configurations in abelian categories ii ringel hall algebras abstract this is the second in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of i configurations in a using the theory of artin stacks it showed well behaved moduli stacks obj a m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod k q of representations of a quiver q write cf obj a for the vector space of q valued constructible functions on the stack obj a motivated by the idea of ringel hall algebras we define an associative multiplication on cf obj a using pushforwards and pullbacks along 1 morphisms between configuration moduli stacks so that cf obj a is a q algebra we also study representations of cf obj a the lie subalgebra cf ind obj a of functions supported on indecomposables and other algebraic structures on cf obj a then we generalize all these ideas to stack functions sf obj a a universal generalization of constructible functions containing more information when ext i x y 0 for all x y a and i 1 or when a coh p for p a calabi yau 3 fold we construct lie algebra morphisms from stack algebras to explicit algebras which will be important in the sequels on invariants counting semistable objects in a

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

moduli stacks and invariants of semistable objects on k3 surfaces abstract for a k3 surface x and its bounded derived category of coherent sheaves d x we have the notion of stability conditions on d x in the sense of t bridgeland in this paper we show that the moduli stack of semistable objects in d x with a fixed numerical class and a phase is represented by an artin stack of finite type over c then following d joyce s work we introduce the invariants counting semistable objects in d x and show that the invariants are independent of a choice of a stability condition

new regularity conditions for strong and total fenchel lagrange duality in infinite dimensional spaces abstract we give new regularity conditions for convex optimization problems in separated locally convex spaces we completely characterize the stable strong and strong fenchel lagrange duality then we give similar statements for the case when a solution of the primal problem is assumed as known obtaining complete characterizations for the so called total and stable totalfenchel lagrange duality respectively for particular settings the conditions that we consider turn into some constraint qualifications already used by different authors like farkas minkowski cq locally farkas minkowski cq and basic cq and we rediscover and improve some recent results from the literature

stability conditions and extremal contractions we study extremal contractions from smooth projective varieties via a moduli theoretic approach in the two dimensional case we show that any extremal contraction appears as a moduli space of bridgeland stable objects in the derived category of coherent sheaves in the three dimensional case we show that a a similar result holds with respect to conjectural bridgeland stability conditions

moduli of pt semistable objects i abstract we show boundedness for pt semistable objects of any chern classes on a smooth projective three fold x then we show that the stack of objects in the heart coh 1 x coh 2 x 1 satisfies a version of the valuative criterion for completeness in the remainder of the paper we give a series of results on how to compute cohomology with respect to this heart

limit stable objects on calabi yau 3 folds the notion of limit stability is introduced in 31 to give a limiting degeneration of bridgeland stability in this article we give the survey of the results in 31 30 we introduce the limit stability counting invariants of limit stable objects and see that the wall crossing phenomena yields the rationality conjecture of pandharipandethomas generating functions

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

deformation obstruction theory for complexes via atiyah and kodaira spencer classes we give a universal approach to the deformation obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family we recover and generalise the obstruction class of lowen and lieblich and prove that it is a product of atiyah and kodaira spencer classes this allows us to obtain deformation invariant virtual cycles on moduli spaces of objects of the derived category on threefolds

constraint qualifications for extended farkas s lemmas and lagrangian dualities in convex infinite programming for an inequality system defined by a possibly infinite family of proper functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions under the new constraint qualifications we obtain characterizations of those reverse convex inequalities which are a consequence of the constrained system and we provide necessary and or sufficient conditions for a stable farkas lemma to hold similarly we provide characterizations for constrained minimization problems to have the strong or strong stable lagrangian dualities several known results in the conic programming problem are extended and improved

on strong and total lagrange duality for convex optimization problems abstract we give some necessary and sufficient conditions which completely characterize the strong and total lagrange duality respectively for convex optimization problems in separated locally convex spaces we also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals in the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so called farkas minkowski and locally farkas minkowski conditions for systems of convex inequalities recently used in the literature moreover we show that our new results extend some existing ones in the literature

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

on stability conditions for the quintic threefold we study the clifford type inequality for a particular type of curves c 2 2 5 which are contained in smooth quintic threefolds this allows us to prove some stronger bogomolov gieseker type inequalities for chern characters of stable sheaves and tilt stable objects on smooth quintic threefolds employing the previous framework by bayer bertram macri stellari and toda we construct an open subset of stability conditions on every smooth quintic threefold in mathbf p 4 mathbb c

smooth random change point models change point models are used to describe processes over time that show a change in direction an example of such a process is cognitive ability where a decline a few years before death is sometimes observed a broken stick model consists of two linear parts and a breakpoint where the two lines intersect alternatively models can be formulated that imply a smooth change between the two linear parts change point models can be extended by adding random effects to account for variability between subjects a new smooth change point model is introduced and examples are presented that show how change point models can be estimated using functions in r for mixed effects models the bayesian inference using winbugs is also discussed the methods are illustrated using data from a population based longitudinal study of ageing the cambridge city over 75 cohort study the aim is to identify how many years before death individuals experience a change in the rate of decline of their cognitive ability copyright 2010 john wiley sons ltd

riemann hilbert problems from donaldson thomas theory we study a class of riemann hilbert problems arising naturally in donaldson thomas theory in certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions we briefly explain connections with gromov witten theory and exact wkb analysis

configurations in abelian categories iii stability conditions and identities this is the third in a series on configurations in an abelian category given a finite poset i an i configuration s p is a finite collection of objects s j and morphisms j k or in satisfying some axioms where j k are subsets of i configurations describe how an object x in decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in using the theory of artin stacks it showed well behaved moduli stacks of objects and configurations in exist when is the abelian category coh p of coherent sheaves on a projective scheme p or of representations of a quiver q the second studied algebras of constructible functions and stack functions on this paper introduces weak stability conditions t t on we show the moduli spaces of t semistable indecomposable t semistable and t stable objects in class a are constructible sets in and some associated configuration moduli spaces constructible in so their characteristic functions and are constructible we prove many identities relating these constructible functions and their stack function analogues under pushforwards we introduce interesting algebras of constructible and stack functions and study their structure in the fourth paper we show are independent of t t and construct invariants of

the space of stability conditions on abelian threefolds and on some calabi yau threefolds we describe a connected component of the space of stability conditions on abelian threefolds and on calabi yau threefolds obtained as the crepant resolution of a finite quotient of an abelian threefold our proof includes the following essential steps 1 we simultaneously strengthen a conjecture by the first two authors and toda and prove that it follows from a more natural and seemingly weaker statement this conjecture is a bogomolov gieseker type inequality involving the third chern character of tilt stable two term complexes on smooth projective threefolds we extend it from complexes of tilt slope zero to arbitrary tilt slope 2 we show that this stronger conjecture implies the so called support property of bridgeland stability conditions and the existence of an explicit open subset of the space of stability conditions 3 we prove our conjecture for abelian threefolds thereby reproving and generalizing a result by maciocia and piyaratne important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall crossing closely related to the support property

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities abstract in this paper we present constraint qualifications which completely characterize the farkas minkowski and the locally farkas minkowski convex possibly infinite inequality systems posed in topological vector spaces the number of constraints and the dimension of the linear space are arbitrary possibly infinite the constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex linear systems and the uniform strong duality or the uniform min max duality relative to the lagrange haar dual problems of suitable convex linear parametric optimization problems

an alternative formulation for a new closed cone constraint qualification abstract we give an alternative formulation for the so called closed cone constraint qualification cccq related to a convex optimization problem in banach spaces recently introduced in the literature this new formulation allows to prove in a simple way that cccq is weaker than some generalized interior point constraint qualifications given in the past by means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

on constraint qualification for an infinite system of convex inequalities in a banach space for a general infinite system of convex inequalities in a banach space we study the basic constraint qualification and its relationship with other fundamental concepts including various versions of conditions of slater type the mangasarian fromovitz constraint qualification as well as the pshenichnyi levin valadier property introduced by li nahak and singer applications are given in the restricted range approximation problem constrained optimization problems as well as in the approximation problem with constraints by conditionally positive semidefinite functions