论文标题
偏光品种的几何形状
Geometry of polarised varieties
论文作者
论文摘要
在本文中，我们调查了射影品种的几何形状，由充足和更一般的nef和大魏尔除数两极化。首先，我们研究线性系统的生育界限。 We show that if $X$ is a projective variety of dimension $d$ with $ε$-lc singularities for $ε>0$, and if $N$ is a nef and big Weil divisor on $X$ such that $N-K_X$ is pseudo-effective, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d,ε$.这是证明其他各种结果的关键。例如，这意味着，如果$ n $是Klt calabi-yau的多种尺寸$ d $的大魏尔除数（不一定是nef），那么线性系统$ | mn | $定义了仅在$ d $上的一些自然数量$ m $的birational Map。例如，它还提供了一些已知结果的新证明，例如，如果$ x $是$ε$ -lc fano dimension $ d $的各种$ n = -k_x $，我们恢复了$ | -MK_X | $的birationality，用于有限的$ m $。 当$ k_x $ and $ k_x $和$ n-k_x $均为伪有效时，我们证明了Nef和Big Weil Divisors $ n $的Big Weil Divisors $ n $ $ n $的类似的Birational Bounded结果（在此不假定$ X $ $ X $）。 使用上述情况，我们在某些自然条件下显示了极化品种的界限。我们将其扩展到半gog典型的calabi-yau对的界限，这是有效的不含LC中心的有效的Weil除数极化的。我们将简要讨论这种偏光性卡拉比对的投射粗模量空间的应用。
In this paper we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if $X$ is a projective variety of dimension $d$ with $ε$-lc singularities for $ε>0$, and if $N$ is a nef and big Weil divisor on $X$ such that $N-K_X$ is pseudo-effective, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d,ε$. This is key to proving various other results. For example, it implies that if $N$ is a big Weil divisor (not necessarily nef) on a klt Calabi-Yau variety of dimension $d$, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d$. It also gives new proofs of some known results, for example, if $X$ is an $ε$-lc Fano variety of dimension $d$ then taking $N=-K_X$ we recover birationality of $|-mK_X|$ for bounded $m$. We prove similar birational boundedness results for nef and big Weil divisors $N$ on projective klt varieties $X$ when both $K_X$ and $N-K_X$ are pseudo-effective (here $X$ is not assumed $ε$-lc). Using the above, we show boundedness of polarised varieties under some natural conditions. We extend these to boundedness of semi-log canonical Calabi-Yau pairs polarised by effective ample Weil divisors not containing lc centres. We will briefly discuss applications to existence of projective coarse moduli spaces of such polarised Calabi-Yau pairs.
