Relative proportionality for subvarieties of moduli spaces of \(K3\) and abelian surfaces.

*(English)*Zbl 1222.14020The proportionality theorem of Hirzebruch and Höfer states that if \(Y\) is a Hilbert modular surface or a ball quotient and \(\bar Y=Y\coprod S_{\bar Y}\) is a compactification with \(S_{\bar Y}\) a normal crossings divisor, then if \(\bar S\subset \bar Y\) is a smooth curve and \(S_{\bar C}=\bar C \cap S_{\bar Y}\) (as sets, i.e. the reduced intersection) then
\[
\alpha \bar C\bar C+\alpha\deg S_{\bar C} \geq -K_{\bar Y}\bar C +(\alpha-1)S_{\bar Y}\bar C \eqno{(*)}
\]
where \(\alpha=2\) for Hilbert modular surfaces and \(\alpha=3\) in the ball quotient case. Moreover, if \(\bar Y\) satisfies \(\Omega^1_{\bar Y}(\log S_{\bar Y})\) nef and \(\omega_{\bar Y}(S_{\bar Y})\) ample on \(Y\) – for instance, \(\bar Y\) is a sutable toroidal compactification – then equality in (*) implies that \(C=\bar C\cap Y\) is a subball quotient.

This paper extends these results to Shimura varieties of ball quotient type (i.e. associated with \({{\text{SU}}}(1,n)\)) or \({\text{SO}}(2,n)\) type. The latter include the moduli of polarised \(K3\) and abelian surfaces. In these generalisations, the curve \(C\) is replaced by a submanifold \(Z\subset Y\).

If \(C\) is a Shimura curve and \(Y\) is an arbitrary projetive surface contained in a Shimura variety \(M\) of \({\text{SO}}(2,n)\) type one gets, after compactification, inequalities as in \((*)\) but with the sign reversed: \[ \alpha \bar C\bar C+\alpha\deg S_{\bar C} \leq -K_{\bar Y}\bar C +(\alpha-1)S_{\bar Y}\bar C \eqno{(†)} \] with \(\alpha=2\) if \(\theta_{\bar C}^{(2)}\neq 0\) and \(\alpha=2\) if \(\theta_{\bar Y}^{(2)}= 0\). Here \(\theta^{(2)}\) denotes the Griffiths-Yukawa coupling: there is also the possibility that \(\theta_{\bar C}^{(2)}= 0\) and \(\theta_{\bar Y}^{(2)}\neq 0\), which occurs when \(Y\) is a product of \(C\) and another curve.

These inequalities also generalise to higher dimension. Using them, the authors formulate a condition that forces \(Y\) to be a Shimura variety of Hodge type: in the case above, if there are sufficiently many curves \(C_i\) satisfying \[ \alpha \bar C_i\bar C_i+\alpha\deg S_{\bar C_i} \leq -K_{\bar Y}\bar C_i +(\alpha-1)S_{\bar Y}\bar C_i \eqno{(‡)} \] then \(Y\) is a Hilbert modular surface or ball quotient, according to the value of \(\alpha\).

This is motivated by the observation that the André-Oort conjecture (which holds under GRH) implies that any subvariety with a dense set of CM points must be of Hodge type.

The results in their full generality are too complicated to state here, but can be described quickly. First the proportionality: a Shimura variety \(M\) of \({\text{SO}}(2,n)\) or \({\text{SU}}(1,n)\) type and a dimension \(d\) submanifold \(Z\subset M\), and suitable compactifications, there are inequalities satisfied by the degrees of certain sheaves. For instance, for \({\text{SO}}(2,n)\) type \[ d\deg_{\omega_{\bar Z}}(S_{\bar Z})({\mathcal N}^*_{\bar Z/\bar M})+(n-d)\deg_{\omega_{\bar Z}}(S_{\bar Z})\big(\Omega^1_{\bar Z}(\log S_{\bar Z})\big)\geq 0 \] if \(\theta_{\bar Z}^{(2)}\neq 0\) and \(S_{\bar Z}=\bar Z\setminus Z\) is a normal crossings divisor: equivalently \[ \mu_{\omega_{\bar Z}}(S_{\bar Z})({\mathcal N}^*_{\bar Z/\bar M})\geq \mu_{\omega_{\bar Z}}(S_{\bar Z})\big(T_{\bar Z}(-\log S_{\bar Z})\big). \] If these inequalities are equalities then (for \(d>1\)) \(Z\) is a Shimura variety.

Second, the characterisations of Shimura varieties as having many subvarieties satisfying the above equalities are obtained from \((†)\) and \((‡)\) by replacing \(Y\) by \(Z\) and \(C_i\) by codimension \(1\) subvarieties \(W_i\) of type \({\text{SO}}(2,d-1)\). If there are at least \(\varsigma(\bar Z)\) such \(W_i\) satisfying the slope condition \[ \mu_{\omega_{\bar W_i}}(S_{\bar W_i})({\mathcal N}^*_{\bar W_i/\bar Z})\geq \mu_{\omega_{\bar W_i}}(S_{\bar W_i})\big(T_{\bar W_i}(-\log S_{\bar W_i})\big), \] where \(\varsigma(\bar Z)=\rho(\bar Z)^2+\rho(\bar Z)+1\) (slightly modified if \(d<4\)), then \(Z\) is a Shimura variety of type \({\text{SO}}(2,d)\).

The proofs are mainly Hodge theory. Ingredients include the results of the last two authors on Arakelov-type inequalities and Shimura varieties, in [J. Differ. Geom. 77, No. 2, 291–352 (2007; Zbl 1133.14010)], and the results of Simpson and others on Higgs bundles and local systems. However, there is also a stage that requires some details of toroidal compactification, because one needs to be able to blow up at the boundary in such a way that the embedding of \(W_i\) in the moduli space extends to the closure.

This paper extends these results to Shimura varieties of ball quotient type (i.e. associated with \({{\text{SU}}}(1,n)\)) or \({\text{SO}}(2,n)\) type. The latter include the moduli of polarised \(K3\) and abelian surfaces. In these generalisations, the curve \(C\) is replaced by a submanifold \(Z\subset Y\).

If \(C\) is a Shimura curve and \(Y\) is an arbitrary projetive surface contained in a Shimura variety \(M\) of \({\text{SO}}(2,n)\) type one gets, after compactification, inequalities as in \((*)\) but with the sign reversed: \[ \alpha \bar C\bar C+\alpha\deg S_{\bar C} \leq -K_{\bar Y}\bar C +(\alpha-1)S_{\bar Y}\bar C \eqno{(†)} \] with \(\alpha=2\) if \(\theta_{\bar C}^{(2)}\neq 0\) and \(\alpha=2\) if \(\theta_{\bar Y}^{(2)}= 0\). Here \(\theta^{(2)}\) denotes the Griffiths-Yukawa coupling: there is also the possibility that \(\theta_{\bar C}^{(2)}= 0\) and \(\theta_{\bar Y}^{(2)}\neq 0\), which occurs when \(Y\) is a product of \(C\) and another curve.

These inequalities also generalise to higher dimension. Using them, the authors formulate a condition that forces \(Y\) to be a Shimura variety of Hodge type: in the case above, if there are sufficiently many curves \(C_i\) satisfying \[ \alpha \bar C_i\bar C_i+\alpha\deg S_{\bar C_i} \leq -K_{\bar Y}\bar C_i +(\alpha-1)S_{\bar Y}\bar C_i \eqno{(‡)} \] then \(Y\) is a Hilbert modular surface or ball quotient, according to the value of \(\alpha\).

This is motivated by the observation that the André-Oort conjecture (which holds under GRH) implies that any subvariety with a dense set of CM points must be of Hodge type.

The results in their full generality are too complicated to state here, but can be described quickly. First the proportionality: a Shimura variety \(M\) of \({\text{SO}}(2,n)\) or \({\text{SU}}(1,n)\) type and a dimension \(d\) submanifold \(Z\subset M\), and suitable compactifications, there are inequalities satisfied by the degrees of certain sheaves. For instance, for \({\text{SO}}(2,n)\) type \[ d\deg_{\omega_{\bar Z}}(S_{\bar Z})({\mathcal N}^*_{\bar Z/\bar M})+(n-d)\deg_{\omega_{\bar Z}}(S_{\bar Z})\big(\Omega^1_{\bar Z}(\log S_{\bar Z})\big)\geq 0 \] if \(\theta_{\bar Z}^{(2)}\neq 0\) and \(S_{\bar Z}=\bar Z\setminus Z\) is a normal crossings divisor: equivalently \[ \mu_{\omega_{\bar Z}}(S_{\bar Z})({\mathcal N}^*_{\bar Z/\bar M})\geq \mu_{\omega_{\bar Z}}(S_{\bar Z})\big(T_{\bar Z}(-\log S_{\bar Z})\big). \] If these inequalities are equalities then (for \(d>1\)) \(Z\) is a Shimura variety.

Second, the characterisations of Shimura varieties as having many subvarieties satisfying the above equalities are obtained from \((†)\) and \((‡)\) by replacing \(Y\) by \(Z\) and \(C_i\) by codimension \(1\) subvarieties \(W_i\) of type \({\text{SO}}(2,d-1)\). If there are at least \(\varsigma(\bar Z)\) such \(W_i\) satisfying the slope condition \[ \mu_{\omega_{\bar W_i}}(S_{\bar W_i})({\mathcal N}^*_{\bar W_i/\bar Z})\geq \mu_{\omega_{\bar W_i}}(S_{\bar W_i})\big(T_{\bar W_i}(-\log S_{\bar W_i})\big), \] where \(\varsigma(\bar Z)=\rho(\bar Z)^2+\rho(\bar Z)+1\) (slightly modified if \(d<4\)), then \(Z\) is a Shimura variety of type \({\text{SO}}(2,d)\).

The proofs are mainly Hodge theory. Ingredients include the results of the last two authors on Arakelov-type inequalities and Shimura varieties, in [J. Differ. Geom. 77, No. 2, 291–352 (2007; Zbl 1133.14010)], and the results of Simpson and others on Higgs bundles and local systems. However, there is also a stage that requires some details of toroidal compactification, because one needs to be able to blow up at the boundary in such a way that the embedding of \(W_i\) in the moduli space extends to the closure.

Reviewer: G. K. Sankaran (Bath)

##### MSC:

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

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

14J28 | \(K3\) surfaces and Enriques surfaces |

14G35 | Modular and Shimura varieties |

14K10 | Algebraic moduli of abelian varieties, classification |

14M27 | Compactifications; symmetric and spherical varieties |