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Introduction to the theory of quasi-log varieties
Abstract
This paper is a gentle introduction to the theory of quasi-log varieties by Ambro. We explain the fundamental theorems for the log minimal model program for log canonical pairs. More precisely, we give a proof of the base point free theorem for log canonical pairs in the framework of the theory of quasi-log varieties. Comment: 17 pages; comments welcome, v2: typos were corrected, v3: many errors were corrected, subsection 2.1 is new, v4: minor modifications, v5: typos were corrected, v6: title changed, a completely revised version, v7: minor modifications following referee's comments, v8: a typo was corrected
... For the details of the theory of quasi-log schemes, see [Fuj6,Chapter 6]. For a gentle introduction to the theory of quasi-log canonical pairs, we recommend the reader to see [Fuj3]. For the standard notations and conventions of the minimal model program, see [Fuj4]. ...
We prove Koll\'ar-type effective basepoint-free theorems for quasi-log canonical pairs.
... where each E i is an lc place of (X, D) with center in Z plus a surjective proper morphism E ?? Z and use the theory of quasi log varieties (see [1] und [2]. This possibly very much simplifies the theory because e.g. it avoids the necessity of proving vanishing theorems etc. on quasi log varieties. ...
In this paper, we give a valuation formula for rational top differential forms of function fields in characteristic zero for arbitrary Abhyankar places generalizing the classical valuation at prime divisors. This enables us to define log discrepancies for log pairs for arbitrary Abhyankar places. If the Abhyankar place has dimension greater than zero we restrict rational top differential forms with valuation zero to the residue field of the Abhyankar place, generalizing the classical restriction of a top differential form with a simple pole along a smooth divisor. This opens up the door to generalize the classical adjunction machinery to arbitrary Abhyankar places.
... 6]. For a gentle introduction to the theory of quasi-log canonical pairs, we recommend the reader to see [6]. For the standard notations and conventions of the minimal model program, see [7]. ...
We prove Kollár-type effective basepoint-free theorems for quasi-log canonical pairs.
... We note that a scheme means a separated scheme of finite type over C. For the details of the theory of quasi-log schemes, see [Fuj14a, Chapter 6] and [Fuj14b]. We also note that [Fuj11a] is a gentle introduction to the theory of quasi-log schemes. ...
We prove the Angehrn-Siu Type effective freeness and effective point
separation for quasi-log canonical pairs. As a natural consequence, we obtain
that these two results hold for semi-log canonical pairs. One of the main
ingredients of our proof is the inversion of adjunction for quasi-log canonical
pairs, which is established in this paper.
... Let us start the proof of Theorem 1.2 (see [F4,Corollary 2.9] when ∆ = 0). ...
We prove the Kodaira vanishing theorem for log-canonical and
semi-log-canonical pairs. We also give a relative vanishing theorem of
Reid--Fukuda type for semi-log-canonical pairs.
... We recommend the reader to see [F3] for some basic applications of the theory of quasi-log schemes. The adjunction and vanishing theorem (see, for example, [F3,Theorem 3.6]) is a key result for qlc pairs. ...
One of the main purposes of this paper is to make the theory of quasi-log
schemes more flexible and more useful. More precisely, we prove that the
pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a
natural quasi-log structure after clarifying the definition of quasi-log
schemes. We treat some applications to singular Fano varieties. This paper also
contains a proof of the simple connectedness of log canonical Fano varieties.
We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use -harmonic forms on noncompact K\"ahler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.
We introduce the notion of quasi-log complex analytic spaces and establish various fundamental properties. Moreover, we prove that a semi-log canonical pair naturally has a quasi-log complex analytic space structure.
The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro's paper hold true. The proof is Kawamata's X-method with the aid of the theory of basic slc-trivial fibrations. For the reader's convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.
We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.
We formulate and establish a generalization of Koll'ar's injectivity theorem for adjoint bundles twisted by a suitable multiplier ideal sheaf. As applications, we generalize Koll'ar's vanishing theorem, Koll'ar's torsion-freeness, generic vanishing theorem, and so on, for pseudo-effective line bundles. Our approach is not Hodge theoretic but is analytic since we treat singular hermitian metrics with arbitrary singularities. For the proof of the main injectivity theorem, we use the theory of harmonic integrals on noncompact K"ahler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems, which seems to be of independent interest.
We introduce various new operations for quasi-log structures. Then we prove
the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes as an
application.
We prove some injectivity, torsion-free, and vanishing theorems for simple
normal crossing pairs. Our results heavily depend on the theory of mixed Hodge
structures on compact support cohomology groups. We also treat several basic
properties of semi divisorial log terminal pairs.
We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kähler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kahler manifolds, flopping contractions, and so on, in order to clarify our results.
Kollár's effective base point free theorem for kawamata log terminal
pairs is very important and was used in Hacon-McKernan's proof of pl flips. In
this paper, we generalize Kollár's theorem for log canonical
pairs.
We give a short and almost self-contained proof of generalizations of Kollár’s vanishing and torsion-free theorems. Although they are contained in Ambro's much more general results on embedded normal crossing pairs, we give an alternate and direct reduction argument to the mixed Hodge theory. In this sense, this paper gives a more readable account of the application to the log minimal model program for log canonical pairs.
We prove some injectivity theorems. Our proof depends on the theory of mixed
Hodge structures on cohomology groups with compact support. Our injectivity
theorems would play crucial roles in the minimal model theory for
higher-dimensional algebraic varieties. We also treat some applications.
We discuss the variations of mixed Hodge structure for cohomology with
compact support of quasi-projective simple normal crossing pairs. We show that
they are graded polarizable admissible variations of mixed Hodge structure.
Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.
We prove that every quasi-projective semi log canonical pair has a quasi-log
structure with several good properties. It implies that various vanishing
theorems, torsion-free theorem, and the cone and contraction theorem hold for
semi log canonical pairs.
We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-dimensional. We also investigate the Kleiman-Mori cones of weak log Fano pairs with log canonical singularities. Comment: 18 pages, v2:some errors were corrected, title has been changed
In this paper, we prove the cone theorem and the contraction theorem for pairs , where X is a normal variety and B is an effective -divisor on X such that is -Cartier. Comment: 64 pages. This is a slightly revised version of Kyoto-Math 2009-25, v2: Theorem 10.5 is new, minor modifications, v3: minor modifications, Theorem 18.10 is new
We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties. Comment: 13 pages, v2: minor revisions, v3: minor revisions following referee's comments
We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves. Comment: 21 pages; v2: mistakes were corrected, v3: the proof of the main theorem was revised, v4: minor modifications, v5: minor revision following referee's comments
We reformulate base point free theorems. Our formulation is flexible and has
some important applications. One of the main purposes of this paper is to prove
a generalization of the base point free theorem in Fukuda's paper: On
numerically effective log canonical divisors.
We treat Kollár's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kolla ́r type cohomology injectivity theorems. Our main theorem is formulated for a compact Kähler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kähler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence. © Osaka University and Osaka City University, Departments of Mathematics.
We prove Angehrn-Siu type effective base point freeness and point separation for log canonical pairs. Comment: 12 pages, v2: minor modifications, v3: very minor modifications, v4: title changed, v5: revision following referee's comments, v6: very minor modifications, to appear in Michigan Math. J
We prove that the canonical ring of a smooth projective variety is finitely generated.
The theory of Mixed Hodge Structures (M.H.S.) on the cohomology of an algebraic variety X over complex numbers was found by Deligne in 1970. The case when A" is a Normal Crossing Divisor is fundamental. When the variety X is embedded in a smooth ambient space we get the Mixed Hodge Structure using standard exact sequences in topology. This technique uses resolution of singularities one time for a complete variety and 2 times for a quasi-projective one. As applications to the study of local cohomology we give the spectral sequence to the Mixed Hodge Structure on cohomology with support on a subspace Y.
Kollár's effective base point free theorem for kawamata log terminal
pairs is very important and was used in Hacon-McKernan's proof of pl flips. In
this paper, we generalize Kollár's theorem for log canonical
pairs.
We prove some injectivity theorems. Our proof depends on the theory of mixed
Hodge structures on cohomology groups with compact support. Our injectivity
theorems would play crucial roles in the minimal model theory for
higher-dimensional algebraic varieties. We also treat some applications.
In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Koll'ar's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita-Kawamata's semi-positivity theorem. By this semi-positivity theorem, we generalize Kawamata's positivity theorem and apply it to the study of a log canonical bundle formula. The final section is an appendix, which is a result of Morihiko Saito.
We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves. Comment: 21 pages; v2: mistakes were corrected, v3: the proof of the main theorem was revised, v4: minor modifications, v5: minor revision following referee's comments
We extend the Cone Theorem of the Log Minimal Model Program to log varieties with arbitrary singularities.
We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular (n+1)-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension . Comment: 14 pages; v2: completely revised and expanded version, v3: Section 5 in v2 was removed because it contained a conceptual mistake
We prove Angehrn-Siu type effective base point freeness and point separation for log canonical pairs. Comment: 12 pages, v2: minor modifications, v3: very minor modifications, v4: title changed, v5: revision following referee's comments, v6: very minor modifications, to appear in Michigan Math. J
On injectivity, vanishing and torsion-free theorems for algebraic varieties [F7] O. Fujino, Introduction to the log minimal model program for log canonical pairs
- O Fujino
O. Fujino, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 8, 95–100. [F7] O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint 2009.
Birational geometry of algebraic varieties With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original Kyoto 606-8502 Japan E-mail address: fujino@math.kyoto-u
- J Kollár
- S Mori
J. Kollár, S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan E-mail address: fujino@math.kyoto-u.ac.jp
Effective base point free theorem for log canonical pairs II— Angehrn–Siu type theorems—, to appear in Michigan Math Theory of non-lc ideal sheaves—basic properties—, preprint
- O Fujino
O. Fujino, Effective base point free theorem for log canonical pairs II—
Angehrn–Siu type theorems—, to appear in Michigan Math. J.
[F4]
O. Fujino, Theory of non-lc ideal sheaves—basic properties—, preprint
2007.
[F5]