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

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

the dt pt correspondence for smooth curves we show a version of the dt pt correspondence relating local curve counting invariants encoding the contribution of a fixed smooth curve in a calabi yau threefold we exploit a local study of the hilbert chow morphism about the cycle of a smooth curve we compute via quot schemes the global donaldson thomas theory of a general abel jacobi curve of genus 3

motivic donaldson thomas invariants of the conifold and the refined topological vertex abstract we compute the motivic donaldson thomas theory of the resolved conifold in all chambers of the space of stability conditions of the corresponding quiver the answer is a product formula whose terms depend on the position of the stability vector generalizing known results for the corresponding numerical invariants our formulae imply in particular a motivic form of the dt pt correspondence for the resolved conifold the answer for the motivic pt series is in full agreement with the prediction of the refined topological vertex formalism

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

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

proper holomorphic mappings of balanced domains in mathbb c n we extend a well known result about the unit ball by h alexander to a class of balanced domains in mathbb c n n 1 specifically we prove that any proper holomorphic self map of a certain type of balanced finite type domain in mathbb c n n 1 is an automorphism the main novelty of our proof is the use of a recent result of opshtein on the behaviour of the iterates of holomorphic self maps of a certain class of domains we use opshtein s theorem together with the tools made available by finiteness of type to deduce that the aforementioned map is unbranched the monodromy theorem then delivers the result

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

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

on the c version of the reflection principle for mappings between cr manifolds we prove a smooth version of the classical schwarz reflection principle for cr mappings between an abstract cr manifold m and a generic cr manifold embedded in euclidean complex space as a consequence of our results we settle a conjecture of x huang formulated in 1994

multiple cover formula of generalized dt invariants i parabolic stable pairs abstract in this paper we introduce the notion of parabolic stable pairs on calabi yau 3 folds and invariants counting them by applying the wall crossing formula developed by joyce song kontsevich soibelman we see that they are related to generalized donaldson thomas invariants counting one dimensional semistable sheaves on calabi yau 3 folds consequently the conjectural multiple cover formula of generalized dt invariants is shown to be equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants the application of this result to the multiple cover formula will be pursued in the subsequent paper

on the proper holomorphic equivalence for a class of pseudoconvex domains a complete and explicit description of the holomorphic proper mappings between weakly pseudoconvex domains of the class ap see below is given the study of proper holomorphic mappings i e holomorphic mappings which antitransform compact sets into compact sets 1 f d d2 d1 d2 c cn n 1 between bounded pseudoconvex domains with smooth boundary has had recently a considerable impulse and has been applied mainly to investigate a under which hypothesis there is no obstruction for the existence of such an f b the regularity up to the boundary of 1 c for which domains proper means in effect biholomorphic the first class of problems has been taken into consideration for instance in 8 5 4 and a typical result is for example that a strictly pseudoconvex domain cannot be mapped properly onto a weakly pseudoconvex domain about b see for instance 4 2 3 and the bibliography there included the property enunciated in c has been proved in the case d d2 unit ball see 1 and 9 when d and d2 are strictly pseudoconvex domains with in addition d2 simply connected see 7 and 4 and in the case d d2 and strictly pseudoconvex see 7 the aim of this note is to look at questions of type a and c above posed for the following class of weakly pseudoconvex domains ap z 1j z2p lzn 2pn p pn e z n precisely we prove theorem in order for a proper holomorphic mapping from ap onto a q to exist it is necessary and sufficient that plq p llql pnlqn e z furthermore assuming the only proper holomorphic mapping between a p and a q is up to biholomorphisms of a q f z zfl l zpnqn received by the editors october 25 1982 and in revised form march 29 1983 1980 mathematics subject classification primary 32h99 secondary 32f15 work performed with finanical aid from m p i di1984 american mathematical society 0002 9947 84 1 00 25 per page

curve counting and dt pt correspondence for calabi yau 4 folds abstract recently cao maulik toda defined stable pair invariants of a compact calabi yau 4 fold x their invariants are conjecturally related to the gopakumar vafa type invariants of x defined using gromov witten theory by klemm pandharipande in this paper we consider curve counting invariants of x using hilbert schemes of curves and conjecture a dt pt correspondence which relates these to stable pair invariants of x after providing evidence in the compact case we define analogous invariants for toric calabi yau 4 folds we formulate a vertex formalism for both theories and conjecture a relation between the fully equivariant dt pt vertex which we check in several cases this relation implies a dt pt correspondence for toric calabi yau 4 folds with primary insertions

representations of robustly stabilizing controllers for systems with parameter uncertainty in this paper we consider the problem of robust stabilization for systems with parameter uncertainty we first formulate a design methodology and then demonstrate how it leads to analytical and graphical solutions we show that in the case of siso plant families with linear uncertainty and two parameter compensators one can obtain a representation of all robustly stabilizing controllers of a fixed order by introducing a rectangular overbound of value sets in the complex plane we also show how one can obtain representations of m parameter robustly stabilizing compensators of a fixed order

on reducing submodules of hilbert modules with g n invariant kernels fix a bounded domain omega subset of c n and a positive definite kernel k on omega both invariant under g n the permutation group on n symbols let h subset of hol omega be the hilbert module determined by k we show that h splits into orthogonal direct sum of subspaces pph indexed by the partitions p of n we prove that each submodule pph is a locally free hilbert module of rank equal to square of the dimension chi p 1 of the irreducible representation corresponding to p given two partitions p and q we show that if chi p 1 not equal chi q 1 then the sub modules pph and pqh are not unitarily equivalent we prove that if h is a contractive analytic hilbert module on omega then the taylor joint spectrum of the n tuple of multiplication operators by elementary symmetric polynomials on pph is clos s omega where s c n c n is the symmetrization map it is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials cz gn in n variables equipped with the sup norm on clos s omega c 2018 elsevier inc all rights reserved

proper self maps of weakly pseudoconvex domains this theorem was proved by burns and shnider i 5 in the case that d is strongly pseudoconvex in the more general case pin6uk 1 9 corollary 2 has reduced the problem of showing that a proper mapping f d d is biholomorphic to proving that f is unbranched i e that det f 0 our interest in the theorem above stems from the fact that if f d t d 2 is a proper holomorphic mapping of a weakly pseudoconvex domain d 1 onto a second domain dz it is possible for f to branch if d1 d 2 this is contrary to what may occur if d 1 is assumed to be strongly pseudoconvex see alexander 1 pin6uk 9 10 fornaess 1 8 bell 1 3 diederich and fornaess 1 7 it has recently been proved in 1 4 and in i 6 that the mapping f in the theorem extends smoothly to this fact is basic to our proof the theorem may be shown to remain valid if 112 is replaced by a stein manifold see 2 but si ace the arguments of 4 and 6 do not carry over automatically to this case the methods used in 1 2 are more technical

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

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

configurations in abelian categories iv invariants and changing stability conditions this is the last 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 are subsets of i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in a using artin stacks it showed well behaved moduli stacks obja 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 kq of representations of a quiver q the second studied algebras of constructible functions and stack functions on obja the third introduced stability conditions t on a and showed the moduli space objss of semistable objects in class is a constructible subset in obja so its characteristic function ss is a constructible function it formed algebras h pa h to h pa h to of constructible and stack functions on obja and proved many identities in them in this paper if t and t are stability conditions on a we write ss in terms of the ss and deduce the algebras h pa h to are independent of t we study invariantsiss or iss i counting semistable objects or configurations in a which satisfy additive and multiplicative identities we compute them completely when a mod kq or a coh p for p a smooth curve we also find invariants with special properties when a coh p for p a smooth surface with kp 1 nef or a calabi yau 3 fold

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

a construction of inner functions on the unit ball in c p in their recent paper 1 hakim and sibony came very close to constructing inner functions on the unit ball in c e in fact only some minor modifications of their arguments are necessary to produce bona fide inner functions the present paper carries out these modifications since we need to point out some information not explicitly stated but clear from the context in their paper and to make the exposition self contained we repeat most of the material in their paper our lemma 1 for instance is identical to their lemma 3 and our application of lemma 1 in the proof of lemma 2 is the same as theirs in the proof of theorem 1 the following notations are used b is the unit ball in c p br z cp izl r we use the distance

on the regularity of cr mappings between cr manifolds of hypersurface type we prove smooth and analytic versions of the classical schwarz reflection principle for transversal cr mappings between two levi nondegenerate cr manifolds of hypersurface type

local embeddability of cr manifolds into spheres in this paper we solve local cr embeddability problem of smooth cr manifolds into spheres under a certain nondegeneracy condition on the chern moser s curvature tensor we state necessary and sufficient conditions for the existence of cr embeddings as finite number of equations and rank conditions on the chern moser s curvature tensors and their derivatives we also discuss the rigidity of those embeddings

algebraic cobordism revisited we define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities the main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by levine and morel double point cobordism provides a simple geometric presentation of algebraic cobordism theory as a corollary the lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

gromov witten pairs correspondence for the quintic 3 fold we use the gromov witten pairs descendent correspondence for toric 3 folds and degeneration arguments to establish the gw p correspondence for several compact calabi yau 3 folds including all cy complete intersections in products of projective spaces a crucial aspect of the proof is the study of the gw p correspondence for descendents in relative geometries projective bundles over surfaces relative to a section play a special role the gw p correspondence for calabi yau complete intersections provides a structure result for the gromov witten invariants in a fixed curve class after change of variables the gromov witten series is a rational function in the variable q eiu invariant under q q 1

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

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

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

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

some regularity theorems for cr mappings the purpose of this article is to study lipschitz cr mappings from an h extendible or semi regular hypersurface in mathbb c n under various assumptions on the target hypersurface it is shown that such mappings must be smooth a rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

embedding strictly pseudoconvex domains into balls every relatively compact strictly pseudocc nvex domain d with c2 boundary in a stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball however for each n 2 there exist bounded strictly pseudoconvex domains d in cn with real analytic boundary such that no proper holomorphic map from d into any finite dimensional ball extends smoothly to d 0 introduction in this paper we study the representations of bounded strictly pseudoconvex domains d c cn if the boundary of d is of class ck k e 2 3 oo then by a theorem of fornaess 8 and khenkin 13 d can be mapped biholomorphically onto the intersection xn q of a bounded strictly convex domain q c cn with ck boundary and a closed complex submanifold x defirxed in a neighborhood of q in cn x intersecting the boundary of q transversally moreover the map f d x n q extends to a holomorphic map on a neighborhood of d the convex domain q depends on d hence a natural question is whether a similar result holds with q replaced by the unit ball b z z1l zn e cn z 2 e zjl2 n that intersects bbn transversally such that d is biholomorphically equivalent to xn e3ff this question has been mentioned by lempert 14 pinduk 20 bedford 2 and others our main result is that the answer to this question is negative in general if d is as above then every biholomorphism of d onto xne3n extends smoothly to d according to 3 however we will show that not all such domains d admit a proper holomorphic map into a finite dimensional ball that is smooth on d theorem 1 1 a similar local result was obtained independently by faran 7 we shall show that the answer to the question q is positive if we allow the intersections of complex submanifolds with strictly convex domains q c cn with real analytic boundaries theorem 1 2 thetheoremsoffornaess 8 andkhenkin 13 onlygiveanqwith smooth boundary received by the editors april 4t 1985 1980 mathematics subject classification primary 32h05 1research supported by a fellowship from the alfred p sloan foundation t 1986 american mathematical society 0002 9947 86 1 00 25 per page

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

boundary regularity of proper holomorphic mappings it s main result is formulated in theorem 2 below although the answer to the conjecture in the general case is still unknown it has been shown to hold in many special cases a fairly detailed history of this subject will be contained in 14 one class of positive answers deals with cases in which one knows that f21 and 2 posses fairly large classes of holomorphic automorphisms that extend holomorphically across the boundaries like for instance reinhardt domains see e g h cartan 8 w kaup 16 and st bell 6 in these situations one even often does not need a smoothness assumption on the boundaries of 1 and 22 on the other hand the first result without a priori knowledge on automorphisms which gave c extendability was proved by ch fefferman in 1974 in 15 and says

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

configurations in abelian categories i basic properties and moduli stacks abstract this is the first in a series of papers on configurations in an abelian category a given a finite partially ordered set 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 are subsets of i configurations describe how an object x in a decomposes into subobjects and are useful for studying stability conditions on a we define and motivate the idea of configurations and explain some natural operations upon them subconfigurations quotient configurations substitution refinements and improvements then we study moduli spaces of i configurations in a and natural morphisms between them using the theory of artin stacks we prove well behaved moduli stacks exist when a is the abelian category of coherent sheaves on a projective scheme p or of representations of a quiver q in the sequels given a stability condition t on a we will show the moduli spaces of semi stable objects or configurations are constructible subsets in the moduli stacks of all objects or configurations we associate infinite dimensional algebras of constructible functions to a quiver q using the method of ringel hall algebras and define systems of invariants of p that count semi stable coherent sheaves on p and satisfy interesting identities

to stabilize a k real parameter affine family of plants it suffices to simultaneously stabilize 4 4 polynomials a controller stabilizes an entire family of plants with affine uncertainty if it simultaneously stabilizes a finite number of polynomials an upper bound for this number is 4k

stable pair invariants under blow ups we use the degeneration formula and the absolute relative correspondence tostudy the change of stable pair invariants of threefold under blow ups and obtain some closed blow up formulae which imply some interesting relations among bps state counts our results provide positive evidence for the gw dt p correspondence and also give a partial correspondence for varieties not necessarily toric or complete intersections