Research of higher dinendional algebraic varaieties

The purpose of this research was to investigate The structure of higher dimensio We considered Thefollowing questions from the view point of numerical geometry，studies divisors on varieties by the calculation of intersection numbers.(I) What kind ofgeometric conclusions which are non-linear in nature can be obtained from the numerical conditions什么是自然的？（II）There are several important cones in the real vector space of numerical classes of divisors on agiven variety. What can we say the properties of these cones？for (i),Let X be a smooth projective variety of dimension n我们关注following Fujita conjectures.and H an ample divisor on X.Takao Fujita of Tokyo Institute of Technology conjectured that thelinear system |mH + Kx|is free if m >= n + 1，and very ample if m >= n + 2. This conjecture follows from the following stronger conjecture：1 . If (H^n) > n^n and (H^d·W) >= n^d holds for any subvariety W of any dimension d，then |H + Kx|is fr…More ee；（H^n） > (n + 1)^n and (H^d·W) >= (n + 1)^d holds for any subvariety W of any dimension d，then |H + Kx|is very ample. For example，one can ily check the above conjectures for X = P^n. They are true if n = 1 by the rieman - rochtheorem,and if n = 2 by Reider's results.I considered the freeness conjectures （the first part of the）因为base point free theorem which I proved before can be used （conjectures）to this问题，and obtained the following results：the stronger conjecture is true if n = 3，and the weaker one if n = 4. the result for n = 3 is based on the previous results by in- lazarsfeldand fujita,and uses results of in- lazarsfeld and Helmke.In the course of the proof of the above results，the analysis of the singularities of the minimal center of log canonical singularities is important。I realized that problem is related to the adjunction process of the canonical divisors，并提供了minimal center只能在日志中使用，如果它的资料是在大多数情况下2 by using the moduli space of pointed stable curves by knudson . for (II)，I investigated certain cones of divisors for Calabi-Yau fiber spaces.Accrding to the Minimal ModelConjecture, which is already verified in dimension 3any algebraic variety X0 has a birational model X with an algebraic fiber space *：X * S such that the canonical divisor Kx is expressed as Kx = *^<**> H for an ample Q-divisor H ons.t is fiber space *：X * S是一个Calabi-Yau fiber space example of a Calabi-Yau fiber space.它是一个通用的声明of a Calabi-Yau manifold这也是在理论物理中适用的。divisors on X relatively over S in the case in which dim X = 3. Such classes from a finitedimensional real vector space N^1 (X/S). The numerical classes of ample (resp. movable，divisors from a cone a (X/S) (resp. M （X/S）B (X/S)) inside N^1 (X/S). The faces of A (X/S) correspond to birational contractions or fiber spacestructures of X,and the decomposition of M (X/S) into subcones A (X'/S) corresponds to differentninimal models X'which are birationally equivalent to X. inspired by the Mirror Symmetry Conjecturefor Calabi-Yau manifolds， D.Morrison raised the following conjecture：there exist only finitely many equivalence classes of faces of A (X/S) under the action of thebiregular automorphism group Aut (X)and there exist only finitely many equivalence classes of subcones A (X/S) of M (X/S) under theaction of the birational automorphism group Bir (X/S). I studied these cones previously in a paperCrepant blowing-up of 3-dimensional canonical singularities and its application to degenerations ofsurfaces, Ann. of Math. 127 (1988), 93-163.In the paper [10]，I introduced a concept of marked minimal models，并提供了一个finiteness theorem for the part inside the cone B (X/S) by using some results in theabove paper and a later paper Termination of log-flips for algebraic 3-folds，Intl. J.Math. 3 (1992), 653-659. Moreover，I gave an affirmative answer to the above conjecture in the case in which dim S > 0 as the maintheorem. As a corollary，I proved that there exist only finitely many minimal models up to isomorphisms in a fixed birationalalgebraic 3-folds with positive Kodaira dimension.The reports of the abovewere much appreciated at the international conferences held at the University of Warwick inthe United Kingdom, and the Johns Hopkins University in the United States. Less

The purpose of this research was to investigate the structure of higher dimensio We considered the following questions from the view point of numerical geometry, which studies divisors on varieties by the calculation of intersection numbers.(I) What kind of geometric conclusions which are non-linear in nature can be obtained from the numerical conditions which are linear in nature?(II)There are several important cones in the real vector space of numerical classes of divisors on a given variety. What can we say about the properties of these cones?For (I), we considered the following Fujita conjectures. Let X be a smooth projective variety of dimension n and H an ample divisor on X.Takao Fujita of Tokyo Institute of Technology conjectured that the linear system |mH + Kx|is free if m >= n + 1, and very ample if m >= n + 2. This conjecture follows from the following stronger conjecture : (1) If (H^n) > n^n and (H^d・W) >= n^d holds for any subvariety W of any dimension d, then |H + Kx|is fr … More ee ; (2) (H^n) > (n + 1)^n and (H^d・W) >= (n + 1)^d holds for any subvariety W of any dimension d, then |H + Kx|is very ample. For example, one can easily check the above conjectures for X = P^n. They are true if n = 1 by the Riemann-Roch theorem, and if n = 2 by Reider's results.I considered the freeness conjectures (the first part of the conjectures) because the argument of the base point free theorem which I proved before can be used to this problem, and obtained the following results : the stronger conjecture is true if n = 3, and the weaker one if n = 4. The result for n = 3 is based on the previous results by Ein-Lazarsfeld and Fujita, and uses results of Ein-Lazarsfeld and Helmke.In the course of the proof of the above results, the analysis of the singularities of the minimal center of log canonical singularities is important. I realized that this problem is related to the adjunction process of the canonical divisors, and proved that the minimal center has only log terminal singularties if its codimension is at most 2 by using the moduli space of pointed stable curves by Knudsen.For (II), I investigated certain cones of divisors for Calabi-Yau fiber spaces.Accrding to the Minimal Model Conjecture, which is already verified in dimension 3, any algebraic variety X0 has a birational model X with an algebraic fiber space * : X * S such that the canonical divisor Kx is expressed as Kx = *^<**> H for an ample Q-divisor H on S.This fiber space * : X * S is an example of a Calabi-Yau fiber space. It is a generalized notion of a Calabi-Yau manifold which is also important in theoretical physics.I considered numerical equivalence classed of divisors on X relatively over S in the case in which dim X = 3. Such classes from a finite dimensional real vector space N^1 (X/S). The numerical classes of ample (resp. movable, big) divisors from a cone A (X/S) (resp. M (X/S), B (X/S)) inside N^1 (X/S). The faces of A (X/S) correspond to birational contractions or fiber space structures of X,and the decomposition of M (X/S) into subcones A (X'/S) corresponds to different ninimal models X'which are birationally equivalent to X.Inspired by the Mirror Symmetry Conjecture for Calabi-Yau manifolds, D.Morrison raised the following conjecture : there exist only finitely many equivalence classes of faces of A (X/S) under the action of the biregular automorphism group Aut (X/S), and there exist only finitely many equivalence classes of subcones A (X/S) of M (X/S) under the action of the birational automorphism group Bir (X/S). I studied these cones previously in a paper Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988), 93-163.In the paper [10], I introduced a concept of marked minimal models, and proved a finiteness theorem for the part inside the cone B (X/S) by using some results in the above paper and a later paper Termination of log-flips for algebraic 3-folds, Intl. J.Math. 3 (1992), 653-659. Moreover, I gave an affirmative answer to the above conjecture in the case in which dim S > 0 as the main theorem. As a corollary, I proved that there exist only finitely many minimal models up to isomorphisms in a fixed birational equivalence class of algebraic 3-folds with positive Kodaira dimension.The reports of the above results were much appreciated at the international conferences held at the University of Warwick in the United Kingdom, and at the Johns Hopkins University in the United States. Less