Narrow Results

This paper studies the (small) quantum homology and cohomology of fibrations p:P→S² whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M, ω). It gives a proof that the rational cohomology splits additively as the vector space tensor product H*(M)⊗H*(S²), and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde–McDuff–Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space P and of the fiber M, whose properties reflect the relations between the Gromov–Witten invariants of P and M. In order to establish these properties we further develop the language introduced in [22] to describe the virtual moduli cycle (defined by Liu–Tian, Fukaya–Ono, Li–Tian, Ruan and Siebert).

We prove transformation formulae for generating functions of Gromov–Witten invariants on general toric Calabi–Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi–Yau threefolds.

In this paper, we study the Gromov–Witten theory of the Hilbert schemes X^[n] of points on a smooth projective surface X with positive geometric genus p_g. For fixed distinct points x₁, …, x_n-1 ∈ X, let β_n be the homology class of the curve {ξ + x₂ + ⋯ + x_n-1 ∈ X^[n] | Supp(ξ) = {x₁}}, and let β_KX be the homology class of {x + x₁ + ⋯ + x_n-1 ∈ X^[n] | x ∈ K_X}. Using cosection localization technique due to Y. Kiem and J. Li, we prove that if X is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor, then all the Gromov–Witten invariants of X^[n] defined via the moduli space of stable maps vanish except possibly when β is a linear combination of β_n and β_KX. When n = 2, the exceptional cases can be further reduced to the Gromov–Witten invariants: with and d ≤ 3, and with d ≥ 1. When , we show that which is consistent with a well-known formula of C. Taubes. In addition, for an arbitrary surface X and d ≥ 1, we verify that .

Let $X$ be a smooth projective toric variety with $k$ ample line bundles. Let $Z$ be the zero locus of $k$ generic sections. It is well known that the ambient quantum ${𝒟}$-module of $Z$ is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of $X$ and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum ${𝒟}$-module.

In this paper, we prove formulas that represent two-pointed Gromov–Witten invariant ${〈{{𝒪}}_{h^a}{{𝒪}}_{h^b}〉}_{0,d}$ of projective hypersurfaces with $d=1,2$ in terms of Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, the moduli spaces of stable maps from genus $0$ stable curves to projective space ${\mathbb{P}}^{N-1}$. Our formulas are based on representation of the intersection number $w{({{𝒪}}_{h^a}{{𝒪}}_{h^b})}_{0,d}$, which was introduced by Jinzenji, in terms of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$, the moduli space of quasi maps from ${\mathbb{P}}^1$ to ${\mathbb{P}}^{N-1}$ with two marked points. In order to prove our formulas, we use the results on Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, that were derived by Mustaţǎ and Mustaţǎ. We also present explicit toric data of ${\widetilde{Mp}}_{0,2}(N,d)$ and prove relations of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$.

We describe the extent to which Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants (GW-invariants) sharpens the usual symplectic sum formula. The key product operation on the target spaces for the refined invariants is specified in terms of abelian covers of symplectic divisors, making it suitable for studying from a topological perspective. We give several qualitative applications of this refinement, which include vanishing results for GW-invariants.

In this paper, we study the Gromov–Witten theory of the Hilbert scheme $X^{[2]}$ of two points on an elliptic surface $X$. Assume that $\left|K_X\right|$ contains an element supported on the smooth fibers of $X$. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the $1$-point genus-$0$ Gromov–Witten invariant ${〈w〉}_{0,d({\beta }_f-2{\beta }_2)}^{X[2]}$ up to some rational number $m(d,X)$ depending only on $d$ and $X$, where $w\in H^4(X^{[2]},ℂ)$, $d\ge 1$, $f$ is a smooth fiber of $X$, ${\beta }_f=x_0+f\subset X^{[2]}$ with $x_0\in X-f$ being a fixed point, and ${\beta }_2=\{\xi \in X^{[2]}|\mathrm{\text{Supp}}(\xi )=\{x_0\}\}$. Moreover, we propose a conjecture regarding $m(d,X)$, and prove that the conjecture is true for $X=C\times E$ where $E$ is an elliptic curve and $C$ is a smooth curve.

In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss–Manin system.

A homology class $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ of a complex flag variety $X=G∕P$ is called a line degree if the moduli space ${\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ of 0-pointed stable maps to X of degree d is also a flag variety $G∕P^{\prime }$. We prove a quantum equals classical formula stating that any n-pointed (equivariant, $\mathrm{\text{K}}$-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety $G∕P^{\prime }$. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags $G∕B$. Our formulas make it straightforward to compute the big quantum $\mathrm{\text{K}}$-theory ring ${\mathrm{\text{QK}}}^{\mathrm{\text{big}}}(X)$ modulo the ideal $〈Q^d〉$ generated by degrees d larger than line degrees.

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.

We relate the quantum Steenrod square to Seidel’s equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in $\mathbb{Z}/2$-equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant pair-of-pants product, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of $S{H}^{\ast }(T^{\ast }{S}^n)$. We finish by calculating the symplectic square for the negative line bundles $M=\text{Tot}({𝒪}(-1)\to {ℂ\mathbb{P}}^m)$, proving an equivariant version of a result due to Ritter.

We describe some recent progress and open problems in Gromov-Witten theory of Calabi-Yau 3-folds, focusing on the quintic 3-fold and toric Calabi-Yau 3-folds.

Hurwitz numbers were introduced by A. Hurwitz in the end of the nineteenth century. They enumerate ramified coverings of two-dimensional surfaces. They also have many other manifestations: as connection coefficients in symmetric groups, as numbers enumerating certain classes of graphs, as Gromov–Witten invariants of complex curves. Hurwitz numbers belong to a tribe of numerical sequences that penetrate the whole body of mathematics, like multinomial coefficients. They are indexed by partitions, or, more generally, by tuples of partitions, which does not allow one to overview all of them simultaneously. Instead, we usually deal with some of their specific subsequences. The Cayley numbers N^N-1 enumerating rooted trees on N marked vertices is may be the simplest such instance. The corresponding exponential generating series has been considered by Euler and he gave it the name of Lambert function. Certain series of Hurwitz numbers can be expressed by nice explicit formulas, and the corresponding generating functions provide solutions to integrable hierarchies of mathematical physics. The paper surveys recent progress in understanding Hurwitz numbers.