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 \(\epsilon \)-lc singularities for \(\epsilon >0\), 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 \(d,\epsilon \). 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 \(\epsilon \)-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 \(\epsilon \)-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.

在本文中，我们研究了由充足且更普遍的 nef 和大 Weil 因数极化的射影簇的几何。首先我们研究线性系统的双有理有界性。我们表明，如果\(X\)是维度\(d\)的射影变体，其中\(\epsilon \) -lc 奇点为\(\epsilon >0\)，并且如果\(N\)是一个 nef和\(X\)上的大 Weil 除数使得\(N-K_{X}\)是伪有效的，那么线性系统\(|mN|\)定义了某个自然数\(m\)的双有理映射)仅取决于\(d,\epsilon \). 这是证明其他各种结果的关键。例如，它意味着如果\(N\)是维度\(d\)的 klt Calabi-Yau 变体上的大 Weil 除数（不一定是 nef） ，则线性系统\(|mN|\)定义了一个某些自然数\(m\)的双有理映射仅取决于\(d\)。它还给出了一些已知结果的新证明，例如，如果\(X\)是一个\(\epsilon \) -lc Fano 维数\(d\)的变体，则取\(N=-K_{X}\ )我们恢复\(|-mK_{X}|\)对于有界\(m\)的双有理性。

当\(K_{X}\)和\ ( N - K_{X}\)都是伪有效（此处\(X\)未假定为\(\epsilon \) -lc）。

使用以上内容，我们显示了在某些自然条件下极化品种的有界性。我们将这些扩展到由不包含 lc 中心的有效充足 Weil 因子极化的半对数规范 Calabi-Yau 对的有界性。我们将简要讨论这种极化 Calabi-Yau 对的射影粗模空间存在性的应用。