Singularities on complex varieties

在最小模型猜想中，证明了代数簇上奇点存在典范、最小和对数典范模型。这是证明了2和3维的情况下，但没有证明更高的维的情况下。本文证明了非退化超曲面奇点存在正则、极小和对数正则模型。然后我们得到了一个整体化的命题：一个完备复曲面簇的Delta正则超曲面具有典范模型和极小模型。证明了曲面法向奇异点的不变量-K^2达到最小值1/3，并且从上到下不存在聚点。这意味着，如果-K^2的值严格下降，那么从一个奇点构造一个新奇点的过程在有限阶段就停止了。另一方面，我们证明了存在自下而上的聚点，并且聚点都是有理数。证明了Kawachi-Masek关于法向曲面上伴随丛基点的结果推广到有边界的情形，证明了如果真实的解析曲线的分支数为零，则该曲线与非奇点是blow-analytic等价的，构造了一种特殊的Lagrange 3-torus，研究了范畴上的zeta函数.特别是我们研究的分布的拉普拉斯运营商的类别。

In the Minimal Model Conjecture, it is conjectured that there exist the canonical, minimal and log-canonical models for a singularity on an algebraic variety. This is proved for 2 and 3-dimensional cases, however not proved for the higher dimensional case. In this project we proved that there exist the canonical, minimal and log-canonical models for a non-degenerate hypersurface singularity. We then obtained the globalized statement : a DELTA-regular hypersurface of a complete toric variety has the canonical and minimal models. We also got the algorithm to construct these models.We proved that the invariant -K^2 for a normal surface singularity attains the minimum value 1/3 and these set has no accumulation point from above. This implies that the procedure to construct a new singularity from a singularity stops at finite stage, if the value of -K^2 goes strictly down by this procedure. On the other hand, we prove that there exist accumulation points from below and the accumulation poins are all rational number. And all positive integers are accumulation points.We prove a generalization of Kawachi-Masek's result on base points of adjoint bundles on a normal surface to the case with a boundary.We proved that if the number of branches of real analytic curve, then the curve is blow-analytically equivalent to a non-singular point.We construct a special Lagrangian 3-torus.We study zeta-function on categories. In particular we study the distribution of the Laplace operator on categories.