---
title: K-Moduli of Quasimaps
url: https://www.emergentmind.com/topics/k-moduli-of-quasimaps
type: topic
---

# K-Moduli of Quasimaps

A quasimap is a generalized notion of a map from a source (typically a curve) to a target variety, allowing for basepoints, which interpolates between stable maps and more flexible enumeration theories, especially when considering GIT targets and certain degenerations. The K-moduli of quasimaps is the moduli stack (and its corresponding coarse space) parameterizing K-(semi/poly)stable quasimaps equipped with additional data and structures, central to modern approaches in K-stability, enumerative geometry, and moduli theory. Recent advances have established the projectivity and structure of K-moduli spaces of quasimaps under log Fano conditions and revealed deep links with moduli spaces of fibrations, such as Calabi-Yau varieties over curves of negative Kodaira dimension [2504.21519].

## 1. Definition and Structure of Quasimaps

Let $X$ be a projective normal variety embedded as $\iota: X \hookrightarrow \mathbb{P}^N$ and let $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$ denote its affine cone. A family of quasimaps of fixed degree $m$ over a base $S$ consists of:
- A proper, flat family of nodal curves $\pi_C: C \to S$.
- A principal $G_{m, S}$-bundle $\mu: P \to C$ with $G_{m, S} = \mathbb{G}_m \times S$ acting diagonally on $\mathbb{A}_S^{N+1}$.
- An equivariant morphism $P \to \mathbb{A}_S^{N+1}$.

The associated line bundle $L$ on $C$ is obtained as the $G_m$-quotient, and the degree of the quasimap is $\deg q = \deg(L_S)$, locally constant over $S$. A log Fano quasimap of degree $m$ and weight $u$ is a tuple $(C, B, q)$ with an effective $\mathbb{Q}$-divisor $B$ on $C$ such that:
- $K_C + B + u D \equiv 0$ in $\mathbb{Q}$ and $(C, B + u D)$ is klt for general $D \in |L|$,
- $\deg(B) + u m < 2$.

The condition that $q$ factors through $[\operatorname{Cone}(X)/\mathbb{G}_m]$ ensures the map lands in $X \subset \mathbb{P}^N$.

## 2. K-Stability and Log Fano Condition

K-stability of quasimaps is formalized via the Donaldson–Futaki (DF) invariant for test configurations:
Given a log Fano quasimap $(C,B) \to [\operatorname{Cone}(X)/G_m]$ of weight $u$, a test configuration is a flat family $\pi: (\mathscr{C}, \mathscr{D}, \mathcal{L}) \to \mathbb{A}^1$ with $G_m$-action such that the general fiber is isomorphic to $(C,B+uD,-(K_C+B+uD))$. The DF invariant is
$$
\mathrm{DF}_{\Delta,LF}(\mathscr{C}, \mathcal{L}) = \frac{(K_{\mathscr{C}/\mathbb{P}^1} + \mathscr{D}) \cdot \mathcal{L}^d}{\operatorname{Vol}(L)} - \frac{d}{d+1} \frac{\mathcal{L}^{d+1}}{\operatorname{Vol}(K+B)},
$$
where $d = \dim C$, and $\mathscr{D}$ is the closure of $B + u D$.

A log Fano condition requires $(C, B + u D)$ is klt and $-(K_C + B + u D)$ is ample ($\deg < 2$). This ensures boundedness, separatedness, and properness of the moduli, as well as well-defined DF invariants. The $\delta$-invariant,
$$
\delta(q) = 2 \cdot \min_{p \in C}(1-\operatorname{mult}_p(B+uD)) / \deg(-K_C - B - uD),
$$
detects K-stability in the sense of Blum–Jonsson.

## 3. The K-Moduli Stack of Quasimaps

For fixed discrete invariants $(m, r, u, v, \iota)$, the stack $\mathcal{M}_{m,r,u,v,\iota}$ of K-semistable log Fano quasimaps is an Artin stack of finite type. Its construction leverages:
- Boundedness via embedding quasimaps into a universal Hilbert scheme and tracking $B$ via Quot schemes.
- $S$-completeness and $\Theta$-reductivity via extension of families over DVRs, exploiting semistable reduction and K-semistability.
- Separatedness through unique extension arguments and finiteness of automorphisms of K-polystable quasimaps.

The tautological family $(\mathcal{C}, B, \mathcal{L}) \to \mathcal{M}_{m,r,u,v,\iota}$ admits a CM (Chow–Mumford) line bundle defined by the Knudsen–Mumford-type decomposition:
$$
\lambda_{\mathrm{CM}} := \lambda_0^{-1}/M, \quad \text{where} \quad \det(\pi_*(\mathcal{O}_\mathcal{C}(M(K+B)+M\mathcal{L}))) \simeq \lambda_0 \otimes \lambda_1 \otimes \lambda_2.
$$
$\lambda_{\mathrm{CM}}$ is nef, positive on families of maximal variation, hence ample on the coarse moduli space $\mathcal{M}_{m,r,u,v,\iota}^{ps}$ by the Nakai–Moishezon criterion.

The main existence theorem asserts: For $0 < u < 1 - v/2$, $\mathcal{M}_{m, r, u, v, \iota}^{ss}$ is of finite type with a projective good moduli space $\mathcal{M}_{m, r, u, v, \iota}^{ps}$, and $\lambda_{\mathrm{CM}}$ descends to an ample $\mathbb{Q}$-line bundle [2504.21519].

## 4. Comparison with K-Moduli of Calabi–Yau Fibrations

Let $M_{d,v,u,r,V}^{CYfib}$ denote the stack of adiabatically K-stable klt-trivial fibrations $f:(X, A)\rightarrow C$ whose generic fiber lies in a fixed connected component $V$ of the klt Calabi–Yau moduli. A quasi-finite morphism, the period/quasimap map,
$$
\alpha : M_{d,v,u,r,V}^{CYfib} \to M_{l(2-u),l,r,u,\iota}^{ps},
$$
relates the moduli of Calabi–Yau fibrations to the moduli of quasimaps, with $l = k l (2-u)$ and $\iota: V \to \mathbb{P}^N$ the Baily–Borel embedding. The map $\alpha$ preserves the main discrete invariants and is compatible with the CM line bundle structures.

As a key structural result, the CM line bundle $\Lambda_{CM,t}$ on $M_{d,v,u,r,V}^{CYfib}$, associated to the polarization $A + t f^*\mathcal{O}_C(1)$, converges to $\alpha^*(\Lambda^{quasimap}_{CM})$ as $t\rightarrow\infty$. The adiabatic limit, relating the CM invariants of fibers as $t$ grows, identifies the CM class of Calabi–Yau fibrations with that of the associated quasimaps [2504.21519].

## 5. Projectivity and Ampleness of the Moduli Spaces

Projectivity of the K-moduli space of log Fano quasimaps follows from ampleness of the CM line bundle and the boundedness/separatedness properties conferred by the log Fano condition. The map $\alpha$ is quasi-finite, and passage to the coarse moduli and ampleness of the limiting CM class yields the quasi-projectivity of the K-moduli space of Calabi–Yau fibrations as a corollary.

Explicitly, on $M_{d,v,u,r,V}^{CYfib}$,
$$
\Lambda_{CM,t} \to \Lambda_{CM,\infty} = \alpha^*(\Lambda_{CM}^{quasimap}),
$$
and the ampleness of $\Lambda_{CM,t}$ for $t \gg 0$ and quasi-finiteness of $\alpha$ together establish the whole quasi-projectivity of $M_{d,v,u,r,V}^{CYfib}$ in this setting.

## 6. Moduli-Theoretic, Enumerative, and Mirror Symmetry Aspects

The K-moduli of quasimaps provides a modular interpretation for enumerative invariants (e.g., quantum $K$-theoretic invariants) by supplying a projective ambient moduli space compatible with both Gromov–Witten and Landau–Ginzburg phase theories. Through wall-crossing techniques and derived enhancements, one can compare stable map and stable quasimap invariants, as exemplified in the case of projective spaces and their hypersurfaces [2302.12947, 2012.01401, 2210.11386]. Combinatorics of boundary strata in the moduli of quasimaps is reflected in generalized hypergeometric mirror periods, as in the I-function formalism, which encapsulates recursive and closed-form formulas for two-point gravitational invariants [2302.12947].

The construction of K-moduli spaces establishes foundational infrastructure for further exploration of wall-crossing, mirror phenomena, and arithmetic aspects in enumerative theories, with the CM line bundle playing a universal role in ampleness and projectivity for moduli of interest. Further directions include extensions to higher genus, more general targets, non-commutative and derived settings, and the interplay with K-theoretic quantum difference modules and quantum Adams operations [2510.09335].

## 7. Summary Table: Key Features of K-Moduli of Quasimaps

| Feature                | Description                                          | Source                         |
|------------------------|------------------------------------------------------|--------------------------------|
| Stack type             | Artin stack of finite type                           | [2504.21519]                   |
| Stability notion       | K-(semi,poly)stability via DF invariant              | [2504.21519]                   |
| Projectivity           | Coarse space is projective via CM line bundle        | [2504.21519]                   |
| Comparison             | Quasi-finite morphism from K-moduli of CY fibrations | [2504.21519]                   |
| CM line bundle         | Ample on coarse moduli, governs positivity           | [2504.21519]                   |
| Enumerative applications| Mirror symmetry, wall-crossing, quantum K-theory    | [2012.01401, 2302.12947]        |

Source: https://www.emergentmind.com/topics/k-moduli-of-quasimaps