arXiv subjects
Publications and source records attributed to Hui-Wen Lin.
For a non-smooth toric flop of the simplest type, we provide the $\textit{quantum product formulas}$ and construct the $\textit{degree-preserving}$ quantum correspondence via $\textit{generalized}$ analytic continuation which is achieved through a $\textit{regularization}$ map. Moreover, the generating functions of Gromov--Witten invariants with ancestors are invariant for $\textit{all genera}$.
Let $\Sigma$ and $\Sigma'$ be two refinements of a fan $\Sigma_0$ and $f \colon X_{\Sigma} \dashrightarrow X_{\Sigma'}$ be the birational map induced by $X_{\Sigma} \rightarrow X_{\Sigma_0} \leftarrow X_{\Sigma'}$. We show that the graph closure $\overline{\Gamma}_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality. In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_{\Sigma}\mathop{\times}\limits_{X_{\Sigma_0}}X_{\Sigma'}$ is in general not toric, though it is still irreducible and $X_{\rm red} = \overline{\Gamma}_f$. A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_\Sigma$ has at most terminal singularities. In this case, we further conclude that $X$ is normal.
We study analytic continuations of quantum cohomology under simple flips $f: X \dashrightarrow X'$ along the extremal ray quantum variable $q^\ell$. The inverse correspondence $\Psi = [\Gamma_f]^*$ by the graph closure gives an embedding of Chow motives $[\hat{X}'] \hookrightarrow [\hat{X}]$ which preserves the Poincar\'e pairing. We construct a deformation $\widehat{\Psi}$ of $\Psi = [\Gamma_f]^*$ which induces a non-linear embedding $$QH(X') \hookrightarrow QH(X)$$ in the category of $F$-manifolds into the regular integrable loci of $QH(X)$ near $q^\ell = \infty$. This provides examples of functoriality of quantum cohomology beyond $K$-equivalent transformations. In this paper, we focus on the case when $X$ and $X'$ are (projective) local models.
For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.
This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.
This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.
Via a natural degeneration of Grassmannian manifolds $G(k,n)$ to Gorenstein toric Fano varieties $P(k,n)$ with conifold singularities, we suggest an approach to study the relation between the tautological system on $G(k,n)$ and the extended GKZ system on the small resolution $\hat{P}(k,n)$ of $P(k,n)$. We carry out the simplest case $(k,n)=(2,4)$ to ensure its validity and show that the extended GKZ system can be regarded as a tautological system on $\hat{P}(2,4)$.
For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.