Wild automorphisms of varieties with Kodaira dimension \(0\). (English) Zbl 1205.14053
Summary: An automorphism \(\sigma \) of a projective variety \(X\) is said to be wild if \(\sigma (Y) \neq Y\) for every non-empty subvariety \(Y \subsetneq X\). Z. Reichstein, D. Rogalski and J.J. Zhang [Adv. Math. 203, No. 2, 365–407 (2006; Zbl 1120.16040)] conjectured that if \(X\) is an irreducible projective variety admitting a wild automorphism then \(X\) is an abelian variety, and proved this conjecture for \(dim(X) \leq 2\). As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension \(0\) case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.
MSC:
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
Citations:
Zbl 1120.16040References:
| [1] | Barth, W., Peters, C., de Ven, A.V.: Compact complex surfaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, vol. 4, pp. 1-288. Springer, Berlin (1984) · Zbl 0718.14023 |
| [2] | Beauville, A.: Some remarks on Kähler manifolds with c1 = 0. Classification of algebraic and analytic manifolds (Katata, 1982). Progr. Math., vol. 39, pp. 1-2. Birkhäuser Boston, Boston, MA (1983). 86c:32031 |
| [3] | Birkenhake C., Lange H.: Complex Abelian Varieties. Springer, Berlin (2004) · Zbl 1056.14063 |
| [4] | Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). MR0463157 (57#116) · Zbl 0367.14001 |
| [5] | Miyaoka Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic Geometry Sendai 10, 449-476 (1985) · Zbl 0648.14006 |
| [6] | Oguiso K.: On the finiteness of fiber-space structures on a Calabi-Yau 3-fold. J. Math. Sci. 106(5), 3320-3335 (2001) · Zbl 1087.14509 · doi:10.1023/A:1017959510524 |
| [7] | Reichstein, Z., Rogalski, D., Zhang, J.J.: Projectively simple rings. Adv. Math. 203(2), 365-407 (2006). MR2227726 (2007i:16071) · Zbl 1120.16040 |
| [8] | Wilson P.: The Kähler cone on Calabi-Yau threefolds. Inventiones mathematicae 107(1), 561-583 (1992) · Zbl 0766.14035 · doi:10.1007/BF01231902 |
| [9] | Wilson, P.: The Role of c2 in Calabi-Yau Classification—a Preliminary Survey. Mirror Symmetry II, AMS/IP 1, 381-392 (1997) · Zbl 0936.14029 |
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