On partially ample divisors (Q2929424)

Let \(X\) be a \(\mathbb{Q}\)-factorial normal projective variety over an algebraically closed field of characteristic \(0\). If \(D\) is a \(\mathbb{Z}\)-divisor on \(X\), then Bs\(|D|\) denotes the base locus of the complete linear system \(|D|\). If \(D\) is a \(\mathbb{Q}\)-divisor on \(X\), then the stable base locus \(\mathrm{SB}(D)=\bigcap \mathrm{Bs}|mD|\), where the intersection is over the positive integers \(m\) such that \(mD\) are \(\mathbb{Z}\)-divisors. \(D\) is cohomologically \(k\)-ample for an integer \(k\) with \(0\leq k< d=\dim X\), if for any coherent sheaf \(\mathcal{F}\) on \(X\), there is a positive integer \(m_1\in \mathbb{N}\) (depending on the sheaf) such that for any \(i>k\) and any \(m>m_1\), NEWLINE\[NEWLINE h^i(X, {\mathcal{F}}\otimes {\mathcal{O}}_X(mm_0D))=0, NEWLINE\]NEWLINE where \(m_0\in \mathbb{N}\) and \(m_0D\) is integral.NEWLINENEWLINEIf \(D\) is a \(\mathbb{R}\)-divisor on \(X\), then the non-ample locus (or augmented base locus) of \(D\) is defined to be NEWLINE\[NEWLINE {\mathbf{B}}_+(D)= \bigcap SB (D-A), NEWLINE\]NEWLINE where the intersection is taken over all ample divisors \(A\) such that \(D-A\) are \(\mathbb{Q}\)-divisors. \(D\) is numerically \(k\)-ample if \(\dim {\mathbf{B}}_+(D)\leq k\) for some \(k\in \mathbb{Z} \) such that \(0\leq k< d=\dim X\). \(D\) is cohomologically \(k\)-ample if \(D\) is numerically equivalent to \(cD'+A\), where \(c>0\), \(D'\) is a cohomologically \(k\)-ample \(\mathbb{Q}\)-divisor and \(A\) is an ample divisor. \(D\) is asymptotically \(k\)-ample if for all integer \(i>k\), \(0\leq k< d=\dim X\) and all divisors \(D'\) in some neighborhood of \(D\) in \(N^1(X)_{\mathbb{R}}\), NEWLINE\[NEWLINE \hat{h}^i(X, D')= \lim \sup_{m\rightarrow \infty}\frac{h^i(X, mD')}{m^d/d!}=0. NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, the author proves the following two main theorems.NEWLINENEWLINETheorem 1.1. For a big and nef divisor \(D\) on \(X\), all three notions of partial ampleness coincide.NEWLINENEWLINETheorem 1.3. If \(D\) is a big divisor on a variety of dimension \(d\geq 2\) and \(D=P+N\) be the good Fujita-Zariski decomposition. Then the following three conditions hold. {\parindent=6mm \begin{itemize}\item[(1)] \(D, P\) are numerically \((d-1)\)-ample. \item[(2)] If \(D\) is cohomologically \(k\)-ample, then \(P\) is cohomologically \(k\)-ample. \item[(3)] If \(D\) is asymptotically \(k\)-ample, then \(P\) is asymptotically \(k\)-ample.NEWLINENEWLINE\end{itemize}}