Moduli Space Of Stable Maps Research Articles

Gromov-Witten Invariants in Complex and Morava-Local K-Theories

AbstractGiven a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum $\mathbb{K}$-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

Quantum Lefschetz property for genus two stable quasimap invariants

AbstractBy the reduced component in a moduli space of stable quasimaps to n-dimensional projective space $$\mathbb {P}^n$$ P n we mean the closure of the locus in which the domain curves are smooth. As in the moduli space of stable maps, we prove the reduced component is smooth in genus 2, degree$$\ge 3$$ ≥ 3 . Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersectionXin $$\mathbb {P}^n$$ P n of genus 2, degree $$\ge 3$$ ≥ 3 is explicitly expressed in terms of the fundamental cycle of the reduced component of $$\mathbb {P}^n$$ P n and virtual cycles of lower genus $$<2$$ < 2 moduli spaces of X.

Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.

Moduli Spaces of Codimension-One Subspaces in a Linear Variety and their Tropicalization

We study the moduli space of $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove that these moduli spaces are linear subspaces themselves, and thus their tropicalization is completely determined by their associated (valuated) matroids. We show that these matroids can be interpreted as the matroid of lines of the hyperplane arrangement corresponding to $X$, and generically are equal to a Dilworth truncation of the free matroid. In this way, we can describe combinatorially tropicalized Fano schemes and tropicalizations of moduli spaces of stable maps of degree $1$ to a plane.

On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of CP1 with prescribed ramification profile over ∞, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers.In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of r-spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of r-spin Hurwitz numbers.

Open [formula omitted] descendent theory I: The stationary sector

We define stationary descendent integrals on the moduli space of stable maps from disks to (CP1,RP1). We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus 0 disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.

Gromov-Witten invariants of [formula omitted] coupled to a KdV tau function

We consider the pull-back of a natural sequence of cohomology classes Θg,n∈H2(2g−2+n)(M‾g,n,Q) to the moduli space of stable maps M‾g,n(P1,d). These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via ZBGW(ħ,t0,t1,...)=exp⁡∑ħ2g−2n!∫M‾g,nΘg,n⋅∏j=1nψjkj∏tkj. Insertions of the pull-backs of the classes Θg,n into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target P1 are given by a random matrix integral and satisfy the Toda equation.

Curve counting in genus one: Elliptic singularities and relative geometry

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.

Curve neighborhoods and minimal degrees in quantum products

Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be an arbitrary parabolic subgroup of G. Let be the G-homogeneous projective space attached to this situation. We consider the (small) quantum cohomology ring attached to X. Postnikov proved that any quantum product of two Schubert classes admits a unique minimal degree. In particular, this result implies that there exists a unique degree d which is minimal with the property that qd occurs with nonzero coefficient in the quantum product of two point classes. This unique minimal degree in is denoted by dX . We use the technique of curve neighborhoods by Buch-Mihalcea to compute dX explicitly in terms of Kostant’s cascade of orthogonal roots. Moreover, we prove orthogonality relations in greedy decompositions of minimal degrees. Such orthogonality relations can be seen as a generalization of the computation of dX from one minimal degree to every minimal degree, and were successfully applied to prove quasi-homogeneity of the moduli space of stable maps.

Higher genus relative Gromov–Witten theory and double ramification cycles

We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.

Connecting Hodge Integrals to Gromov–Witten Invariants by Virasoro Operators

In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety X can be connected to the generating function for Gromov–Witten invariants of X by a series of differential operators $$\{ L_m \mid m \ge 1 \}$$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich–Witten tau-function in the point case. This result is a generalization of the work in Liu and Wang [Commun. Math. Phys. 346(1):143–190, 2016] for the point case which solved a conjecture of Alexandrov.

Double ramification cycles with target varieties

Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S) has a meromorphic section with zeroes and poles exactly at the marked points x_i with orders prescribed by the integers a_i. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X. The main result of the paper is an explicit formula (in tautological classes) for the push-forward of the virtual fundamental class of the moduli space of stable maps to rubber over X via the forgetful morphism to the moduli space of stable maps to X. In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle. Applications of the new formula, viewed as calculating double ramification cycles with target X, are given.

Zero cycles on the moduli space of curves

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

Reduced invariants from cuspidal maps

We consider genus one enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the reduced genus one invariants of the quintic threefold, providing a modular interpretation of the latter.

Relations in the tautological ring of the moduli space of $K3$ surfaces

We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler’s relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal K3 surface. The decomposition has terms supported on Noether–Lefschetz loci which are not visible in the Beauville–Voisin decomposition for a fixed K3 surface. As a result of our universal decomposition, we prove the conjecture of Marian–Oprea–Pandharipande: the full tautological ring of the moduli space of K3 surfaces is generated in Chow by the classes of the Noether–Lefschetz loci. Explicit boundary relations are constructed for all \kappa classes. More generally, we propose a connection between relations in the tautological ring of the moduli spaces of curves and relations in the tautological ring of the moduli space of K3 surfaces. The WDVV relation in genus 0 is used in our proof of the MOP conjecture.

Uniqueness of minimal morphisms of logarithmic schemes

We give a sufficient condition under which the moduli space of morphisms between logarithmic schemes is quasifinite under the moduli space of morphisms between the underlying schemes. This implies that the moduli space of stable maps from logarithmic curves to a target logarithmic scheme is finite over the moduli space of stable maps, and therefore that it has a projective coarse moduli space when the target is projective.

The fundamental solution matrix and relative stable maps

Givental’s Lagrangian cone {mathscr {L}}_X is a Lagrangian submanifold of a symplectic vector space which encodes the genus-zero Gromov–Witten invariants of X. Building on work of Braverman, Coates has obtained the Lagrangian cone as the push-forward of a certain class on the moduli space of stable maps to . This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. We recast this construction in its natural context, namely the moduli space of stable maps to relative the divisor . We find that the resulting push-forward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental’s quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov–Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.

Stable maps in higher dimensions

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich’s definition when the domain is a curve and Tian-Donaldson’s definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau–Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical Kahler metrics.

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincare polynomials of the moduli spaces in low degrees.

Pixton’s double ramification cycle relations

We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on Mbar_{g,n} vanishes in codimension beyond g. This yields a collection of tautological relations in the Chow ring of Mbar_{g,n}. We describe, furthermore, how these relations can be obtained from Pixton's 3-spin relations via localization on the moduli space of stable maps to an orbifold projective line.