Analytic rigidity of K-trivial extremal contractions of smooth threefolds

Csilla Tamas, Purdue University

Abstract

In the classification theory of higher dimensional algebraic varieties, the study of certain types of birational morphisms, called “extremal contractions”, is of central importance. In 1982, S. Mori classified the birational extremal contractions ϕ : X → Y of smooth 3-folds where the canonical bundle K of X is negative along the curves contracted by ϕ. His study revolutionized the birational geometry of algebraic varieties. In this thesis we will discuss the problem of classifying birational extremal contractions of smooth 3-folds where the canonical bundle is trivial along the curves contracted, in the case when a divisor is contracted to a point. We prove the analytic rigidity of the contraction in the case when this exceptional divisor D is normal with (ωD2) ≥ 5, i.e. we show that the analytic structure of the contraction is completely determined by the isomorphism class of the exceptional locus and its normal bundle in X. This was previously known only in the case when the exceptional divisor was nonsingular. We also construct, for each possible exceptional divisor D (a normal rational del Pezzo surface of degree d ≥ 5), an embedding of D into a smooth threefold X with the required normal bundle, and hence obtain analytic contractions.

Degree

Ph.D.

Advisors

Matsuki, Purdue University.

Subject Area

Mathematics

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