Adjoint pluricanonical systems on varieties of general type
Let \(X\) be a smooth complex projective variety of general type of dimension \(d\) and let \(K_X\) be the canonical bundle. Then, for any sufficiently high integer \(n\): a) the plurigenera \(P_n=h^0(nK_X)\) grow like \(n^d\) and b) the rational map \(\varphi_n\) defined by \(|nK_X|\) is birational. This leads to wonder if it is possible to determine an explicit number \(n_d\), potentially the minimum, depending only on \(d\), such that: a) \(P_n > 0\) or b) \(\varphi_n\) is birational, for every \(n \geq n_d\). E.\ g., for surfaces, \(P_n > 0\) for \(n \geq 2\) and \(\varphi_n\) is birational for \(n \geq 5\). In higher dimensions recent advances have been obtained by several authors. In a similar perspective, one can also replace \(K_X\) with other line bundles, in particular big adjoint bundles. In line with this, the author investigates non-vanishing and birationality for big adjoint bundles like \(K_X + \ell(K_X+L)\) (adjoint pluricanonical bundles) where \(L\) is a big line bundle of sufficiently large volume, on a variety \(X\) of general type. In particular, he proves results of the following type for pairs \((X,L)\) as above, as corollaries of very precise estimates: 1) if \(d=2\), then \(h^0(2K_X+L)>0\) and \(|3K_X+2L|\) gives a birational map; 2) if \(d=3\), then \(h^0(6K_X+5L)>0\) and \(|8K_X+7L|\) gives a birational map. Similar results, with larger numbers, are obtained also for \(d=4\).
scientific article; zbMATH DE number 6202446