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The aim of this series of notes is to introduce fibrations in algebraic geometry (the classification theory of complex algebraic/analytic varieties). Fibrations are among the most powerful tools for the classification of varieties, and fibration structures are particularly well suited to inductive arguments. In these notes, we focus mainly on applications of fibrations to classification results.
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The aim of this series of notes is to study the classical paper BCHM in detail. We aim to complete the proof of the main theorem and summarize some important applications of BCHM. We do not strictly follow the structure of the original paper; instead, we divide the material into several thematic topics, including the existence of flips, existence of minimal models, termination problems, the non-vanishing conjecture, and the finite generation problem.
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In this series of notes, we summarize several standard metric methods that are useful in birational geometry.
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The aim of this note is to introduce the minimal model program for Kähler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kähler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kähler varieties, (2) The base point free theorem and cone theorem: Since Kähler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kähler manifold, (3) Contraction theorem: In the classical MMP, we require the base point free theorem to produce some semi-ampleness divisor. This semi-ample divisor will induce the negative contraction morphism, and this approach is not very clear under Kaehler setting.
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The purpose of this series of notes is to discuss the recent paper Moishezon morphism by Professor Kollár in detail. Let us first briefly answer the question WHY Moishezon?
First, Moishezon spaces have more functorial behavior (compared with projective varieties), as we will see in my notes. Secondly, from almost any projective variety we can construct a Moishezon space via bimeromorphic modification, making Moishezon spaces versatile in birational geometry. Thirdly, by Artin’s fundamental theorem, the category of Moishezon spaces appears naturally in moduli theory. Another compelling reason to consider the Moishezon category is that it allows cutand-paste operations similar to those we can perform in topology.
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In this part of the notes, we focus on the properties of moduli spaces. The guiding philosophy behind moduli theory is that
The moduli of a geometric object completely determine the object itself. In birational geometry, one of the central themes is that the geometry of a variety is largely governed by the curves and divisors it contains.
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The aim of this note is to introduce adjunction theory in birational geometry. Among the various techniques in birational geometry, adjunction theory is one of the most powerful tools. It allows us to relate the geometry—particularly the singularities—of an ambient variety to those of suitable subvarieties.
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The aim of this series notes is to give a brief introduction to rationality questions in birational geometry.