Reduction conjecture for arbitrary smooth GIT quotients and quantum D-module decomposition for arbitrary standard flips

Gu, Yu and Yu (arXiv:2508.15770, 2025) proved that for a simple wall-crossing $X_-\dashrightarrow X_+$ of GIT (geometric invariant theory) quotients of a smooth variety with reductive group action, with wall $S$, the genus-zero quantum $D$-module of $X_-$ decomposes into that of $X_+$ plus copies of that of $S$. Two conjectures that would extend this picture remain open in their stated generality. First, the reduction conjecture of Iritani (with Sanda), restated as Conjecture 1.11 by Gu–Yu–Yu: a discrete Fourier transform maps the equivariant $J$-function of the ambient space into the Givental cone of any smooth GIT quotient — proved so far only for toric quotients (Givental's mirror theorem), weight-$\pm 1$ fixed divisors, projective-bundle quotients, and highest/lowest quotients. Second, Gu–Yu–Yu's Conjecture 1.8: the same quantum $D$-module decomposition holds for arbitrary global standard flips in birational geometry — proved so far only for local models, blowups, and $D$-flips satisfying auxiliary algebraic hypotheses. The problem is to prove both conjectures in full generality, together with the further question of when the Reid–Thaddeus master space of a flip is smooth with a simple torus action. As of August 2026, neither conjecture has been resolved in its stated generality.

Background

Genus-zero Gromov–Witten theory organizes the enumerative invariants of a smooth complex projective variety $X$ (counts of holomorphic maps from curves, packaged by curve class with formal Novikov variables $Q^d$ keeping track of degrees) into the quantum $D$-module $\operatorname{QDM}(X)$: the cohomology $H^{*}(X)$ tensored with a $z$-formal variable, equipped with the Dubrovin connection $z\nabla$ whose flat sections are encoded by the $J$-function $J_X(\tau)$. The Givental Lagrangian cone $\mathcal{L}_X$ is the formal Lagrangian submanifold of Givental's symplectic space swept out by the $z J$-type solutions; it determines all genus-zero Gromov–Witten invariants of $X$, and maps between cones induce maps of quantum $D$-modules. Two varieties can then be compared at the level of the full big (i.e. all-parameter) quantum theory, not just the small quantum cohomology ring.

Geometric invariant theory (GIT) constructs quotients of a variety $W$ by a reductive group $G$ starting from a choice of $G$-linearized ample line bundle. Thaddeus showed that varying the linearization across a wall produces different quotients $X_- = W /\!/_{L_-} G$ and $X_+ = W /\!/_{L_+} G$ connected by a birational transformation, with a wall $S = W /\!/ G(0)$; this is a variation of GIT (VGIT) wall-crossing. Gu–Yu–Yu call such a wall-crossing simple when there is a single connected wall locus on which all stabilizers are $\mathbb{C}^{*}$, the linearization weights at wall points are coprime, the nonzero weights of the action on normal directions are $\pm 1$, and the quotients are smooth and projective. Their main theorem (Theorem 1.2) shows that for a simple wall-crossing with $r_+ < r_-$ (where $r_\pm$ measure the relative dimensions of the exceptional loci), the quantum $D$-module of $X_-$ is, after a formal change of variables over a localized Novikov ring, a direct sum of the quantum $D$-module of $X_+$ and $r_- - r_+$ copies of that of $S$, compatibly with connections and pairings; the case $r_+ = r_-$ recovers the invariance of quantum cohomology under ordinary flops proved by Lee, Lin, Qu and Wang.

The proof rests on Fourier analysis of equivariant quantum cohomology developed by Iritani and Iritani–Koto, inspired by Teleman's conjecture, which predicts that the equivariant quantum cohomology of a variety with torus action is identified with the quantum cohomology of its GIT quotient via the Seidel representation (the action of shift operators on equivariant parameters). For a smooth projective variety $W$ with a $\mathbb{C}^{*}$-action, the $\mathbb{C}^{*}$-equivariant Gromov–Witten theory (with equivariant parameter $\lambda$) is related to the non-equivariant theories of the fixed components and of the GIT quotients by two kinds of transformations: continuous Fourier transformations onto fixed components, which are known to preserve Givental cones, and a discrete Fourier transformation $\mathrm{F}_X$, which rewrites an equivariant series in $(\lambda, Q)$ as a series in a shift-operator variable (the Seidel operator) and is expected to land in the quantum theory of a smooth quotient $X = W /\!/ \mathbb{C}^{*}$. Iritani's reduction conjecture, formulated with Sanda and stated for a general torus, asserts precisely that this expectation holds: for $J_W(\theta)$ the equivariant big $J$-function of $W$, the transformed series $z\mathrm{F}_X(J_W(\theta))$ is well defined (supported on the correct dual GIT chamber) and lies on the Givental cone $\mathcal{L}_X$, thereby inducing a map of quantum $D$-modules from the equivariant theory of $W$ to the theory of $X$. It is proved only in special positions: for toric quotients (Givental's mirror theorem), for projective-bundle quotients $P(V)$ (Iritani–Koto), for quotients equal to a fixed divisor of $\mathbb{C}^{*}$-weight $\pm 1$ (Iritani), and for the highest and lowest GIT quotients, meaning the quotients adjacent to the extreme chambers, which are projective bundles over the extreme fixed components (Gu–Yu–Yu, Theorem 4.15). In all these cases the discrete transform can be identified with a cone-preserving continuous transform — an identification unavailable for quotients in intermediate chambers.

On the birational-geometric side, the minimal model program modifies varieties along flips: a birational map $X_- \dashrightarrow X_+$ that replaces a $q_-$-negative by a $q_+$-positive contraction over a common base $X_0$. A standard flip of type $(r_+ - 1, r_- - 1)$ has exceptional loci that are projective bundles $\mathbb{P}(V_\pm) \to S$ of ranks $r_\pm$ over a common smooth wall $S$; the flip is called simple when $S$ is a point. Gu–Yu–Yu's Conjecture 1.8 asserts that the conclusion of their main theorem extends verbatim to every such flip with $r_+ < r_-$: the reduced big quantum $D$-module of $X_-$ decomposes into that of $X_+$ plus $r_- - r_+$ copies of that of $S$ after formal changes of variables. Verified cases are: blowups ($r_+ = 1$, by Iritani), the local models $X_{\pm,\mathrm{loc}} = \mathbb{P}_{\mathbb{P}(V_\pm)}(\psi_\pm^{*}(V_\mp)(-1) \oplus \mathcal{O})$ (Gu–Yu–Yu, Corollary 6.8, extending Lee–Lin–Wang for simple flips), and $D$-flips — flips whose two contractions over the common base are induced by a relatively ample $\mathbb{Q}$-Cartier divisor $D$ on $X_-$ and its pushforward, in the sense of Thaddeus — whose associated graded section algebra satisfies the generation, primality and ampleness conditions of Lemma 6.9 (Corollary 6.12). The graded algebra construction of Reid and Thaddeus realizes such flips as VGIT wall-crossings when the master space is smooth with a simple action: for a $\mathbb{Q}$-Cartier divisor $D$ (a divisor some positive integral multiple of which is Cartier) defining the flip, the master space $W = \operatorname{Spec} \bigoplus_n H^0(X_-, \mathcal{O}(nD))$ carries a $\mathbb{C}^{*}$-action with quotients $X_\pm$; but this $W$ is in general neither smooth nor quasi-projective, and characterizing when it is smooth with a simple action is itself open (Section 6.5 of the paper). A weaker adjacent result, the decomposition of the quantum spectrum (the asymptotic spectrum of quantum multiplication) and the Gamma structure (the integral structure defined by Iritani's Gamma class) of standard flips, is due to Shen–Shoemaker.

Problem Statement

The common objective is to extend the quantum-cohomology wall-crossing theorem from its proved special cases to the two general statements conjectured in the literature, so that genus-zero quantum cohomology is controlled across arbitrary GIT chambers and arbitrary minimal-model-program flips. The problem consists of two central conjectures and one further open question.

(i) Reduction conjecture (Iritani, with Sanda; restated for $\mathbb{C}^{*}$-quotients as Conjecture 1.11 of Gu–Yu–Yu): Let $W$ be a smooth projective variety with a $\mathbb{C}^{*}$-action and let $X = W /\!/ \mathbb{C}^{*}$ be an arbitrary smooth GIT quotient. Let $J_W(\theta)$ be the $\mathbb{C}^{*}$-equivariant big $J$-function of $W$ and let $\mathrm{F}_X$ denote the discrete Fourier transformation associated with the quotient. Prove that $z \mathrm{F}_X(J_W(\theta))$ is defined and lies on the Givental cone $\mathcal{L}_X$ of $X$, inducing a map from the equivariant quantum $D$-module of $W$ to the quantum $D$-module of $X$. In the stronger general-torus form of Iritani–Sanda, the conjecture asserts that the discrete Fourier sum is supported on the dual chamber of the GIT chamber of $X$ and that multiplying by $z$ yields a point of $\mathcal{L}_X$; that form implies the $\mathbb{C}^{*}$ case stated here, which is the operative formulation of Conjecture 1.11. The conjecture must be established for every smooth quotient, not only toric quotients, weight-$\pm 1$ fixed divisors, projective-bundle quotients, or highest/lowest quotients.

(ii) Flip decomposition conjecture (Gu–Yu–Yu, Conjecture 1.8): Let $f \colon X_- \dashrightarrow X_+$ be an arbitrary standard flip of type $(r_+ - 1, r_- - 1)$ with $r_+ < r_-$ between smooth projective varieties, with wall $S$. Prove that there exist formal changes of variables $\tau_{X_+}$ and $\zeta_j$, $0 \le j \le r_- - r_+ - 1$, and an isomorphism of $\mathbb{C}[z]$-modules over the appropriately localized Novikov ring,

$$\Psi \colon \operatorname{QDM}(X_-)_{L^a,\mathrm{red}} \longrightarrow (\tau_{X_+})^{*}\operatorname{QDM}(X_+)_{L^a,\mathrm{red}} \oplus \bigoplus_{j=0}^{r_- - r_+ - 1} (\zeta_j)^{*}\operatorname{QDM}(S)_{L^a,\mathrm{red}},$$

compatible with the quantum connections and the pairings; here $\operatorname{QDM}(-)_{L^a,\mathrm{red}}$ denotes the quantum $D$-module after localization at Novikov monomials and reduction (elimination of the redundant Novikov variables), in the conventions of Gu–Yu–Yu. The statement must hold for arbitrary global standard flips, not only blowups, local models $X_{\pm,\mathrm{loc}}$, or $D$-flips satisfying the auxiliary conditions of Lemma 6.9.

(iii) From Section 6.5 of Gu–Yu–Yu: characterize the geometric hypotheses on a flip $f$ (or on the defining divisor $D$) under which the Reid–Thaddeus master space $W = \operatorname{Spec} \bigoplus_{n \in \mathbb{Z}} H^0(X_-, \mathcal{O}(nD))$ is smooth and quasi-projective and the induced $\mathbb{C}^{*}$-action satisfies the weight-$(\pm 1)$ simpleness conditions.

Conjectures (i) and (ii) are the two central open statements; (iii), from Section 6.5 of Gu–Yu–Yu, is a further open question about the Reid–Thaddeus master space of a flip.

References

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