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Calabi-Yau metric on the Fermat surface. Isometries and totally geodesic submanifolds. (English) Zbl 0749.53028
The Fermat surface \(F\) is the non-singular quartic in \(\mathbb{C}\mathbb{P}_ 3\) defined by \(x^ 4_ 0+x^ 4_ 1+x^ 4_ 2+x^ 4_ 3=0\). This surface \(F\) is simply-connected with vanishing first Chern class and is the basic example of a K3 surface. The Fubini-Study metric on \(\mathbb{C}\mathbb{P}_ 3\) induces a Kähler metric on \(F\) and hence, by Yau’s proof of the Calabi conjecture, there is a unique Kähler-Einstein metric \(m\) on \(F\) with Kähler form cohomologous to that induced by the Fubini-Study metric. In this paper, the authors study the automorphism group of this metric. Even though the metric is not known explicitly it is still possible to identify this group. In fact, there are some obvious automorphisms induced by automorphisms of \(\mathbb{C}\mathbb{P}_ 3\) which preserve \(F\). These are given by permuting the coordinates and multiplying any coordinate by a fourth root of unity. Together with complex conjugation, this gives some automorphisms of \(m\). The authors show that there are no more. They go on, using an interesting blend of differential and algebraic geometry, to investigate the fine structure of these automorphisms describing their action on the parallel complex structures and identifying their fixed point sets. The resulting geometry is surprisingly rich.

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI
[1] Piatetskii-Shapiro, I.; Shafarevitch, I., A Torelli theorem for algebraic surfaces of type K3, Math. USSR isvestiya, 35, 530-572, (1971)
[2] Tiurina, G.N., Algebraic surfaces, Trudy MIAN, v. 75, (1965), chapt. 9.
[3] Griffiths, Ph.; Harris, J., Principles of algebraic geometry, (1978) · Zbl 0408.14001
[4] Geometric des surfaces K3, Asterisque, (1985), N. 126
[5] Besse, A., Geometrie riemannianne en dimension, 4, (1981)
[6] Kobayashi, Sh., Transformation groups in differential geometry, (1972)
[7] Barth, W.; Peters, C.; Van De Ven, A., Compact complex surfaces, (1984) · Zbl 0718.14023
[8] Mumford, D., Tata lecture on theta, I, II, (1983)
[9] Hitchin, N., On compact four-dimensional Einstein manifolds, J. differ. geom., 9, N 3, 435-442, (1974) · Zbl 0281.53039
[10] Besse, A., Einstein manifolds, (1986)
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