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You-Cheng Chou

Publications and source records attributed to You-Cheng Chou.

12 recordsLinked to original sources
arXiv2026

Virasoro Recursions for Simplicially Stable Curves with Colliding Markings

We study descendant integrals on moduli stacks of simplicially stable curves. For each finite collision complex, we factor the reduction morphism into elementary wall crossings and obtain a reconstruction formula in terms of ordinary Witten-Kontsevich correlators. We use the formula to derive corrected Virasoro recursions. We then package all finite-support theories in a completed disjoint-support Fock space, where the extended potential is a cumulant translate of the Witten-Kontsevich potential.

math.AG↗
arXiv2026

On the $K$-theoretic logarithmic double ramification class

The logarithmic double ramification cycle is the virtual fundamental class of the locus where a line bundle on a family of curves is fiberwise trivial. We construct a K-theoretic logarithmic double ramification class and prove a product formula and a \(\mathrm{GL}_r(\mathbb Z)\)-invariance property. We also give an explicit formula for this class in terms of a Grothendieck polynomial via a novel $K$-theoretic Thom--Porteous formula for vector bundles on algebraic stacks.

math.AG↗
arXiv2026

A theory of generalized Lam\'e curves

We study the generalized Lam'e equation (GLE) on an elliptic curve $E$ with multiple regular singularities $\mathbf{p} = (p_i)_{i = 1}^r$ of weights $\mathbf{n} = (n_i)_{i = 1}^r$. By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves: (i) The generalized Lam'e curve (GLC), $\mathcal{Y}_{\mathbf{n}, \mathbf{p}}$, which lies in an affine bundle over $\operatorname{Sym}^n E$ for total weight $n:=\sum n_i \in \mathbb{Z}_{\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions. (ii) The log-free curve, $V_{\mathbf{n}, \mathbf{p}}$, a non-complete intersection variety arising when all $n_i \in \frac{1}{2}\mathbb{N}$, which we prove is a reduced curve, confirming a conjecture of Wang. We analyze the GLC as an algebraic family over the pole configuration space. By studying the addition map$$\sigma \colon \operatorname{Sym}^n E \longrightarrow E,$$where we establish a generically finite, universal degree formula, we show that the geometry of boundary degenerations under pole collisions perfectly mirrors the tensor algebra of $\mathfrak{sl}_2(\mathbb{C})$-modules within the BGG category $\mathcal{O}$. This provides the local structural limits needed to establish the global flatness of the GLC. Furthermore, we develop a framework of twisted isomonodromic deformations and construct $(\mathbf{n}, \mathbf{p})$-deformed pre-modular forms parameterized by twisted monodromy data $(t,s)$. Their vanishing solves the underlying monodromy problem and factorizes along boundary strata, allowing an arbitrary configuration to be continuously deformed down to the classical Lam'e equation. Finally, using an asymptotic scaling technique, we completely solve the Treibich conjecture for $r=2$ symmetric pairs, extend it to $r \leq 4$, and propose a general formula enumerating symmetric finite-gap KdV potentials for all $r$.

math.AG↗
arXiv2024

Characterization and enumeration on Lam\'e equations with finite monodromy

We give a complete characterization of the classical Lam\'e equations $y'' = (n(n + 1)\wp(z) + B)y$, $n \in \Bbb R$, $B \in \Bbb C$ on flat tori $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z\,\tau)$ with finite monodromy groups $M$. Beuker--Waall had shown that such $n$ must lie in a finite number of arithmetic progressions $n_i + \Bbb N \subset \Bbb Q$ and they determined all corresponding $M$. By combining the theory of dessin d'enfants with the geometry of spherical tori, we prove the existence of $(B, \tau)$ for each such $n$ and provide a description of all such $(n, B, \tau, M)$. In particular, for a given $(n, M)$ with $n \not\in \tfrac{1}{2} + \Bbb Z$, we prove the finiteness of $(B, \tau)$ and derive an explicit counting formula of them. (The case $n \in \tfrac{1}{2} + \Bbb Z$ is a classical result due to Brioschi--Halphen--Crawford.) The main ingredients in this work are (1) the definition and classification of basic spherical triangles with finite monodromy and (2) the process of attaching cells corresponding to $n \mapsto n + 1$ which reduces the problem to the basic case.

math.DG↗
arXiv2023

Higher Genus Quantum $K$--theory

We prove genus $g$ invariants in quantum $K$-theory are determined by genus zero invariants of a smooth stack in the spirit of K.~Costello's result in Gromov--Witten theory.

math.AG↗
arXiv2023

Quantum K-invariants and Gopakumar-Vafa invariants II. Calabi-Yau threefolds at genus zero

This is the second part of our ongoing project on the relations between Gopakumar-Vafa BPS invariants (GV) and quantum K-theory (QK) on the Calabi--Yau threefolds (CY3). We show that on CY3 a genus zero quantum K-invariant can be written as a linear combination of a finite number of Gopakumar--Vafa invariants with coefficients from an explicit ``multiple cover formula''. Conversely, GV can be determined by QK in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr. This result is consistent with the ``virtual Clemens conjecture'' for the Calabi--Yau threefolds. A heuristic derivation of the relation between QK and GV via the virtual Clemens conjecture and the multiple cover formula is also given.

math.AG↗
arXiv2020

The Log Product Formula in quantum $K$-theory

We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. The proof requires introducing log virtual fundamental classes in $K$-theory and verifying their various functorial properties. We introduce a log version of $K$-theory and prove the formula there as well.

math.AG↗
arXiv2016

Generalized Lamé equation with finite monodromy

In this paper, we study the algebraic form of the symmetric generalized Lamé equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes the form \begin{equation*} \begin{split} \frac{d^2 y}{dz^2}-\left[n_1(n_1+1)(\wp(z+a) +\wp(z-a))\right.\left. +A_1(ζ(z+a) - ζ(z-a))+n_0(n_0+1) \wp(z)+B\right]y=0, \end{split} \end{equation*} where $n_1, n_0 \in \Bbb R$, $A_1,B \in \Bbb C$, and $\wp$ is the Weierstrass elliptic function. We give a complete list of all the group types that occur as the finite projective monodromy groups on the algebraic form and give the corresponding parameters $n_0$ and $n_1$. For equations with only $1$ or $2$ regular singular points, we further determine their monodromy group types. The main tool used is the Grothendieck correspondence which gives a bijection between Belyi pairs and dessin d'enfants. By Klein's theorem, we may regard generalized Lamé equations with finite monodromy as pullbacks of the hypergeometric equations. In this paper, we will restrict our cases with the assumption that the pullback maps are Belyi functions. Under this setup, our main results consist of a systematic construction on the required dessin. In particular a gluing procedure of dessin will be developed to enable inductive constructions of the dessin.

math.AG↗