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The aim of the notes is try to see how Hodge theory can be applied in Birational Geometry.
1. Canonical bundle formula and semi-positivity theorem,
2. DuBois singularity and Hodge theory,
3. Log canonical threshold and the Bernstein-Sato polynomial,
1 minute read
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In this series of notes we will summarize some standard metric methods which will be useful in birational geometry.
1 minute read
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The aim of this series of note is to introduce the fibration in Algebraic geometry (classification theory of complex algebraic/analytic varieties). Fibration is the most powerful tool for the classification of varieties and fibration structure is very suitable for the induction process. In this notes we will major focus on the possible applications of the fibrations in the classification results. Fibration is central in Algebraic Geometry.
In algebraic geometry (birational geometry), we understand variety using fibration.
1 minute read
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The aim of this note is to introduce the minimal model program for Kahler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kahler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kahler varieties, (2) The base point free theorem and cone theorem: Since Kahler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kahler 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.
2 minute read
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This aim of this series of notes is to study the classical paper BCHM in details. We will try to finish the proof of the main theorem and summarize some interesting applications of BCHM. We will not strictly follow the structure of the original paper, instead we will divide the paper into several topics discussion, including: Existence of flip, existence of minimal model, termination problem, non-vanishing conjecture and finite generation problem etc.
