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Relative proportionality for subvarieties of moduli spaces of $$K3$$ and abelian surfaces. (English) Zbl 1222.14020
The 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.

##### 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
##### Keywords:
Hirzebruch proportionality; Shimura variety; Higgs bundle
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