On the formal principle for curves on projective surfaces
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Mathematische Annalen
On the formal principle for curves on projective surfaces Jorge Vitório Pereira1 · Olivier Thom1 Received: 1 October 2019 / Revised: 12 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.
1 Introduction In this paper, we investigate pairs (X , Y ) of complex varieties where Y is a compact subvariety of the complex variety X . We are particularly interested in the analytic classification of such pairs when X is a smooth projective surface. Definition 1.1 A pair (X , Y ) satisfies the formal principle if for any other pair (X , Y ) such that the formal completion Y of X along Y is formally isomorphic to the formal completion Y of X along Y then the germ of X along Y is biholomorphic to the germ of X along Y . If Y is a smooth compact curve on a smooth surface X with non-zero selfintersection then the pair (X , Y ) satisfies the formal principle. Indeed, if Y 2 < 0 then [7, Section 4, Satz 6] implies that (X , Y ) satisfies the formal principle. When Y 2 > 0, then the result is implied by [6], see also the discussion in [13, Section 4]. The case of zero self-intersection is in sharp contrast. To the best of our knowledge, the first example of a pair (X , Y ) for which the formal principle does not hold is due to V.I. Arnold: it consists of a germ of surface X containing an elliptic curve of zero self-intersection and non-torsion normal bundle obtained through the suspension of a germ of non-linearizable biholomorphism, see [2]. Even if one restricts to neighbor-
Communicated by Ngaiming Mok.
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Olivier Thom [email protected] Jorge Vitório Pereira [email protected]
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Instituto Nacional de Matematica Pura e Aplicada Rio de Janeiro, Rio de Janeiro, Brazil
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J. V. Pereira, O. Thom
hoods of elliptic curves with trivial normal bundles, the analytic classification differs considerably from the formal classification, see [15, Theorem 5]. There are many more works investigating the formal principle. We invite the reader to consult the recent paper [11] and the surveys in [13] and [8, Section VII.4] to get a view of different directions of research on the subject. 1.1 Projective formal principle The results just mentioned provide an abundance of pairs for which the formal principle does not hold. They are based on local analytic construction and do not globalize. Indeed, Neeman in [17, Article 1, Theorem 6.12] shows that a smooth elliptic curve Y with trivial normal bundle on a projective surface X either is a fiber of a fibration, or X is birationally equivalent to P(E), the projectivization of the unique rank two vector bundle over Y obtained as a non-trivial extension of the trivial line-bundle by itself, and Y corresponds to the natural section Y → P(E). Taking into account this result, it seems natural to consider the following restricted version of the formal principle
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