Twistor spaces with a pencil of fundamental divisors

B. Kreußler

Documenta mathematica, Tome 4 (1999), pp. 127-166

Résumé

In this paper simply connected twistor spaces Z containing a pencil of fundamental divisors are studied. The Riemannian base for such spaces is diffeomorphic to the connected sum nCP2. We obtain for n≥5 a complete description of the set of curves intersecting the fundamental line bundle K−21​ negatively. For this purpose we introduce a combinatorial structure, called blow-up graph. We show that for generic S∈∣−21​K∣ the algebraic dimension can be computed by the formula a(Z)=1+κ−1(S). A detailed study of the anti Kodaira dimension κ−1(S) of rational surfaces permits to read off the algebraic dimension from the blow-up graphs. This gives a characterisation of Moishezon twistor spaces by the structure of the corresponding blow-up graphs. We study the behaviour of these graphs under small deformations. The results are applied to prove the main existence result, which states that every blow-up graph belongs to a fundamental divisor of a twistor space. We show, furthermore, that a twistor space with dim∣−21​K∣=3 is a LeBrun space [LeB2]. We characterise such spaces also by the property to contain a smooth rational non-real curve C with C.(−21​K)=2−n.

Détail

DOI : 10.4171/dm/56

Classification : 14M20, 32J17, 32L25
Mots-clés : moishezon manifold, algebraic dimension, self--dual, twistor space