arXiv subjects
Publications and source records attributed to Hua-Zhong Ke.
We prove the Virasoro constraints for the relative Gromov--Witten theory of all smooth projective effective orbifold curves, allowing relative conditions at ordinary points. The absolute Jiang--Tseng Virasoro conjecture for smooth projective effective orbifold curves follows as a corollary.
For a Fano manifold $X$, Gamma conjecture II aims to use $\mathcal{D}_{\rm{coh}}^b(X)$ to describe the asymptotic behavior of its Dubrovin connection via $\widehatΓ$-integral structure. It was proposed by Galkin, Golyshev and Iritani, and can be regarded as a quantitative refinement of Dubrovin's conjecture on Fano manifolds with semisimple big quantum cohomology. As a step toward Gamma conjecture II, we define the Gamma-I property at points satisfying the (SR) condition, arising from the original Gamma conjecture I. We prove that the property holds globally in the following sense: if it holds at one such point, then it holds throughout the connected component of the (SR)-region containing that point. Based on this global Gamma-I property, we establish a strategy-type theorem relating Gamma conjecture II to the Gamma-I property at a possibly non-semisimple point, together with an analysis of small quantum cohomology. We further apply this theorem to prove Gamma conjecture II for del Pezzo surfaces; the proof combines Iritani's Galois action with addtional elementary operations on exceptional collections, and its most technically involved step consists in verifying the required global Gamma-I property.
We investigate power series satisfying the exponential concentration property, and show that suitable modifications of hypergeometric series respect this property. As a geometric application, we prove that the quantum period of a Fano manifold possesses the same property, whenever the manifold admits a convenient weak Landau-Ginzburg model with non-negative coefficients.
We show a quantum version of Chern character homomorphism from the small quantum K-theory to the small quantum cohomology in the cases of projective spaces and incidence varieties, whose classical limit gives the classical Chern character homomorphism. We also provide a ring presentation of the small quantum K-theory of Milnor hypersurfaces.
The Gamma conjecture II for the quantum cohomology of a Fano manifold $F$, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of $\mathcal D^b(F)$ and the $\widehatΓ$-integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and the Gamma conjecture II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin's theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov-Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov's exceptional collections and an application of our criterion.
Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture $\mathcal{O}$ by deriving a generalized Perron-Frobenius theorem on eigenvalues of real matrices and a vanishing result of certain Gromov-Witten invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying mirror techniques proposed by Galkin-Iritani together with the study of Gamma conjecture I for weighted projective spaces.
For a symplectic manifold M without boundary (not necessarily compact), we prove Poincare type duality in filtered cohomology rings of differential forms on M.
We prove that Fano complete intersections in projective spaces satisfy Conjecture $\mathcal O$ proposed by Galkin-Golyshev-Iritani.
For a $\mathbb P^1$-orbifold $\mathscr C$, we prove that its big quantum cohomology is generically semisimple. As a corollary, we verify a conjecture of Dubrovin for orbi-curves. We also show that the small quantum cohomology of $\mathscr C$ is generically semisimple iff $\mathscr C$ is Fano, i.e. it has positive orbifold Euler characteristic.
Let $X$ and $X'$ be nonsingular projective $3$-folds related by a flop of a disjoint union of $(-2)$-curves. We prove a flop formula relating the Donaldson-Thomas invariants of $X$ to those of $X'$, which implies some simple relations among BPS state counts. As an application, we show that if $X$ satisfies the GW/DT correspondence for primary insertions and descendants of the point class, then so does $X'$. We also propose a conjectural flop formula for general flops.
In this note, we describe a a systematic procedure to find toric crepant resolutions of orbifold vertex, and show that the generating series of certain disc invariants of the orbifold vertex can be suitably identified with the generating series of certain disc invariants of its toric crepant resolutions.
We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example $[\mathbb C^3 /\mathbb Z_5 (1, 1, 3)]$.
In this note, we present a recursive formula for the full partition function Z of descendent integrals over moduli spaces of open and closed Riemann surfaces, assuming the conjecture recently proposed by Pandharipande, Solomon and Tessler that Z satisfies the open KdV and Virasoro equations.
We use degeneration formula to study the change of stable pair invariants of 3-folds under blow-ups and obtain some closed blow-up formulae. Related results on Donaldson-Thomas invariants are also discussed. Our results give positive evidence for GW/DT/P correspondence, and also give partial correspondence for varieties not necessarily toric or complete intersections.
Using the degeneration formula and absolute/relative correspondence, one studied the change of Gromov-Witten invariants under blow-up for six dimensional symplectic manifolds and obtained closed blow-up formulae for high genus Gromov-Witten invariants. Our formulae also imply some relations among generalized BPS numbers introduced by Pandharipande.