global smoothing of calabi yau threefolds yoshinori namikawa i j h m steenbrink 2 i department of mathematics sophia university kioi cho tokyo 102 japan 2 mathematical institute university of nijmegen toernooiveld 6525 ed nijmegen the netherlands oblatum 10 x 1994 13 vi 1995 introduction friedman fr has studied the relationship between local and global deforma tions of a threefold z with isolated hypersurface singularities which admits small resolutions one of his main results is as follows let z be a moishezon threefold with only ordinary double points pl p assume that the canon ical line bundle

on the pluricanonical maps of varieties of intermediate kodaira dimension in this paper we will prove a uniformity result for the iitaka fibration f x rightarrow y provided that the generic fiber has a good minimal model and the prokhorov shokurov conjecture holds in particular the result holds if the variation of f is zero or kappa x text dim x 1

kodaira dimension of irregular varieties let f x y be an algebraic fiber space with general fiber f if y is of maximal albanese dimension we show that x y f

guaranteed stability with subspace methods abstract we show how stability of models can be guaranteed when using the class of identification algorithms which have become known as subspace methods in many of these methods the a matrix is obtained or can be obtained as the product of a shifted matrix with a pseudo inverse we show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position then a stable model results the cost of this is some possibly large distortion of the results but in some applications that is outweighed by the advantage of guaranteed stability

on uniruled degenerations of algebraic varieties with trivial canonical divisor we give a uniruledness criterion on degenerate fibers in a family of varieties whose general fiber has a numerically trivial canonical divisor the criterion is related with the so called non nef locus of the canonical divisor of the total space

automorphism groups of positive entropy on minimal projective varieties abstract we determine the geometric structure of a minimal projective threefold having two independent and commutative automorphisms of positive topological entropy and generalize this result to higher dimensional smooth minimal pairs x g as a consequence we give an effective lower bound for the first dynamical degree of these automorphisms of x fitting the boundary case

nef dimension of minimal models we reduce the abundance conjecture in dimension 4 to the following numerical statement if the canonical divisor k is nef and has maximal nef dimension then k is big from this point of view we classify in dimension 2 nef divisors which have maximal nef dimension but which are not big

on birational geometry of minimal threefolds with numerically trivial canonical divisors for a minimal threefold x with k x equiv 0 and a nef and big weil divisor l on x we investigate the birational geometry inspired by l we prove that ml and k x ml give birational maps for all m ge 17 the result remains true under weaker assumption that l is big and has no stable base components

subadditivity of kodaira dimensions for fibrations of three folds in positive characteristics abstract in this paper we will prove subadditivity of kodaira dimensions for a fibration with possibly singular geometric generic fiber under certain nefness and relative semi ampleness conditions as an application for a fibration f x y of a smooth projective threefold over an algebraically closed field of characteristic p 5 under the assumption that y is of general type and non uniruled we prove subadditivity of kodaira dimensions when general fibers are smooth or when k x y is relatively big over y

on algebraic fiber spaces over varieties of maximal albanese dimension we study algebraic fiber spaces f x y where y is of maximal albanese dimension in particular we give an effective version a theorem of kawa mata if pm x 1 for some m 2 then the albanese map of x is surjective combining this with ch it follows that x is birational to an abelian variety if and only if p2 x 1 and q x dim x

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

viehweg s hyperbolicity conjecture for families with maximal variation we use the theory of hodge modules to construct viehweg zuo sheaves on base spaces of families with maximal variation and fibers of general type and more generally families whose geometric generic fiber has a good minimal model combining this with a result of campana p un we deduce viehweg s hyperbolicity conjecture in this context namely the fact that the base spaces of such families are of log general type this is approached as part of a general problem of identifying what spaces can support hodge theoretic objects with certain positivity properties

on ueno s conjecture k we show that if x is a smooth complex projective variety with kodaira dimension 0 then the kodaira dimension of a general fiber of its albanese map is at most h 0 omega 1 x

bergman kernels and the pseudoeffectivity of relative canonical bundles a criterion for the pseudoeffectivity of twisted relative canonical bundles of surjective projective maps

projectivity of the moduli space of stable log varieties and subadditivity of log kodaira dimension reference epfl article 231100doi 10 1090 jams 871 url http dx doi org 10 1090 jams 871 record created on 2017 09 19 modified on 2017 10 02

pluricanonical maps of varieties of maximal albanese dimension let x be a smooth complex projective algebraic variety of maximal albanese dimension we give a characterization of kappa x in terms of the set v 0 x omega x p in mbox rm pic 0 x h 0 x omega x otimes p ne 0 an immediate consequence of this is that the kodaira dimension kappa x is invariant under smooth deformations we then study the pluricanonical maps varphi m x dashrightarrow mathbb p h 0 x mk x we prove that if x is of general type varphi m is generically finite for m geq 5 and birational for m geq 5 mbox rm dim x 1 more generally we show that for m geq 6 the image of varphi m is of dimension equal to kappa x and for m geq 6 kappa x 2 varphi m is the stable canonical map

uniformisation in dimension four towards a conjecture of iitaka let x be a compact kahler manifold whose universal cover is mathbb c n a conjecture of iitaka claims that some finite etale cover of x is a torus we prove this conjecture in various cases in dimension four we also show that in the projective case iitaka s conjecture is a consequence of the non vanishing conjecture

on the bicanonical map of primitive varieties with q x mathrm dim x the degree and the euler number let x be a variety of maximal albanese dimension and of general type assume that q x mathrm dim x the albanese variety mathrm alb x is a simple abelian variety and the bicanonical map is not birational we prove that the euler number chi x omega x is equal to 1 and 2k x separates two distinct points over the same general point on mathrm alb x via mathrm alb x theorem 1 1

on the abundance theorem in the case of numerical kodaira dimension zero we present a shorter proof of the abundance theorem in the case of numerical kodaira dimension 0 proved by nakayama and its log generalization

building blocks of etale endomorphisms of complex projective manifolds etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true they are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak calabi yau varieties

potential density of rational points on the variety of lines of a cubic fourfold we prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field where sufficiently general means that a condition of terasoma type is satisfied these varieties have trivial canonical bundle and have geometric picard number equal to one

varieties fibered by good minimal models let f x y be an algebraic fiber space such that the general fiber has a good minimal model we show that if f is the iitaka fibration or if f is the albanese map with relative dimension no more than three then x has a good minimal model