and ample with respect to \((M_{h})_{\mathrm{red}}\). (2) Let \(Y_{0}\) be reduced and \(\varphi:Y_{0}\rightarrow M_{h}\) induced by a family \(f_{0}:X_{0}\rightarrow Y_{0}\) in \(\mathcal{M}_{h}(Y_{0})\). Then \(\varphi^{*}\lambda_{\nu}^{(p)}=f_{0*}\omega^{p\nu}_{X_{0}/Y_{0}}\). (3) Let \(\zeta:C\rightarrow\overline{M}_{h}\) be a morphism from a nonsingular curve C, with \(C_{0}=\zeta^{-1}(M_{h})\) dense in C and such that \(C_{0}\rightarrow M_{h}\) is induced by a family \(h_{0}:S_{0}\rightarrow C_{0}\). If \(h_{0}\) extends to a semistable family \(h:S\rightarrow C\), then \(\zeta^{*}\lambda_{\nu}^{(p)}=h_{*}\omega^{p\nu}_{S/C}\).

By using Theorem 5, one obtains the uniform boundedness of families of canonically polarized manifolds in: [S. J. Kovács and M. Lieblich, “Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”, Ann. Math. (2) 172, No. 3, 1719–1748 (2010; Zbl 1223.14040), republished in “Erratum: ‘Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”, ibid. 173, No. 1, 585–617 (2011; Zbl 1234.14029)].

Reviewer: Vasile Brînzănescu (Bucureşti)

### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14J10 | Families, moduli, classification: algebraic theory |

14J40 | \(n\)-folds (\(n>4\)) |

14F18 | Multiplier ideals |

### References:

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