counting representations of quivers over finite fields abstract by counting the numbers of isomorphism classes of representations indecomposable or absolutely indecomposable of quivers over finite fields with fixed dimension vectors we obtain a multi variable formal identity if the quiver has no edge loops this identity turns out to be a q analogue of the kac denominator identity modulus a conjecture of kac

a computational criterion for the kac conjecture abstract we give a criterion for the kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated kac moody algebra our criterion suits very well for computer tests

the dynamics of pseudographs in convex hamiltonian systems we study the evolution under convex hamiltonian flows on cotangent bundles of compact manifolds of certain distinguished subsets of the phase space these subsets are generalizations of lagrangian graphs we call them pseudographs they emerge in a natural way from fathi s weak kam theory by this method we find various orbits which connect prescribed regions of the phase space our study is inspired by works of john mather as an application we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem we hope that our method will have applications to more examples

a generic property of families of lagrangian systems we prove that a generic lagrangian has finitely many minimizing measures for every cohomology class

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

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 gaussian distribution for refined dt invariants and 3d partitions we show that the refined donaldson thomas invariants of mathbb c 3 suitably normalized have a gaussian distribution as limit law combinatorially these numbers are given by weighted counts of 3d partitions our technique is to use the hardy littlewood circle method to analyze the bivariate asymptotics of a q deformation of macmahon s function the proof is based on that of e m wright who explored the single variable case

on the motivic class of the stack of bundles let g be a split connected semisimple group over a field we give a conjectural formula for the motivic class of the stack of g bundles over a curve c in terms of special values of the motivic zeta function of c the formula is true if or g sln if upon applying the poincare or called the serre characteristic by some authors the formula reduces to results of teleman and atiyah bott on the gauge group if upon applying the counting measure it reduces to the fact that the tamagawa number of g over the function field of c is p1 g

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

bps state counting on singular varieties we define new partition functions for theories with targets on toric singularities via products of old partition functions on crepant resolutions we compute explicit examples and show that the new partition functions turn out to be homogeneous on macmahon factors

existence of c 1 critical subsolutions of the hamilton jacobi equation let m be a c second countable manifold without boundary we denote by tm the tangent bundle and by tm m the canonical projection a point in tm will be denoted by x v with x m and v tx m 1 x in the same way a point of the cotangent space t m will be denoted by x p with x m and p t x m a linear form on the vector space tx m we will suppose that g is a complete riemannian metric on m for v tx m the norm v is g v v 1 2 we will denote by the dual norm on t x m we will also use the notations v x for v tx m and p x for v t x m we will assume in the whole paper that h t m r is a function of class at least c2 which satisfies the following three conditions 1 uniform superlinearity for every k 0 there exists c k r such that

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

volume comparison and its applications in riemann finsler geometry abstract we introduce a new geometric quantity the mean covariation for finsler metrics and establish a volume comparison theorem as an application we obtain some precompactness and finiteness theorems for finsler manifolds

bakry emery curvature dimension condition and riemannian ricci curvature bounds the aim of the present paper is to bridge the gap between the bakry emery and the lott sturm villani approaches to provide synthetic and abstract notions of lower ricci curvature bounds we start from a strongly local dirichlet form mathcal e admitting a carre du champ gamma in a polish measure space x mathfrak m and a canonical distance mathsf d mathcal e that induces the original topology of x we first characterize the distinguished class of riemannian energy measure spaces where mathcal e coincides with the cheeger energy induced by mathsf d mathcal e and where every function f with gamma f le1 admits a continuous representative in such a class we show that if mathcal e satisfies a suitable weak form of the bakry emery curvature dimension condition mathrm be k infty then the metric measure space x mathsf d mathfrak m satisfies the riemannian ricci curvature bound mathrm rcd k infty according to duke math j 163 2014 1405 1490 thus showing the equivalence of the two notions two applications are then proved the tensorization property for riemannian energy spaces satisfying the bakry emery mathrm be k n condition and thus the corresponding one for mathrm rcd k infty spaces without assuming nonbranching and the stability of mathrm be k n with respect to sturm gromov hausdorff convergence

large time behavior of the heat kernel in this paper we study the large time behavior of the heat kernel on complete riemannian manifolds with nonnegative ricci curvature which was studied by p li with additional maximum volume growth assumption following y ding s original strategy by blowing down the metric using cheeger and colding s theory about limit spaces of gromov hausdorff convergence combining with the gaussian upper bound of heat kernel on limit spaces we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure as one application we get the consistent large time limit of heat kernel in more general context which generalizes the former result of p li furthermore by choosing different sequences to blow down the suitable metric we show the first example manifold whose heat kernel has inconsistent limit behavior which answers an open question posed by p li negatively

on diffusion in high dimensional hamiltonian systems the purpose of this paper is to construct examples of diffusion for e hamiltonian perturbations of completely integrable hamiltonian systems in 2d dimensional phase space with d large in the first part of the paper simple and explicit examples are constructed illustrating absence of long time stability for size e hamiltonian perturbations of quasi convex integrable systems already when the dimension 2d of phase space becomes as large as log1e we first produce the example in gevrey class and then a real analytic one with some additional work in the second part we consider again e hamiltonian perturbations of completely integrable hamiltonian system in 2d dimensional space with e small but not too small e exp d with d the number of degrees of freedom assumed large it is shown that for a class of analytic time periodic perturbations there exist linearly diffusing trajectories the underlying idea for both examples is similar and consists in coupling a fixed degree of freedom with a large number of them the procedure and analytical details are however significantly different as mentioned the construction in part i is totally elementary while part ii is more involved relying in particular on the theory of normally hyperbolic invariant manifolds methods of generating functions aubry mather theory and mather s variational methods part i is due to bourgain and part ii due to kaloshin

geometry of arnold diffusion the goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of arnold diffusion we choose to speak of a specific model that of geometric rays in a periodic optical medium this model is equivalent to that of a particle in a periodic potential in mathbb r n with energy prescribed and to the geodesic flow in a riemannian metric on mathbb r n

symbolic dynamics and arnold diffusion we consider hyperbolic tori of three degrees of freedom initially hyperbolic hamiltonian systems we prove that if the stable and unstable manifold of a hyperbolictorus intersec t transversaly then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a bernoulli shift the proof is based on a new geometricodynamical feature of partially hyperbolic systems the transversality torsion phenomenon which produces complete hyperbolicity from partial hyperbolicity we deduce the existence of infinitely many hyperbolicperiodicorbits near the given torus the relevanc e of these results for the instability of near integrable hamiltonian systems is then discussed for a given transition chain we construct chain of hyperbolic periodic orbits then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolicsets r 2002 elsevier science usa all rights reserved

a general mechanism of diffusion in hamiltonian systems qualitative results we present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable hamiltonian systems our approach relies on the scattering map outer dynamics and on the recurrence property of the inner dynamics restricted to a normally hyperbolic invariant manifold we apply topological methods to find trajectories that follow these two dynamics this method differs in several crucial aspects from earlier works there are virtually no assumptions on the inner dynamics as the method does not use at all the invariant objects for the inner dynamics e g primary and secondary tori lower dimensional hyperbolic tori and their stable unstable manifolds aubry mather sets the method applies when the unperturbed hamiltonian is not necessarily convex and of arbitrary degrees of freedom in addition this mechanism is easy to verify analytically or numerically in concrete examples as well as to establish diffusion in generic systems we include several applications such as bridging large gaps in a priori unstable models in any dimension and establishing diffusion in cases when the inner dynamics in a non twist map

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

local spectral convergence in rcd k n spaces abstract in this paper we give necessary and sufficient conditions for the validity of the local spectral convergence in balls in the setting of rcd spaces

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

speed of arnold diffusion for analytic hamiltonian systems for a convex real analytic e close to integrable hamiltonian system with n 5 degrees of freedom we construct an orbit exhibiting arnold diffusion with the diffusion time bounded by exp c epsilon frac 1 2 n 2 this upper bound of the diffusion time almost matches the lower bound of order exp epsilon frac 1 2 n 1 predicted by the nekhoroshev type stability results our method is based on the variational approach of bessi and mather and includes a new construction on the space of frequencies

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

transition tori in the five body problem abstract transition tori are invariant tori connected by transverse heteroclinic orbits as shown by arnold nearby orbits can drift from torus to torus in this paper a family of transition tori is constructed for the planar five body problem

wall crossing of bps states on the conifold from seiberg duality and pyramid partitions in this paper we study the relation between pyramid partitions with a general empty room configuration erc and the bps states of d branes on the resolved conifold we find that the generating function for pyramid partitions with a length n erc is exactly the same as the d6 d2 d0 bps partition function on the resolved conifold in particular kahler chambers we define a new type of pyramid partition with a finite erc that counts the bps degeneracies in certain other chambers

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

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

weak and strong convergence of derivations and stability of flows with respect to mgh convergence abstract this paper is devoted to the study of weak and strong convergence of derivations and of the flows associated to them when dealing with a sequence of metric measure structures x d m n m n weakly convergent to m in particular under curvature assumptions either only on the limit metric structure x d m or on the whole sequence of metric measure spaces we provide several stability results

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

on the number of mather measures of lagrangian systems in 1996 ricardo ricardo mane discovered that mather measures are in fact the minimizers of a universal infinite dimensional linear programming problem this fundamental result has many applications of which one of the most important is to the estimates of the generic number of mather measures mane obtained the first estimation of that sort by using finite dimensional approximations recently we were able with gonzalo contreras to use this method of finite dimensional approximation in order to solve a conjecture of john mather concerning the generic number of mather measures for families of lagrangian systems in the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem we use a notion of countably rectifiable sets of finite codimension in banach and frechet spaces which may deserve independent interest

symplectic aspects of mather theory we prove that the so called aubry and mane sets introduced by john mather in lagrangian dynamics are symplectic invariants in order to do so we introduce a barrier in phase space and propose definitions of aubry and mane sets for non convex hamiltonian systems on montre que les ensembles dits d aubry et de mane introduits par john mather en dynamique lagrangienne sont des invariants symplectiques pour ceci on introduit une barriere sur l espace des phases et on definit des ensembles d aubry et de mather pour des systemes hamiltoniens non convexes

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

convergence of pointed non compact metric measure spaces and stability of ricci curvature bounds and heat flows aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition we propose various definitions show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured gromov hausdorff convergence then we show that the curvature conditions cd k and rcd k are stable under this notion of convergence and that the heat flow passes to the limit as well both in the wasserstein and in the l framework we also prove the variational convergence of cheeger energies in the naturally adapted mosco sense and the convergence of the spectra of the laplacian in the case of spaces either uniformly bounded or satisfying the rcd k condition with k 0 when applied to riemannian manifolds our results allow for sequences with diverging dimensions

the a deformation theory of a point and the derived categories of local calabi yaus abstract let a be an augmented algebra over a semi simple algebra s we show that the ext algebra of s as an a module enriched with its natural a infinity structure can be used to reconstruct the completion of a at the augmentation ideal we use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non compact calabi yau three folds since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators abstract energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy widely extended are those based on piezoelectric materials whose dynamics are hamiltonian submitted to different sources of dissipation damping and coupling these dissipations bring the system to low energy regimes which is not desired in long term as it diminishes the absorbed energy to avoid or to minimize such situations we propose that the coupling of two oscillators could benefit from theory of arnold diffusion such phenomenon studies o 1 energy variations in hamiltonian systems and hence could be very useful in energy harvesting applications this article is a first step towards this goal we consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit by considering the coupling damping and forcing as perturbations we prove that the unperturbed system possesses a 4 dimensional normally hyperbolic invariant manifold with 5 and 4 dimensional stable and unstable manifolds respectively these are locally unique after the perturbation by means of the parameterization method we numerically compute parameterizations of the perturbed manifold its stable and unstable manifolds and study its inner dynamics we show evidence of homoclinic connections when the perturbation is switched on

the refined bps index from stable pair invariants a refinement of the stable pair invariants of pandharipande and thomas for non compact calabi yau spaces is introduced based on a virtual bialynicki birula decomposition with respect to a mathbb c action on the stable pair moduli space or alternatively the equivariant index of nekrasov and okounkov this effectively calculates the refined index for m theory reduced on these calabi yau geometries based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of morrison mozgovoy nagao and szendroi for local mathbb p 1 we explicitly compute refined invariants in low degree for local mathbb p 2 and local mathbb p 1 times mathbb p 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula the modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined pt invariants on appropriate geometries to nekrasov s partition function and a refinement of chern simons theory on a lens space we also relate our product formula to wall crossing