Abstract
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.
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References
F. Ambro, The Adjunction Conjecture and its applications, arXiv:math/9903060v3.
C. Birkar, Boundedness of Fano type fibrations, Ann. Sci. ENS, in press, arXiv:2209.08797.
C. Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. Math., 193 (2021), 347–405.
C. Birkar, Anti-pluricanonical systems on Fano varieties, Ann. Math., 190 (2019), 345–463.
C. Birkar, Singularities on the base of a Fano type fibration, J. Reine Angew. Math., 715 (2016), 125–142.
C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. IHES, 115 (2012), 325–368.
C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468.
C. Birkar and D-Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. IHES, 123 (2016), 283–331.
S. Fukuda, A note on Ando’s paper “Pluricanonical systems of algebraic varieties of general type of dimension \(\le 5\)”, Tokyo J. Math., 14 (1991), 479–487.
C. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1–25.
C. D. Hacon, J. McKernan and C. Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc., 20 (2018), 865–901.
C. D. Hacon, J. McKernan and C. Xu, ACC for log canonical thresholds, Ann. Math. (2), 180 (2014), 523–571.
C. D. Hacon, J. McKernan and C. Xu, On the birational automorphisms of varieties of general type, Ann. Math. (2), 177 (2013), 1077–1111.
C. Jiang, On birational geometry of minimal threefolds with numerically trivial canonical divisors, Math. Ann., 365 (2016), 49–76.
M. Kapustka, G. Mongardi, G. Pacienza and P. Pokora, On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity, arXiv:1911.03367v2.
Y. Kawamata, Subadjunction of log canonical divisors, II, Am. J. Math., 120 (1998), 893–899.
J. Kollár, Families of Varieties of General Type, Cambridge University Press, Cambridge, 2023.
J. Kollár, Singularities of the Minimal Model Program, Cambridge University Press, Cambridge, 2013.
J. Kollár, in Singularities of Pairs, in Algebraic Geometry, Proc. Symp. Pure Math., vol. 62, Santa Cruz 1995, pp. 221–286, Am. Math. Soc., Providence, 1997.
R. Lazarsfeld, Positivity in Algebraic Geometry I, Springer, Berlin, 2004.
D. Martinelli, S. Schreieder and L. Tasin, On the number and boundedness of log minimal models of general type, Ann. Sci. Éc. Norm. Supér., 53 (2020), 1183–1210.
Y. Odaka, On log minimality of weak K-moduli compactifications of Calabi-Yau varieties, arXiv:2108.03832.
Y. Odaka, Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures, arXiv:2011.12748.
K. Oguiso and T. Peternell, On polarized canonical Calabi–Yau threefolds, Math. Ann., 301 (1995), 237–248.
I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math. (2), 127 (1988), 309–316.
V. V. Shokurov, 3-Fold log flips, with an appendix by Yujiro Kawamata, Russ. Acad. Sci. Izv. Math., 40 (1993), 95–202.
S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551–587.
H. Tsuji, Pluricanonical systems of projective varieties of general type I, Osaka J. Math., 43 (2006), 967–995.
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Birkar, C. Geometry of polarised varieties. Publ.math.IHES 137, 47–105 (2023). https://doi.org/10.1007/s10240-022-00136-w
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DOI: https://doi.org/10.1007/s10240-022-00136-w