## Erratum: “Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich’s conjecture”.(English)Zbl 1234.14029

This article was first printed in [Ann. Math. (2) 172, No. 3, 1719–1748 (2010; Zbl 1223.14040)]. The paper under review should be considered the version of record as it corrects a font error that resulted in notational problems throughout the first printing.
Let $$k$$ be an algebraically closed field of characteristic zero. Consider a smooth projective curve $$B$$ of genus $$g$$ over $$k$$ and a finite subset $$\Delta$$ of $$B$$. Then a family over $$B$$ is a flat projective morphism $$f: X\to B$$ with connected fibers, where $$X$$ is a smooth projective variety over $$k$$. Such a family is called isotrivial if any two fibers of $$f$$ are isomorphic for general base points $$a$$ and $$b$$ in $$B$$, while $$f: X\to B$$ is admissible (with respect to the pair $$(B,\Delta)$$) if it is not isotrivial and $$\Delta$$ contains the discriminant locus of $$f$$, i.e., the map $$f: X\setminus f^{-1}(\Delta)\to B\setminus\Delta$$ is smooth.
At the 1962 ICM in Stockholm, I. R. Shafarevich formulated the following conjecture:
Let $$(B,\Delta)$$ be given as above and $$q\geq 2$$ an integer. Then
(1) there exist only finitely many isomorphism classes of admissible families of curves of genus $$q$$ over $$B$$;
(2) there exist no such families if the inequality $$2g-2+ \#\Delta\leq 0$$ holds.
Shafarevich proved (2) in the special case $$B=\mathbb{P}^1_k$$, whereas (1) was confirmed by A. N. Parshin for $$\Delta=\phi$$ [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191–1219 (1968; Zbl 0181.23902)] and by S. J. Arakelov [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1269–1293 (1971; Zbl 0238.14012)] in general. Actually, Parshin and Arakelov used a reformulation of Shafarevich’s conjecture in terms of deformation types of families. More precisely, they proved the following version: Let $$(B,\Delta)$$ be fixed as above and $$q\geq 2$$ an integer. Then the following statements hold.
(B) (Boundedness): There exist only finitely many deformation types of admissible families of curves of genus $$q$$ over $$B$$ with respect to $$B\setminus\Delta$$;
(R) (Rigidity): There exist no non-trivial deformations of admissible families of the above type;
(H) (Hyperbolicity): if $$2g- 2+\#\Delta\leq 0$$, then no admissible families of the above type exist.
As for the study of this version of Shafarevich’s conjecture for families of higher-dimensional varieties, the past decade has seen an avalanche of results concerning both (B) and (H), whereas (R) definitely fails in higher dimensions, due to E. Viehweg [Positivity of direct image sheaves and applications to families of higher-dimensional manifolds. Trieste: The Abdus Salam International Centre for Theoretical Physics. ICTP Lect. Notes 6, 249–284 (2001; Zbl 1092.14044)] and S. Kovacs [in: Higher dimensional varieties and rational points. Lectures of the summer school and conference, Budapest, Hungary, September 3–21, 2001. Berlin: Springer; Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 12, 133–167 (2003; Zbl 1058.14057)].
In the present paper, the authors investigate the validity of the boundedness property (B) for families of higher-dimensional canonically polarized projective varieties from a stack-theoretic point of view. Their approach is based on the following philosophy: If there existed an algebraic stack $${\mathcal D}$$ parametrizing families of canonically polarized varieties over the base $$B\setminus\Delta$$, and if furthermore $${\mathcal D}$$ could be shown to be of finite type, then the boundedness condition (B) would follow for families of canonically polarized varieties.
However, as E. Bedulev and E. Viehweg proved several years ago in their seminal paper [Invent. Math. 139, No. 3, 603–615 (2000; Zbl 1057.14044)], such stack $${\mathcal D}$$ almost never exists. In view of this fact, the bulk of work done in the current paper is devoted to pointing out that a suitable proxy for $${\mathcal D}$$ can be constructed by standard stack-theoretic methods. Their main result (Theorem 1.7.) concerns so-called weakly bounded compactifiable Deligne-Mumford stacks over a quasi-compact and quasi-separated $$\mathbb{Q}$$-scheme $$T$$. At the end of the paper, this general framework is applied to the Deligne-Mumford stack of canonically polarized varieties with given Hilbert polynomial $$h$$, thereby showing that the number of deformation types of canonically polarized varieties over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. Also, the authors obtain a direct generalization of Shafarevich’s original conjecture to certain families of canonically polarized varieties, namely to the newly introduced, so-called infinitesimally rigid families of this type.
As for the precise contents of this paper, the first four sections provide the general stack-theoretic framework, together with the main theorems, while the last section discusses the above-mentioned applications to canonically polarized varieties and to the higher-dimensional analogue of Shafarevich’s conjecture.

### MSC:

 14J10 Families, moduli, classification: algebraic theory 14D15 Formal methods and deformations in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14H10 Families, moduli of curves (algebraic) 14A20 Generalizations (algebraic spaces, stacks) 14D23 Stacks and moduli problems 14D06 Fibrations, degenerations in algebraic geometry
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