By Fabrizio Catanese, Hélène Esnault, Alan Huckleberry, Klaus Hulek, Thomas Peternell

ISBN-10: 3540354794

ISBN-13: 9783540354796

This selection of surveys current an summary of contemporary advancements in advanced Geometry. themes diversity from curve and floor concept via precise forms in better dimensions, moduli thought, Kähler geometry, and team activities to Hodge idea and attribute p-geometry.

Written by means of proven specialists this ebook might be a needs to for mathematicians operating in advanced Geometry

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**Example text**

CV60]). 2) Mostow (cf. [Mos73]) proved that indeed locally symmetric spaces of complex dimension ≥ 2 are strongly rigid, in the sense that any homotopy equivalence is induced by a unique isometry. These varieties are of general type and the moduli space of varieties of general type is deﬁned over Z, and naturally the absolute Galois group ¯ Gal(Q/Q) acts on the set of their connected components. So, in our special ¯ case, Gal(Q/Q) acts on the isolated points which parametrize rigid varieties. In particular, rigid varieties are deﬁned over a number ﬁeld and work of Shimura gives a possible way of computing explicitly their ﬁelds of deﬁnition.

E. all factors are powers of a half twist, with respective exponent 1, 2, 3. Chisini posed the following Conjecture 3. ) Given two generic coverings f : S → P2C , f : S → P2C , one of them of degree d ≥ 5, assume that they have the same branch curve B. Is it then true that f and f are equivalent? Observe that the condition on the degree is necessary, since counterexamples with d = 4 are furnished by the dual curve of a smooth plane cubic (as already known to Chisini, who gave a counterexample with d = 4, d = 3, while counterexamples with d = d = 4 were given in [Cat86b]).

Here is a partial picture: 39 37 IVa1 Ia {{ xx { x { x {{ xx |xx }{{ IVa2 IVb1 36 IVb2 38 IIIa { { {{ {{ }{{ V1 IIIb o || || | | }|| V2 II Ib This picture is partial because up to now it is not known whether all possible arrows are drawn. More precisely, M6,4,0 is connected if and only if one of the two following degenerations is possible: Ia → V1 or Ia → V2 . 26 Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli This picture was done by Horikawa in [Hor78] with the exception of the horizontal line IIIb ← II, recently obtained in [BCP04].

### Global Aspects of Complex Geometry by Fabrizio Catanese, Hélène Esnault, Alan Huckleberry, Klaus Hulek, Thomas Peternell

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