floor diagrams relative to a conic and gw w invariants of del pezzo surfaces abstract we enumerate via floor diagrams complex and real curves in c p 2 blown up in n points on a conic as an application we deduce gromov witten and welschinger invariants of del pezzo surfaces these results are mainly obtained using li s degeneration formula and its real counterpart

stochastic approximation with dependent disturbances i abstract a generalization of robbins monro stochastic approximation is presented in the paper it is shown that if disturbances satisfy a sort of generalized law of large numbers then an appropriate stochastic approximation procedure converges almost surely or only in probability depending on what kind of law of large numbers strong or weak is satisfied by disturbances in that sense theorems presented in the paper generalize robbins monro stochastic approximation schemes because the law of large numbers can be satisfied as is well known by sequences of dependent random variables on the other hand as theorem of gladishev a generalized version of robbins monro theorem can be obtained from the results presented in the paper see theorem 10 one can consider this paper as the one providing new proofs for different versions of stochastic approximation the proofs of the theorems of the paper are different than usual proofs of stochastic approximation procedure in particular they are not based on the martingale convergence theorem roughly speaking the proofs exploit the analogy between the stochastic approximation procedures of robbins monro versions and deterministic numerical iterative procedures seeking zeros of the system of nonliner equations as the results of the paper were thought to be applied to estimation of parameters of discrete stochastic processes so called identification special notation has been introduced this notation is believed to be useful for the above purpose

the number of ramified covering of a riemann surface by riemann surface interpreting the number of ramified covering of a riemann surface by riemann surfaces as the relative gromov witten invariants and applying a gluing formula we derive a recursive formula for the number of ramified covering of a riemann surface by riemann surface with elementary branch points and prescribed ramification type over a special point

the donaldson thomas invariants under blowups and flops using the degeneration formula for doanldson thomas invariants we proved formulae for blowing up a point and simple flops given a smooth projective calabi yau 3 fold x the moduli space of stable sheaves on x has virtual dimension zero donaldson and thomas d t defined the holomorphic casson invariant of x which essentially counts the number of sta ble bundles on x however the moduli space has positive dimension and is singular in general making use of virtual cycle technique see b f and l t thomas showed in thomas that one can define a virtual moduli cycle for some x including calabi yau and fano 3 folds as a consequence one can define donaldson type in variants of x which are deformation invariant donaldson thomas invariants pro vide a new vehicle to study the geometry and other aspects of higher dimensional varieties it is important to understand these invariants much studied gromov witten invariants of x are the counting of stable maps from curves to x in mnop1 mnop2 maulik nekrasov okounkov and pandharipande discovered relations between gromov witten invarints of x and donaldson thomas invariants constructed from moduli spaces of ideal sheaves of curves on x they conjectured that these two invariants can be identified via the equations of partition functions of both theory this suggests that many phe nomena on gromov witten theory have the counterparts in donaldson thomas theory donaldson thomas invariants are deformation independent in the birational geometry of 3 folds we have blowups and flops donaldson thomas invariants couldn t be effective in studying birational geometry unless we understand how invariants change under birational operations li and ruan in l r studied how gromov witten invariants change under a flop for calabi yau 3 fold they proved that one can identify the 3 point functions of x and the flop x f of x up to some transformation of the q variables the same question was aslo studied by liu and yau in l y recently using the j li s degeneration formula from algebraic geometry in hu1 hu2 the first author studied the change of gromov witten

a note on plurisubharmonic defining functions in mathbb c n let omega subset subset mathbb c n n 3 be a smoothly bounded domain suppose that admits a smooth defining function which is plurisubharmonic on the boundary of then a diederich fornaess exponent can be chosen arbitrarily close to 1 and the closure of admits a stein neighborhood basis

weighted blowup correspondence of orbifold gromov witten invariants and applications let mathsf x be a compact symplectic orbifold groupoid with mathsf s being a compact symplectic sub orbifold groupoid and underline mathsf x mathfrak a be the weight mathfrak a blowup of mathsf x along mathsf s with mathsf z being the exceptional divisor we show that there is a weighted blowup correspondence between some certain absolute orbifold gromov witten invariants of mathsf x relative to mathsf s and some certain relative orbifold gromov witten invariants of the pair underline mathsf x mathfrak a mathsf z as an application we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weighted blowup invariant

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

holomorphic curves in exploded manifolds compactness abstract this paper establishes compactness results for the moduli stack of holomorphic curves in suitable exploded manifolds this together with 14 and 13 allows the definition of gromov witten invariants of these exploded manifolds

generalized laws of large numbers and auxiliary results concerning stochastic approximation with dependent disturbances ii in refs 1 3 results have been presented concerning convergence with probability 1 of stochastic approximation procedures the results were quoted with partial or without proofs the aim of this paper is to provide the necessary tools for completion of the proofs and on their basis to be able to generate new results the presentation of the proofs is unified and is based on the notion of a regular discrete m process it turns out that these processes have many interesting properties amongst others almost sure convergence to zero under some simple conditions the interesting fact is that from the main theorem concerning convergence of m processes follows after proper reformulation the generalization of kronecker s lemma see section 4 of this paper which plays an important role in examining the properties of almost sure convergence of random variables as is well known kronecker s lemma has an implication stating that if certain series converge then some related sequences converge to zero our generalization has a form of equivalence i e gives necessary and sufficient conditions between convergence of the series mentioned in kronecker s lemma and convergence of some sequences and some other series thus this generalization can be useful in examining almost sure convergence of series and also strong laws of large numbers with weights and dependent summands see theorem 5 and the following corollaries so although the basic purpose of the paper is practical and auxiliary it can be interesting by itself for those who work with almost sure convergence of random variables

sum formulas for local gromov witten invariants of spin curves holomorphic 2 forms on kahler surfaces lead to local gromov witten invariants of spin curves this paper shows how to derive sum formulas for such local gw invariants from the sum formula for gw invariants of certain ruled surfaces these sum formulas also verify the maulik pandharipande formulas that were recently proved by kiem and li let x be a kahler surface with a holomorphic 2 form the real part of also denoted by then induces an almost complex structure on x j id jk 1j id jk 0 1 here j is the kahler structure on x and k is the endomorphism of tx defined by the formula u k v u v where is the kahler metric on x the almost complex structure j satisfies a remarkable image localization property if f is a j holomorphic map into x that represents a non zero 1 1 class then the image of f lies in the zero set d of the holomorphic 2 form for simplicity assume x is a minimal surface of general type and d is smooth the normal bundle n to d is then a theta characteristic on d i e n is a square root of the canonical bundle of d the pair d n is called a spin curve of genus h where h is the genus of d the total space nd of n has a tautological holomorphic 2 form that induces by the same manner as in 0 1 an almost complex structure j on nd satisfying the image localization property namely mg n nd d d j mg n d d consequently the moduli space of j holomorphic maps is compact so it represents a virtual fundamental class that defines local gw invariants of the spin curve d n these local gw invariants depend only on the genus h and the parity p h0 n mod 2 and gw invariants of kahler surfaces with pg 0 are the sum of local gw invariants associated to spin curves lp1 gw invariants count maps from connected domains while gromov taubes invariants count maps from not necessarily connected domains these gt invariants can be obtained from gw

an obstruction bundle relating gromov witten invariants of curves and kahler surfaces in a previous paper the authors defined symplectic local gromov witten invariants associated to spin curves and showed that the gw invariants of a k ahler surface x with p g 0 are a sum of such local gw invariants this paper describes how the local gw invariants arise from an obstruction bundle in the sense of taubes over the space of stable maps into curves together with the results of our earlier paper this reduces the calculation of the gw invariants of elliptic and general type complex surfaces to computations in the gw theory of curves with additional classes the euler classes of the real obstruction bundles

many closed symplectic manifolds have infinite hofer zehnder capacity we exhibit many examples of closed symplectic manifolds on which there is an autonomous hamiltonian whose associated flow has no nonconstant periodic orbits the only previous explicit example in the literature was the torus t2n n 2 with an irrational symplectic structure the underlying smooth manifolds of our examples include for instance the k3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it infinitely many minimal four manifolds having any given finitely presented group as their fundamental group and simply connected minimal four manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3 the examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the neck in the symplectic sum have no closed characteristics we conjecture that any closed symplectic four manifold with b 1 admits symplectic forms with a similar property

local gromov witten invariants are log invariants abstract we prove a simple equivalence between the virtual count of rational curves in the total space of an anti nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle we conjecture a generalization to direct sums of line bundles

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

a sufficient condition for compactness of the neumann operator we show that the existence of a certain family of plurisubharmonic functions on a bounded domain in cn implies that the neumann operator associated to the domain is compact our condition generalizes previous work of catlin on compactness of the neumann operator

optimal recursive procedures of identification the paper concerns the parameter estimation of a stochastic process we consider recursive estimating procedures based on the idea of stochastic approximation p p f r p 1 where is a sequence of positive reals and f is a sequence of functions depending on observations y of the process and parameters p we try to construct functions f estimating functions in an optimal way we look for the optimal f to be of the form f w f where f is a sequence of some functions depending on the model of the process while matrix in general w is selected in such a way as to make the procedure 1 optimal the notion of optimality of the procedure of identification is based on previous results remarks concerning notation all vectors considered are columns x denotes transposition hence x is a row vector matrices and vectors are multiplied according to the usual rules if f e r m is a vector depending on a vector x e r then

a structure theorem for the gromov witten invariants of kahler surfaces we prove a structure theorem for the gromov witten invariants of compact kahler surfaces with geometric genus pg 0 under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components the theorem shows that the gw invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic euler characteristic of the surface we compute special cases of these universal functions much of the work on the gromov witten invariants of kahler surfaces has focused on rational and ruled surfaces which have geometric genus pg 0 this paper focuses on surfaces with pg 0 a class that includes most elliptic surfaces and most surfaces of general type in this context we prove a general structure theorem that shows with one technical assumption how the gw invariants are completely determined by the local geometry of a generic canonical divisor the structure theorem is a consequence of a simple fact the image localization lemma of section 3 given a kahler surface x and a canonical divisor d kx this lemma shows that the complex structure j on x can be perturbed to a non integrable almost complex structure jd with the property the image of all jd holomorphic maps lies in the support of d this immediately gives some striking vanishing theorems for the gw invariants of kahler surfaces see section 3 more importantly it implies that the gromov witten invariant of x for genus g and n marked points is a sum gwg n x a gw loc g n dk ak over the connected components dk of d of local invariants that count the contribution of maps whose image lies in or after perturbing to a generic moduli space near dk these local invariants have not been previously defined the proof of their existence relies on using non integrable structures and geometric analysis techniques our structure theorem characterizes the local invariants by expressing them in terms of usual gw invariants of certain standard surfaces for this we make the mild assumption that one can deform the kahler structure on x and choose a canonical divisor d so that all components of d are smooth with this assumption the restriction of kx to each component dk of multiplicity mk is the normal bundle nk to dk and nk k kdk partially supported by the n s f

open gromov witten theory on symplectic manifolds and symplectic cutting abstract let x be a symplectic manifold and l be a lagrangian submanifold diffeomorphic to s n r p n or a lens space of a certain type using the symplectic cut and symplectic sum constructions we express the open gromov witten invariants of x l in terms of open gromov witten invariants of a pair x l determined by l and the standard gromov witten invariants of a symplectic manifold x determined by x l we also describe other applications of this approach

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

topological recursive relations in h2g ℳg n we show that any degree at least g monomial in descendant or tautological classes vanishes on g n when g 2 this generalizes a result of looijenga and proves a version of getzler s conjecture the method we use is the study of the relative gromov witten invariants of 1 relative to two points combined with the degeneration formulas of ip1

a sobolev mapping property of the bergman kernel we prove that if d is a pseudoconvex domain with lipschitz boundary having an exhaustion function rho such that rho eta is plurisubharmonic then the bergman projection maps the sobolev space w s boundedly to itself for any s eta 2

a few remarks on orthogonal polynomials knowing a sequence of moments of a given infinitely supported distribution we obtain quickly coefficients of the power series expansion of monic polynomials p n n 0 that are orthogonal with respect to this distribution coefficients of expansion of x n in the series of p j j n two sequences of coefficients of the 3 term recurrence of the family of p n n 0 the so called linearization coefficients i e coefficients of expansion of p n p m in the series of p j j m n conversely assuming knowledge of the two sequences of coefficients of the 3 term recurrence of a given family of orthogonal polynomials p n n 0 we express with their help coefficients of the power series expansion of p n coefficients of expansion of x n in the series of p j j n moments of the distribution that makes polynomials p n n 0 orthogonal further having two different families of orthogonal polynomials p n n 0 and q n n 0 and knowing for each of them sequences of the 3 term recurrences we give sequence of the so called connection coefficients between these two families of polynomials that is coefficients of the expansions of p n in the series of q j j n we are able to do all this due to special approach in which we treat vector of orthogonal polynomials p j x j 0 n as a linear transformation of the vector x j j 0 n by some lower triangular n 1 i n 1 matrix n

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

flops motives and invariance of quantum rings for ordinary ops the correspondence dened by the graph closure is shown to give equivalence of chow motives and to preserve the poincar e pairing in the case of simple ordinary ops this correspondence preserves the big quantum cohomology ring after an analytic continuation over the

comparison of the bergman and szego kernels abstract the quotient of the szego and bergman kernels for a smooth bounded pseudoconvex domains in c n is bounded from above by a constant multiple of log p for any p n where is the distance to the boundary for a class of domains that includes those of d angelo finite type and those with plurisubharmonic defining functions the quotient is also bounded from below by a constant multiple of log p for any p 1 moreover for convex domains the quotient is bounded from above and below by constant multiples of

the diederich fornaess index i for domains of non trivial index abstract we study bounded pseudoconvex domains in complex euclidean space we define an index associated to the boundary and show this new index is equivalent to the diederich fornaess index defined in 1977 this connects the diederich fornaess index to boundary conditions and refines the levi pseudoconvexity we also prove the worm domain is of index 2 it is the first time that a precise non trivial diederich fornaess index in euclidean spaces is obtained this finding also indicates that the diederich fornaess index is a continuum in 0 1 not a discrete set the ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations

blow up formulae of high genus gromov witten invariants for threefolds using the degeneration formula and absolute relative correspondence we study the change of gromov witten invariants under blow ups for smooth projective threefolds and obtain several closed blow up formulae for high genus gromov witten invariants our formulae also imply some simple relations among generalized bps numbers

gw invariants relative to normal crossing divisors in this paper we introduce a notion of symplectic normal crossing divisor v and define the gw invariant of a symplectic manifold x relative to such a divisor our definition includes normal crossing divisors from algebraic geometry the invariants we define in this paper are key ingredients in symplectic sum type formulas for gw invariants and extend those defined in our previous joint work with t h parker 16 which covered the case v was smooth the main step is the construction of a compact moduli space of relatively stable maps into the pair x v x v in the case v is a symplectic normal crossing divisor in x

fredholm hodge and liouville theorems on noncompact manifolds fredholm liouville hodge and l2 cohomology theorems are proved for laplacians associated with a class of metrics defined on manifolds that have finitely many ends the metrics are conformal to ones that are asymptotically translation invariant they are not necessarily complete the fredholm results are of necessity with respect to weighted sobolev spaces embedding and compact embedding theorems are also proved for these spaces two of the most useful facts in analysis on a compact riemannian manifold are that the laplacian is fredholm and its kernel consists of closed and coclosed forms that provide unique representatives for all the de rham cohomology classes naturally one would like to extend these results to noncompact manifolds the first such result in this direction is due to kodaira 9 12 p 165 it is that l2 a m g is the orthogonal direct sum of dco aq 1m dg co aq lm and k2 awm g ae l2 am g ida dga 0 unfortunately if no restriction is made on the manifold or the metric one cannot improve on this the laplacian need not be fredholrn l2 harmonic forms need not be closed or coclosed even if they are closed and coclosed the space of harmonic l2 forms need not be finite dimensional see 7 and even if the laplacian is fredholm and l2 harmonic forms are closed and coclosed those forms need not provide unique or total representation of de rham cohomology thus one of the main questions in analysis on noncompact manifolds is what conditions on m and g allow one to carry over the fredholm and hodge theorems for compact manifolds and if they cannot be carried over completely to what extent can they be in the case of hodge s theorem i e properties of the space of l2 harmonic forms this question has been quite actively investigated recently see 2 8 11 and 13 for instance one of the results of atiyah patodi and singer in 2 is that if a manifold has cylindrical ends then k 2 a m g is naturally isomorphic to the image of hcomp m in hfdr m see 2 proposition 4 9 in 11 muller investigates the spectrum of the laplacian on manifolds that outside a compact set are q x r with received by the editors august 30 1985 and in revised form april 10 1986 1980 mathematics subject classification 1985 revision primary 58gi0 35j05 1987 american mathematical society 0002 9947 87 1 00 25 per page this content downloaded from 157 55 39 45 on tue 19 jul 2016 04 20 24 utc all use subject to http about jstor org terms

virtual moduli cycles and gromov witten invariants of noncompact symplectic manifolds this paper constructs and studies the gromov witten invariants and their properties for noncompact geometrically bounded symplectic manifolds two localization formulas for gw invariants are also proposed and proved as applications we get solutions of the generalized string equation and dilation equation and their variants the more solutions of wdvv equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups to realize these purposes we further develop the language introduced by liu tian to describe the virtual moduli cycle defined by liu tian fukaya ono li tian ruan and siebert

behavior of the bergman projection on the diederich fornaess worm w 1 introduction in this paper we show that the bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by sobolev norms let q be a smooth bounded pseudoconvex domain in c and let p t denote the orthogonal projection from l2 f onto the bergman subspace b ff l2 f3 r with respect to the weighted norm i fll j olf z l 2 e t lzll2dv l 2 let wk t denote the sobolev space consisting of functions whose derivatives of order to k g2 on the other hand there is a large collection of results implying that for certain types of domains the unweighted bergman projection p pr preserves w k for all k 0 see fk for the strictly pseudoconvex case for results on weakly pseudoconvex domains the reader may consult ko2 ca si and the recent bstl ch as well as the references cited therein most of these results are focused on the a neumann operator rather than p see bst2 for the connection also except for kol positive results in this area are typically valid for any choice of smooth positive weight function on 0 the question of whether or not p is similarly well behaved for all weakly pseudoconvex domains has remained open for many years in this paper we show that this is not the case in fact when q is the so called worm domain of diederich and forna ss then p does not map w to w k when k zr total amount of winding this latter quantity is explained in section 4 below where the construction of the worm is reviewed and the main result is proved the proof depends on computations for a piecewise levi flat model domain depending in turn on certain one dimensional computations these are treated in sections 3 and 2 respectively section 5 contains additional remarks and questions

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

a conjectural generating function for numbers of curves on surfaces i give a conjectural generating function for the numbers of nodal curves in a linear system of dimension on an algebraic surface it reproduces the results of vainsencher v for the case le 6 and kleiman piene k p for the case le 8 the numbers of curves are expressed in terms of five universal power series three of which i give explicitly as quasimodular forms this gives in particular the numbers of curves of arbitrary genus on a k3 surface and an abelian surface in terms of quasimodular forms generalizing the formula of yau zaslow for rational curves on k3 surfaces the coefficients of the other two power series can be determined by comparing with the recursive formulas of caporaso harris for the severi degrees in 2 we verify the conjecture for genus 2 curves on an abelian surface we also discuss a link of this problem with hilbert schemes of points

genus decreasing formula for higher genus welschinger invariants we study the changes experienced by higher genus welschinger invariants of real del pezzo surfaces in families undergoing a nodal degeneration and obtain a genus decreasing wall crossing formula

hamiltonian s 1 manifolds are uniruled the main result of this note is that every closed hamiltonian s 1 manifold is uniruled i e it has a nonzero gromov witten invariant one of whose constraints is a point the proof uses the seidel representation of 1 of the hamiltonian group in the small quantum homology of m as well as the blow up technique recently introduced by hu li and ruan it applies more generally to manifolds that have a loop of hamiltonian symplectomorphisms with a nondegenerate fixed maximum some consequences for hofer geometry are explored an appendix discusses the structure of the quantum homology ring of uniruled manifolds

symplectic calabi yau 6 manifolds abstract in this article we construct simply connected symplectic calabi yau 6 manifolds by applying gompf s symplectic fiber sum operation along t 4 using our method we also construct symplectic non kahler calabi yau 6 manifolds with fundamental group z this paper also produces the first examples of simply connected and non simply connected symplectic calabi yau 6 manifolds with fundamental groups z p z q and z z q for any p 1 and q 2 via co isotropic luttinger surgery

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