更新3-folds
Update on 3-folds
摘要：
紧致黎曼曲面分为K_C<0，g=0，或K_C=0，g=1，或K_C>0，g>=2三种情况是众所周知的，并且对应于球面、欧几里德和双曲非欧几里德平面几何中常见的剖分。分类的目的是根据这种剖分来处理所有的射影代数簇。该模型是Castelnuovo和Enriques在1900年左右对表面的处理。因为一个簇的标准类可能没有一个明确的符号，我们通常不得不修改一个品种的极小模型程序（MMP）之前，它是有意义的应用，这涉及到收缩，翻转和纤维空间分解。三重分类是由Mori等人在1980年代完成的。过去5年的新结果为高维分类增加了许多微妙的层次。其中包括：->将MMP扩展到维度4（Shokurov），->有理连通簇（Campana和Kollár-Miyaoka-Mori），-> 3倍的明确分类结果（Corti，Kawakita和其他人），-> Calabi-Yaus，镜像对称性和弦理论应用，-> orbifolds和McKay对应的解决方案，->导出范畴作为变种的特征（Bondal，Orlov，Bridgeland等人），->双有理刚性（Sarkisov，Corti，Pukhlikov等人）。对3-folds的研究也在代数、几何和理论物理的几个不同分支中产生了丰富的应用。我的讲座将探讨其中的一些主题。
英文摘要：
The division of compact Riemann surfaces into 3 cases K_C<0, g=0, or K_C=0, g=1, or K_C>0, g>=2 is well known, and corresponds to the familiar trichotomy of spherical, Euclidean and hyperbolic non-Euclidean plane geometry. Classification aims to treat all projective algebraic varieties in terms of this trichotomy. The model is the treatment of surfaces by Castelnuovo and Enriques around 1900. Because the canonical class of a variety may not have a definite sign, we usually have to modify a variety by a minimal model program (MMP) before it makes sense to apply the trichotomy; this involves contractions, flips and fibre space decompositions. The classification of 3-folds was achieved by Mori and others during the 1980s. New results over the last 5 years have added many layers of subtlety to higher dimensional classification. These include: -> extension of MMP to dimension 4 (Shokurov), -> rational connected varieties (Campana and Kollár--Miyaoka--Mori), -> explicit classification results for 3-folds (Corti, Kawakita and others), -> Calabi-Yaus, mirror symmetry and string theory applications, -> resolution of orbifolds and McKay correspondence, -> the derived category as a characteristic of varieties (Bondal, Orlov, Bridgeland and others), -> birational rigidity (Sarkisov, Corti, Pukhlikov and others). The study of 3-folds also yields a rich crop of applications in several different branches of algebra, geometry and theoretical physics. My lecture surveys some of these topics.
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