hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

gopakumar vafa invariants via vanishing cycles in this paper we propose an ansatz for defining gopakumar vafa invariants of calabi yau threefolds using perverse sheaves of vanishing cycles our proposal is a modification of a recent approach of kiem li which is itself based on earlier ideas of hosono saito takahashi we conjecture that these invariants are equivalent to other curve counting theories such as gromov witten theory and pandharipande thomas theory our main theorem is that for local surfaces our invariants agree with pt invariants for irreducible one cycles we also give a counter example to the kiem li conjectures where our invariants match the predicted answer finally we give examples where our invariant matches the expected answer in cases where the cycle is non reduced non planar or non primitive

matrices in a r s with minimum t term ranks abstract let r and s be two sequences of nonnegative integers in nonincreasing order which have the same sum and let a r s be the class of all 0 1 matrices which have row sums given by r and column sums given by s for a positive integer t the t term rank of a 0 1 matrix a is defined as the maximum number of 1 s in a with at most one 1 in each column and at most t 1 s in each row in this paper we address conditions for the existence of a matrix in a r s that realizes all the minimum t term ranks for t 1

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

on reduced stable pair invariants let x s times e be the product of a k3 surface s and an elliptic curve e reduced stable pair invariants of x can be defined via 1 cutting down the reduced virtual class with incidence conditions or 2 the behrend function weighted euler characteristic of the quotient of the moduli space by the translation action of e we show that 2 arises naturally as the degree of a virtual class and that the invariants 1 and 2 agree this has applications to deformation invariance rationality and a dt pt correspondence for reduced invariants of s times e

on the term rank partition abstract for each positive integer t the t term rank of a 0 1 matrix a is the maximum number of 1 s in a with at most one 1 in each column and at most t 1 s in each row in 5 r brualdi et al 2012 stated several results for the t term rank including a formula for the maximum t term rank over a nonempty class of 0 1 matrices with the same row sum and column sum vectors in this paper we state more results for the t term rank using these results we define and we study the term rank partition we also deduce a formula for the minimal t term rank over a nonempty class of 0 1 matrices with the same row sum and column sum vectors

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

holomorphic anomaly equations and the igusa cusp form conjecture let s be a k3 surface and let e be an elliptic curve we solve the reduced gromov witten theory of the calabi yau threefold s times e for all curve classes which are primitive in the k3 factor in particular we deduce the igusa cusp form conjecture the proof relies on new results in the gromov witten theory of elliptic curves and k3 surfaces we show the generating series of gromov witten classes of an elliptic curve are cycle valued quasimodular forms and satisfy a holomorphic anomaly equation the quasimodularity generalizes a result by okounkov and pandharipande and the holomorphic anomaly equation proves a conjecture of milanov ruan and shen we further conjecture quasimodularity and holomorphic anomaly equations for the cycle valued gromov witten theory of every elliptic fibration with section the conjecture generalizes the holomorphic anomaly equations for elliptic calabi yau threefolds predicted by bershadsky cecotti ooguri and vafa we show a modified conjecture holds numerically for the reduced gromov witten theory of k3 surfaces in primitive classes

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

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

crystal melting and toric calabi yau manifolds we construct a statistical model of crystal melting to count bps bound states of d0 and d2 branes on a single d6 brane wrapping an arbitrary toric calabi yau threefold the three dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of d branes the crystal is composed of atoms of different colors each of which corresponds to a node of the quiver diagram and the chemical bond is dictated by the arrows of the quiver diagram bps states are constructed by removing atoms from the crystal this generalizes the earlier results on the bps state counting to an arbitrary non compact toric calabi yau manifold we point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the donaldson thomas theory

derived algebraic geometry determinants of perfect complexes and applications to obstruction theories for maps and complexes we show how a quasi smooth derived enhancement of a deligne mumford stack x naturally endows x with a functorial perfect obstruction theory in the sense of behrend fantechi this result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety for moduli of maps for x s an algebraic k3 surface g n and 0 in h 2 s z a curve class we construct a derived stack rm red g n s whose truncation is the usual stack m g n s of pointed stable maps from curves of genus g to s hitting the class and such that the inclusion m g s rm red g s induces on m g s a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of okounkov maulik pandharipande thomas o p2 m p m p t the approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory not relying on any result on semiregularity maps but also a new global geometric interpretation we give two further applications to moduli of complexes for a k3 surface s we show that the stack of simple perfect complexes on s is smooth this result was proved with different methods by inaba in for the corresponding coarse moduli space finally we construct a map from the derived stack of stable embeddings of curves into a smooth complex projective variety x to the derived stack of simple perfect complexes on x with vanishing negative ext s and show how this map induces a morphism of the corresponding obstruction theories when x is a calabi yau threefold an important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is pointwise illusie s trace map for perfect complexes we expect that this determinant map might be useful in other contexts as well

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

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

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

vertical d4 d2 d0 bound states on k3 fibrations and modularity an explicit formula is derived for the generating function of vertical d4 d2 d0 bound states on smooth k3 fibered calabi yau threefolds generalizing previous results of gholampour and sheshmani it is also shown that this formula satisfies strong modularity properties as predicted by string theory this leads to a new construction of vector valued modular forms which exhibit some of the features of a generalized hecke transform

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

on the t term rank of a matrix abstract for t a positive integer the t term rank of a 0 1 matrix a is defined to be the largest number of 1 s in a with at most one 1 in each column and at most t 1 s in each row thus the 1 term rank is the ordinary term rank we generalize some basic results for the term rank to the t term rank including a formula for the maximum term rank over a nonempty class of 0 1 matrices with the same row sum and column sum vectors we also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t

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

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

counting curves on surfaces in calabi yau 3 folds motivated by s duality modularity conjectures in string theory we define new invariants counting a restricted class of two dimensional torsion sheaves enumerating pairs z subset h in a calabi yau threefold x here h is a member of a sufficiently positive linear system and z is a one dimensional subscheme of it the associated sheaf is the ideal sheaf of z subset h pushed forward to x and considered as a certain joyce song pair in the derived category of x we express these invariants in terms of the mnop invariants of x

random media and eigenvalues of the laplacian let be a fixed number 1 we remove m balls of centersw1 w m beta with the same radius m from a bounded domain inr3 we consider the asymptotic behaviour of thekth eigenvalue of the laplacian in omega backslash overline m beta balls under the dirichlet condition as a random variable on a probability space w 1 w m beta in omega m beta whenm

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

the thom sebastiani theorem for the euler characteristic of cyclic l algebras abstract let l be a cyclic l algebra of dimension 3 with finite dimensional cohomology only in dimension one and two by transfer theorem there exists a cyclic l algebra structure on the cohomology h l the inner product plus the higher products of the cyclic l algebra defines a superpotential function f on h 1 l we associate with an analytic milnor fiber for the formal function f and define the euler characteristic of l is to be the euler characteristic of the etale cohomology of the analytic milnor fiber in this paper we prove a thom sebastiani type formula for the euler characteristic of cyclic l algebras as applications we prove the joyce song formulas about the behrend function identities for semi schur objects in the derived category of coherent sheaves over calabi yau threefolds a motivic thom sebastiani type formula and a conjectural motivic joyce song formula for the motivic milnor fiber of cyclic l algebras are also discussed

parabolic refined invariants and macdonald polynomials a string theoretic derivation is given for the conjecture of hausel letellier and rodriguez villegas on the cohomology of character varieties with marked points their formula is identified with a refined bps expansion in the stable pair theory of a local root stack generalizing previous work of the first two authors in collaboration with pan haiman s geometric construction for macdonald polynomials is shown to emerge naturally in this context via geometric engineering in particular this yields a new conjectural relation between macdonald polynomials and refined local orbifold curve counting invariants the string theoretic approach also leads to a new spectral cover construction for parabolic higgs bundles in terms of holomorphic symplectic orbifolds

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

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

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

gopakumar vafa type invariants on calabi yau 4 folds via descendent insertions the gopakumar vafa type invariants on calabi yau 4 folds which are non trivial only for genus zero and one are defined by klemm pandharipande from gromov witten theory and their integrality is conjectured in a previous work of cao maulik toda mathop mathrm dt nolimits 4 invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical interpretation of the genus zero gv type invariants in this paper we propose a sheaf theoretical interpretation of the genus one gv type invariants using descendent insertions on the above moduli spaces the conjectural formula in particular implies nontrivial constraints on genus zero gv type equivalently gw invariants of cy 4 folds which can be proved by the wdvv equation

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

topological field theory and rational curves we analyze the quantum field theory corresponding to a string propagating on a calabi yau threefold this theory naturally leads to the consideration of witten s topological non linear model and the structure of rational curves on the calabi yau manifold we study in detail the case of the world sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by candelas de la ossa green and parkes

minimal matrices in the bruhat order for symmetric 0 1 matrices abstract in this paper we study the minimal matrices for the bruhat order on the class of symmetric 0 1 matrices with given row sum vector we will show that when restricted to the symmetric matrices new minimal matrices may appear besides the symmetric matrices for the nonrestricted bruhat order we modify the algorithm presented by brualdi and hwang 2004 which gives a minimal matrix for the bruhat order on the class of 0 1 matrices with given row and column sum vectors in order to obtain a minimal matrix for the bruhat order on the class of symmetric 0 1 matrices with given row sum vector we identify other minimal matrices in some of these classes namely we determine all the minimal matrices when the row sums are constant and equal to 3 we then describe a family of symmetric matrices that are minimal for the bruhat order on the class of 2k by 2k 0 1 matrices with constant row sums equal to k 1 and identify in terms of the term rank of a matrix a class of symmetric matrices that are related in the bruhat order with one of these minimal matrices

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