Singularities on complex varieties

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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.