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We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge theory provides a number of conceptual simplifications to the argument. The essential new ingredient is a volume bound for Griffiths transverse subvarieties of period domains.more » « less
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We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture.more » « less
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Classically, an indecomposable class $$R$$ in the cone of effective curves on a K3 surface $$X$$ is representable by a smooth rational curve if and only if $$R^{2}=-2$$ . We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $$M$$ deformation equivalent to a Hilbert scheme of $$n$$ points on a K3 surface, an extremal curve class $$R\in H_{2}(M,\mathbb{Z})$$ in the Mori cone is the line in a Lagrangian $$n$$ -plane $$\mathbb{P}^{n}\subset M$$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $$(R,R)=-\frac{n+3}{2}$$ , and the primitive such classes are all contained in a single monodromy orbit.more » « less
