# zbMATH — the first resource for mathematics

Pluricanonical systems of projective varieties of general type. I. (English) Zbl 1142.14012
For any complex projective manifold $$X$$ of general type with canonical divisor $$K_X$$ the rational map $$\varphi_m$$ defined by the pluricanonical system $$| mK_X|$$ is birational for $$m\geq m_0$$. It was proved by the author in a subsequent paper [Osaka J. Math. 44, No. 3, 723–764 (2007; Zbl 1186.14043)] and independently by other authors, that $$m_0$$ only depends on $$\dim X$$: For every $$n\in\mathbb N$$ there exists a number $$\nu_n\in\mathbb N$$ such that $$\varphi_m$$ is birational for every $$m\geq \nu_n$$, $$n=\dim X$$.
In the present paper the proof of this statement is still based on the additional assumption that every projective variety of general type has a minimal model, namely a projective variety $$X_m$$ which is birationally equivalent to $$X$$, with only $$\mathbb Q$$-factorial terminal singularities and with $$K_{X_m}$$ a nef $$\mathbb Q$$-Cartier divisor. The subadjunction theorem of Kawamata is also essential for the proof. It is shown that the theorem is equivalent to the following fact: For every $$n\in\mathbb N$$ there exists a positive number $$C_n$$ such that $n!\cdot\overline{\lim\limits_{m\rightarrow\infty}}m^{-n}\dim H^0(X,{\mathcal O}_X(mK_X))\geq C_n,$ for every complex projective manifold $$X$$ of general type with $$\dim X=n$$.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 14E05 Rational and birational maps 14J40 $$n$$-folds ($$n>4$$) 32U05 Plurisubharmonic functions and generalizations 14E25 Embeddings in algebraic geometry
Full Text:
##### References:
 [1] U. Anghern and Y.-T. Siu: Effective freeness and point separation for adjoint bundles , Invent. Math. 122 (1995), 291–308. · Zbl 0847.32035 [2] E. Bombieri: Canonical models of surfaces of general type , Publ. I.H.E.S. 42 (1972), 171–219. · Zbl 0259.14005 [3] T. Fujita: On the structure of polarized varieties with $$\Delta$$-genera zero , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103–115. · Zbl 0333.14004 [4] A. Grothendieck: Fondements de la Géometrie Algébrique, Collected Bourbaki Talks, Paris, 1962. [5] R. Harthshorne: Algebraic Geometry, GTM 52 , Springer, 1977. · Zbl 0367.14001 [6] L. Hörmander: An introduction to complex analysis in several variables. Third edition. North-Holland Mathematical Library 7 , North-Holland Publishing Co., Amsterdam, (1990). [7] Y. Kawamata: Subadjunction of log canonical divisors II, Amer. J. of Math. 120 (1998), 893–899, alg-geom math.AG/9712014. · Zbl 0919.14003 [8] Y. Kawamata: Fujita’s freeness conjecture for $$3$$-folds and $$4$$-folds , Math. Ann. 308 (1997), 491–505. · Zbl 0909.14001 [9] Y. Kawamata: Semipositivity, vanishing and applications, Lecture note in School on Vanishing Theorems and Effective Results in Algebraic Geometry (ICTP, Trieste, May 2000). [10] S. Kobayashi and T. Ochiai: Mappings into compact complex manifolds with negative first Chern class , Jour. Math. Soc. Japan 23 (1971), 137–148. · Zbl 0203.39101 [11] J. Kollár and S. Mori: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., Cambridge University Press, 1998. [12] S. Mori: Flip conjecture and the existence of minimal model for $$3$$-folds , J. Amer. Math. Soc. 1 (1988), 117–253. · Zbl 0649.14023 [13] A.M. Nadel: Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature , Ann. of Math. 132 (1990), 549–596. JSTOR: · Zbl 0731.53063 [14] N. Nakayama: Invariance of plurigenera of algebraic varieties , RIMS preprint 1191 (March 1998). [15] T. Ohsawa and K. Takegoshi: $$L^2$$-extension of holomorphic functions , Math. Z. 195 (1987), 197–204. · Zbl 0625.32011 [16] T. Ohsawa: On the extension of $$L^2$$ holomorphic functions. V @. Effects of generalization , Nagoya Math. J. 161 (2001), 1–21. · Zbl 0986.32002 [17] H. Tsuji: Analytic Zariski decomposition , Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 161–163. · Zbl 0786.14005 [18] H. Tsuji: Existence and applications of analytic Zariski decompositions ; in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends in Math. Birkhäuser Boston, Boston, MA, 1999, 253–272. · Zbl 0965.32022 [19] H. Tsuji; On the structure of pluricanonical systems of projective varieties of general type , preprint (1997). [20] H. Tsuji: Global generation of adjoint bundles , Nagoya Math. J. 142 (1996), 5–16. · Zbl 0861.32018 [21] H. Tsuji: Deformation invariance of plurigenera , Nagoya Math. J. 166 (2002), 117–134. · Zbl 1064.14035 [22] H. Tsuji: Pluricanonical systems of projective varieties of general type , math.AG/9909021 (1999). [23] H. Tsuji: Pluricanonical systems of projective varieties of general type II, math.CV/0409318 (2004).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.