singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

on the birational geometry of fano 4 folds we study the birational geometry of a fano fourfold x from the point of view of mori dream spaces more precisely we study rational contractions of x here a rational contraction is a rational map f x dashrightarrow y where y is normal and projective which factors as a finite sequence of flips followed by a surjective morphism with connected fibers such f is called elementary if x y 1 where is the picard number we first give a characterization of non movable prime divisors in x when x 6 this is related to the study of birational and divisorial elementary rational contractions of x then we study the rational contractions of fiber type on x which are elementary or more generally quasi elementary the main result is that x 11 if x has an elementary rational contraction of fiber type and x 18 if x has a quasi elementary rational contraction of fiber type

on fujita s spectrum conjecture abstract we prove fujita s spectrum conjecture on the discreteness of pseudo effective thresholds for polarized varieties

log canonical models for the moduli space of curves the first divisorial contraction in this paper we initiate our investigation of log canonical models for m g as we decrease from 1 to 0 we prove that for the first critical value 9 11 the log canonical model is isomorphic to the moduli space of pseudostable curves which have nodes and cusps as singularities we also show that 7 10 is the next critical value i e the log canonical model stays the same in the interval 7 10 9 11 in the appendix we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space

on the birational automorphisms of varieties of general type we show that the number of birational automorphisms of a variety of general type x is bounded by c vol x kx where c is a constant that only depends on the dimension of x

on existence of log minimal models and weak zariski decompositions we first introduce a weak type of zariski decomposition in higher dimensions an mathbb r cartier divisor has a weak zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective mathbb r cartier divisor we then prove that there is a very basic relation between zariski decompositions and log minimal models which has long been expected we prove that assuming the log minimal model program in dimension d 1 a lc pair x z b of dimension d has a log minimal model in our sense if and only if kx b has a weak zariski decomposition z

bounding singular surfaces via chern numbers we prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and chern numbers as an application we prove that given r in mathbb r 0 and epsilon in 0 1 the class mathcal f r epsilon of 2 dimensional pairs x d of general type with epsilon klt singularities d with standard coefficients and 4c 2 x d c 1 2 x d le r forms a bounded family

efficient protocols for generating bipartite classical distributions and quantum states we investigate the fundamental problem of generating bipartite classical distributions or quantum states by designing efficient communication protocols and proving their optimality we establish a number of intriguing connections to fundamental measures in optimization convex geometry and information theory 1 to generate a classical distribution p x y we tightly characterize the minimum amount of quantum communication needed by the psd rank of p as a matrix a measure recently proposed by fiorini et al proc 44th acm symp theory comput pp 95 106 2012 in studies of the minimum size of extended formulations of optimization problems such as tsp this echos the previous characterization for the optimal classical communication cost by the nonnegative rank of p the result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it 2 when an approximation is allowed to generate a distribution x y p we present a classical protocol of the communication cost o c x y 1 where c x y is common information a well studied measure in information theory introduced by wyner ieee trans inf theory 21 2 163 179 1975 this also links nonnegative rank and common information two seemingly unrelated quantities in different fields 3 for approximately generating a quantum pure state we completely characterize the minimum cost by a corresponding approximate rank closing a possibly exponential gap left in ambainis etal siam j comput 32 6 1570 1585 2003

quantum game players can have advantage without discord the last two decades have witnessed a rapid development of quantum information processing a new paradigm which studies the power and limit of quantum advantages in various information processing tasks problems such as when quantum advantage exists and if existing how much it could be are at a central position of these studies in a broad class of scenarios there are implicitly or explicitly at least two parties involved who share a state and the correlation in this shared state is the key factor to the efficiency under concern in these scenarios the shared entanglement or discord is usually what accounts for quantum advantage in this paper we examine a fundamental problem of this nature from the perspective of game theory a branch of applied mathematics studying selfish behaviors of two or more players we exhibit a natural zero sum game in which the chance for any player to win the game depends only on the ending correlation we show that in a certain classical equilibrium a situation in which no player can further increase her payoff by any local classical operation whoever first uses a quantum computer has a big advantage over its classical opponent the equilibrium is fair to both players and as a shared correlation it does not contain any discord yet a quantum advantage still exists this indicates that at least in game theory the previous notion of discord as a measure of non classical correlation needs to be reexamined when there are two players with different objectives

self equivalences of stable module categories let p be an abelian p group e a cyclic p group acting freely on p and k an algebraically closed field of characteristic p 0 in this work we prove that every self equivalence of the stable module category of k p rtimes e comes from a self equivalence of the derived category of k p rtimes e work of puig and rickard allows us to deduce that if a block b with defect group p and inertial quotient e is rickard equivalent to k p rtimes e then they are splendidly rickard equivalent that is broue s original conjecture implies rickard s refinement of the conjecture in this case all of this follows from a general result concerning the self equivalences of the thick subcategory generated by the trivial module

on principal blocks of p constrained groups a theorem of k w roggenkamp and l l scott shows that for a finite group g with a normal p subgroup containing its own centralizer any two group bases of the integral group ring mathbb z g are conjugate in the units of mathbb z p g though the theorem presents itself in the work of others and appears to be needed there is no published account there seems to be a flaw in the proof because a theorem appearing in the survey k w roggenkamp the isomorphism problem for integral group rings of finite groups progress in mathematics 95 birkhauser basel 1991 pp 193 220 where the main ingredients of a proof are given is false in this paper it is shown how to close this gap at least if one is only interested in the conclusion mentioned above therefore some consequences of the results of a weiss on permutation modules are stated the basic steps of which any proof should consist are discussed in some detail in doing so a complete yet short proof of the theorem is given in the case that g has a normal sylow p subgroup 2000 mathematical subject classification primary 16u60 secondary 20c05

on picard groups of blocks of finite groups we show that the subgroup of the picard group of ap block ofa finite group given by bimodules with endopermutation sources modulo theautomorphism group of a source algebra is determined locally in terms of thefusion system on a defect group we show that the picard group of a block overa complete discrete valuation ringoof characteristic zero with an algebraicclosurekoffpas residue field is a colimit of finite picard groups of blocks overp adic subrings ofo we apply the results to blocks with an abelian defectgroup and frobenius inertial quotient and specialise this further to blockswith cyclic or klein four defect groups

towards the ample cone of mg n in this paper we study the ample cone of the moduli space m g n of stable n pointed curves of genus g our motivating conjecture is that a divisor on m g n is ample iff it has positive intersection with all 1 dimensional strata the components of the locus of curves with at least 3g n 2 nodes this translates into a simple conjectural description of the cone by linear inequalities and as all the 1 strata are rational includes the conjecture that the mori cone is polyhedral and generated by rational curves our main result is that the conjecture holds iff it holds for g 0 more precisely there is a natural finite map r m0 2g n m g n whose image is the locus rg n of curves with all components rational any 1 strata either lies in rg n or is numerically equivalent to a family e of elliptic tails and we show that a divisor d is nef iff d e 0 and r d is nef we also give results on contractions i e morphisms with connected fibers to projective varieties of m g n for g 1 showing that any fibration factors through a tautological one given by forgetting points and that the exceptional locus of any birational contraction is contained in the boundary

termination of many 4 dimensional log flips we prove that any sequence of 4 dimensional log flips that begins with a klt pair x d such that k x d is numerically equivalent to an effective divisor terminates this implies termination of flips that begin with a log fano pair and termination of flips in a relative birational setting we also prove termination of directed flips with big k x d as a consequence we prove existence of minimal models of 4 dimensional dlt pairs of general type existence of 5 dimensional log flips and rationality of kodaira energy in dimension 4

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

the structure of blocks with a klein four defect group we prove erdmann s conjecture stating that every block with a klein four defect group has a simple module with trivial source and deduce from this that puig s finiteness conjecture holds for source algebras of blocks with a klein four defect group the proof uses the classification of finite simple groups

reduced tame and exotic fusion systems we define here two new classes of saturated fusion systems reduced fusion systems and tame fusion systems these are motivated by our attempts to better understand and search for exotic fusion systems fusion systems which are not the fusion systems of any finite group our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable and that every reduced fusion system which is not tame is the reduction of some exotic nonrealizable fusion system

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

asymptotic base loci on singular varieties we prove that the non nef locus and the restricted base locus of a pseudoeffective divisor coincide on klt pairs we also extend to klt pairs f russo s characterization of nef and abundant divisors by means of asymptotic multiplier ideals

on the birational geometry of spaces of complete forms ii skew forms abstract moduli spaces of complete skew forms are compactifications of spaces of skew symmetric linear maps of maximal rank on a fixed vector space where the added boundary divisor is simple normal crossing in this paper we compute their effective nef and movable cones the generators of their cox rings and for those spaces having picard rank two we give an explicit presentation of the cox ring furthermore we give a complete description of both the mori chamber and stable base locus decompositions of the effective cone of some spaces of complete skew forms having picard rank at most four

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

infinitely generated pseudocompact modules for finite groups and weiss theorem abstract one of the most beautiful results in the integral representation theory of finite groups is a theorem of a weiss that detects a permutation r lattice for the finite p group g in terms of the restriction to a normal subgroup n and the n fixed points of the lattice where r is a finite extension of the p adic integers using techniques from relative homological algebra we generalize weiss theorem to the class of infinitely generated pseudocompact lattices for a finite p group allowing r to be any complete discrete valuation ring in mixed characteristic a related theorem of cliff and weiss is also generalized to this class of modules the existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result the permutation cover is explicitly described

isomorphisms of p adic group rings a long standing problem first posed by graham higman 15 and later by brauer 4 is the isomorphism problem for integral group rings given finite groups g and h is it true that zg zh implies g 2 h many authors have worked on this question but progress has been difficult 30 perhaps the best positive result was that of whitcomb in 1968 37 who showed that the implication g h holds for g metabelian dade 9 showed there were counterexamples even in the metabelian case if z were replaced by the family of all fields some mathematicians came to believe the problem was a kind of grammatical accident that counterexamples for z were surely there if difficult to find we ourselves began this investigation looking for counterexamples but found that they were indeed very difficult to find much more difficult than we had anticipated slowly we began to believe that at least some exciting positive results were possible in this paper we answer the isomorphism problem for finite p groups over the p adic integers z zcp and in a very strong way in the normalized units augmentation 1 of z pg there is only one con gay czu ss of groups of order g 1 this answers the isomorphism problem in the affirmative for finite p groups g over z or zp and in addition computes the entire picard group 2 of the category of z g modules in terms of automorphisms of g similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal picard groups we also treat more general coefficient domains namely integral domains s of characteristic 0 in which no rational prime divisor of the group order is invertible for the sg isomorphism problem and treat similar semilocal dedekind domains for the picard group computations the zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting cf corollary 3 below over z we announce a positive answer to the isomorphism

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

ascending chain condition for log canonical thresholds and termination of log flips we prove that the ascending chain condition acc for log canonical lc thresholds in dimension d existence of log flips in dimension d and the log minimal model program lmmp in dimension d 1 imply termination of any sequence of log flips starting with a d dimensional effective lc pair and also imply termination of flops in dimension d in particular the latter terminations in dimension 4 follow from the alexeev borisov conjecture in dimension 3

boundedness of pluricanonical maps of varieties of general type using the techniques of 20 and 10 we prove that certain log forms may be lifted from a divisor to the ambient variety as a consequence of this result following 22 we show that for any positive integer n there exists an integer rn such that if x is a smooth projective variety of general type and dimension n then phi rk x colon x dasharrow mathbb p h 0 mathcal o x rk x is birational for all r rn

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

some examples of picard groups of blocks abstract we calculate examples of picard groups for 2 blocks with abelian defect groups with respect to a complete discrete valuation ring these include all blocks with abelian 2 groups of 2 rank at most three with the exception of the principal block of j 1 in particular this shows directly that all such picard groups are finite and picent the group of morita auto equivalences fixing the centre is trivial these are amongst the first calculations of this kind further we prove some general results concerning picard groups of blocks with normal defect groups as well as some other cases

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

termination of flops for 4 folds the purpose of this note is to prove the termination of flops for 4 folds i e the second part of the so called flop conjecture in dimension 4 which asserts that there is no infinite sequence of flops with respect to a fixed reference divisor for projective 4 folds having only terminal singularities our idea for the proof stems from crossbreeding the ideas of kollar in ko3 where he proves the termination of flops for 3 folds with those of kawamata in kmm where he shows the termination of flips for 4 folds we should mention that the notion of difficulty due to shokurov si plays the most basic role in the ar

weak solutions to degenerate complex monge ampere flows ii abstract studying the long term behavior of the kahler ricci flow on mildly singular varieties one is naturally led to study weak solutions of degenerate parabolic complex monge ampere equations the purpose of this article the second of a series on this subject is to develop a viscosity theory for degenerate complex monge ampere flows on compact kahler manifolds our general theory allows in particular to define and study the normalized kahler ricci flow on varieties with canonical singularities generalizing results of song and tian

on the existence of f thresholds and related limits we show the existence of f thresholds in full generality in addition we study properties of standard graded algebras over a field for which f pure threshold and f threshold at the irrelevant maxi

on codimension 1 del pezzo foliations on varieties with mild singularities in this paper we extend to the singular setting the theory of fano foliations developed in our previous paper araujo and druel adv math 238 70 118 2013 a mathbb q fano foliation on a complex projective variety x is a foliation fancyscript f subsetneq t x whose anti canonical class k fancyscript f is an ample mathbb q cartier divisor in the spirit of kobayashi ochiai theorem we prove that under some conditions the index i fancyscript f of a mathbb q fano foliation is bounded by the rank r of fancyscript f and classify the cases in which i fancyscript f r next we consider mathbb q fano foliations fancyscript f for which i fancyscript f r 1 these are called del pezzo foliations we classify codimension 1 del pezzo foliations on mildly singular varieties

height zeta functions of equivariant compactifications of unipotent groups we prove manin s conjecture for bi equivariant compactifications of unipotent groups 2015 wiley periodicals inc