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Fenglong You

Publications and source records attributed to Fenglong You.

At least 19 recordsLinked to original sources
arXiv2025

Relative mirror symmetry for non-Fano varieties

Given a smooth projective variety $X$ with a smooth anticanonical divisor $D$, we study mirror symmetry for the log Calabi--Yau pair $(X,D)$ without assuming that $D$ is nef. We consider the mirror proper Landau--Ginzburg model $(\check X,W)$ from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of $X$ and the classical period of $W$, and identify the discrepancy between them as originating from curve counts in $D$, governed by the mirror map associated with $D$. We also obtain an explicit formula for the proper potential $W$ that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential.

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arXiv2025

Relative mirror symmetry, theta functions and the Gamma conjecture

Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef. (2) We study theta functions associated with the pair $(X,D)$. (3) We introduce functions on the mirror that are obtained from the higher-degree part of the big relative quantum cohomology. As an application, we use these new ingredients in relative mirror symmetry to prove a version of the mirror symmetric Gamma conjecture for $X$ for $\mathcal O_X$ and $\mathcal O_{\operatorname{pt}}$ in this setting, where the Landau--Ginzburg potential is defined as a sum of theta functions.

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arXiv2024

Orbifold theta functions and mid-age invariants

We use the orbifold approach to study theta functions in intrinsic mirror symmetry. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule.

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arXiv2024

Gromov--Witten theory beyond maximal contacts

Given a smooth projective variety $X$ and a smooth nef divisor $D$, we identify genus zero relative Gromov--Witten invariants of $(X,D)$ with $(n+1)$ relative markings with genus zero orbifold Gromov--Witten invariants of multi-root stacks over the $\mathbb P^1$-bundle $P:=\mathbb P(\mathcal O_X(-D)\oplus \mathcal O_X)$ with $n$ orbifold markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants of ambient insertions with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.

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arXiv2022

The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map

Given a smooth log Calabi--Yau pair $(X,D)$, we use the intrinsic mirror symmetry construction to define the mirror proper Landau--Ginzburg potential and show that it is a generating function of two-point relative Gromov--Witten invariants of $(X,D)$. We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of \cite{FTY} to compute two-point relative invariants. When $D$ is nef, we compute the proper Landau--Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety $X$, this implies the conjecture of \cite{GRZ} that the proper Landau--Ginzburg potential is the open mirror map. When $X$ is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

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arXiv2022

Relative quantum cohomology under birational transformations

We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.

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arXiv2021

Degenerations, fibrations and higher rank Landau-Ginzburg models

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations.

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arXiv2021

The local-orbifold correspondence for simple normal crossings pairs

For $X$ a smooth projective variety and $D=D_1+\ldots+D_n$ a simple normal crossings divisor, we establish a precise cycle-level correspondence between the genus zero local Gromov-Witten theory of the bundle $\oplus_{i=1}^n \mathcal{O}_X(-D_i)$ and the maximal contact Gromov-Witten theory of the multi-root stack $X_{D,\vec{r}}$. The proof is an implementation of the rank reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

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arXiv2020

A Gromov-Witten theory for simple normal-crossing pairs without log geometry

We define a new Gromov-Witten theory relative to simple normal crossing divisors as a limit of Gromov-Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism, Virasoro constraints (genus zero) and a partial cohomological field theory. Furthermore, we use the degree zero part of the relative quantum cohomology to provide an alternative mirror construction of Gross-Siebert and to prove the Frobenius structure conjecture of Gross-Hacking-Keel.

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arXiv2020

Double ramification cycles on the moduli spaces of admissible covers

We derive a formula for the virtual class of the moduli space of rubber maps to $[\mathbb{P}^1/G]$ pushed forward to the moduli space of stable maps to $BG$. As an application, we show that the Gromov-Witten theory of $[\mathbb{P}^1/G]$ relative to $0$ and $\infty$ are determined by known calculations.

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arXiv2020

A mirror theorem for multi-root stacks and applications

Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an $I$-function, a slice of Givental's Lagrangian cone for Gromov--Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $X_{D,\vec r}$ stabilize for sufficiently large $\vec r$. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau--Ginzburg potentials using orbifold invariants of $X_{D,\vec r}$.

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arXiv2020

Relative Gromov-Witten invariants and the enumerative meaning of mirror maps for toric Calabi-Yau orbifolds

We provide an enumerative meaning of the mirror maps for toric Calabi-Yau orbifolds in terms of relative Gromov-Witten invariants of the toric compactifications. As a consequence, we obtain an equality between relative Gromov-Witten invariants and open Gromov-Witten invariants. Therefore, the instanton corrected mirrors for toric Calabi-Yau orbifolds can be constructed using relative Gromov-Witten invariants.

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arXiv2020

Gromov--Witten invariants of root stacks with mid-ages and the loop axiom

We study orbifold Gromov--Witten invariants of the $r$-th root stack $X_{D,r}$ with a pair of mid-ages when $r$ is sufficiently large. We prove that genus $g$ invariants with a pair of mid-ages $k_a/r$ and $1-k_a/r$ are polynomials in $k_a$ and the $k_a^i$-coefficients are polynomials in $r$ with degree bounded by $2g$. Moreover, genus zero invariants with a pair of mid-ages are independent of the choice of mid-ages. As an application, we obtain an identity for relative Gromov--Witten theory which can be viewed as a modified version of the usual loop axiom.

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arXiv2020

Higher genus relative and orbifold Gromov-Witten invariants of curves

Given a smooth target curve $X$, we explore the relationship between Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved that relative invariants are equal to the $r^0$-coefficient of the corresponding orbifold Gromov-Witten invariants of $r$-th root stack for $r$ sufficiently large. Our result provides a precise relation between relative and orbifold invariants of target curves generalizing the result of Abramovich-Cadman-Wise to higher genus invariants of curves. Moreover, when $r$ is sufficiently large, we proved that relative stationary invariants of $X$ are equal to the orbifold stationary invariants in all genera. Our results lead to some interesting applications: a new proof of genus zero equality between relative and orbifold invariants of $X$ via localization; a new proof of the formula of Johnson-Pandharipande-Tseng for double Hurwitz numbers; a version of GW/H correspondence for stationary orbifold invariants.

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arXiv2020

Higher genus relative and orbifold Gromov-Witten invariants

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-th root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov-Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher genus relative Gromov-Witten invariants of $(X,D)$ are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise \cite{ACW}. When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera.

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arXiv2019

Higher genus relative Gromov--Witten theory and DR-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.

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arXiv2019

Structures in genus-zero relative Gromov--Witten theory

In this paper, we define genus-zero relative Gromov--Witten invariants with negative contact orders. Using this, we construct relative quantum cohomology rings and Givental formalism. A version of Virasoro constraints also follows from it.

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