K-Moduli of Quasimaps

• K-Moduli of Quasimaps is a framework that parameterizes K-stable log Fano quasimaps, ensuring projectivity through the ample CM line bundle.
• It employs advanced techniques in K-stability and log Fano conditions, controlling boundedness and separatedness via test configurations and DF invariants.
• The approach connects moduli of quasimaps with Calabi–Yau fibrations and mirror symmetry, facilitating applications in enumerative geometry and wall-crossing.

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 (Hashizume et al., 30 Apr 2025).

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 $\iota: X \hookrightarrow \mathbb{P}^N$0.

The associated line bundle $\iota: X \hookrightarrow \mathbb{P}^N$1 on $\iota: X \hookrightarrow \mathbb{P}^N$2 is obtained as the $\iota: X \hookrightarrow \mathbb{P}^N$3-quotient, and the degree of the quasimap is $\iota: X \hookrightarrow \mathbb{P}^N$4, locally constant over $\iota: X \hookrightarrow \mathbb{P}^N$5. A log Fano quasimap of degree $\iota: X \hookrightarrow \mathbb{P}^N$6 and weight $\iota: X \hookrightarrow \mathbb{P}^N$7 is a tuple $\iota: X \hookrightarrow \mathbb{P}^N$8 with an effective $\iota: X \hookrightarrow \mathbb{P}^N$9-divisor $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$0 on $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$1 such that:

• $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$2 in $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$3 and $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$4 is klt for general $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$5,
• $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$6.

The condition that $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$7 factors through $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$8 ensures the map lands in $\operatorname{Cone}(X) \subset \mathbb{A}^{N+1}$9.

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 $m$0 of weight $m$1, a test configuration is a flat family $m$2 with $m$3-action such that the general fiber is isomorphic to $m$4. The DF invariant is

$m$5

where $m$6, and $m$7 is the closure of $m$8.

A log Fano condition requires $m$9 is klt and $S$0 is ample ($S$1). This ensures boundedness, separatedness, and properness of the moduli, as well as well-defined DF invariants. The $S$2-invariant,

$S$3

detects K-stability in the sense of Blum–Jonsson.

3. The K-Moduli Stack of Quasimaps

For fixed discrete invariants $S$4, the stack $S$5 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 $S$6 via Quot schemes.
• $S$7-completeness and $S$8-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 $S$9 admits a CM (Chow–Mumford) line bundle defined by the Knudsen–Mumford-type decomposition:

$\pi_C: C \to S$0

$\pi_C: C \to S$1 is nef, positive on families of maximal variation, hence ample on the coarse moduli space $\pi_C: C \to S$2 by the Nakai–Moishezon criterion.

The main existence theorem asserts: For $\pi_C: C \to S$3, $\pi_C: C \to S$4 is of finite type with a projective good moduli space $\pi_C: C \to S$5, and $\pi_C: C \to S$6 descends to an ample $\pi_C: C \to S$7-line bundle (Hashizume et al., 30 Apr 2025).

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

Let $\pi_C: C \to S$8 denote the stack of adiabatically K-stable klt-trivial fibrations $\pi_C: C \to S$9 whose generic fiber lies in a fixed connected component $G_{m, S}$0 of the klt Calabi–Yau moduli. A quasi-finite morphism, the period/quasimap map,

$G_{m, S}$1

relates the moduli of Calabi–Yau fibrations to the moduli of quasimaps, with $G_{m, S}$2 and $G_{m, S}$3 the Baily–Borel embedding. The map $G_{m, S}$4 preserves the main discrete invariants and is compatible with the CM line bundle structures.

As a key structural result, the CM line bundle $G_{m, S}$5 on $G_{m, S}$6, associated to the polarization $G_{m, S}$7, converges to $G_{m, S}$8 as $G_{m, S}$9. The adiabatic limit, relating the CM invariants of fibers as $\mu: P \to C$0 grows, identifies the CM class of Calabi–Yau fibrations with that of the associated quasimaps (Hashizume et al., 30 Apr 2025).

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 $\mu: P \to C$1 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 $\mu: P \to C$2,

$\mu: P \to C$3

and the ampleness of $\mu: P \to C$4 for $\mu: P \to C$5 and quasi-finiteness of $\mu: P \to C$6 together establish the whole quasi-projectivity of $\mu: P \to C$7 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 $\mu: P \to C$8-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 (Jinzenji et al., 2023, Zhang et al., 2020, Kern et al., 2022). 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 (Jinzenji et al., 2023).

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 (Bai et al., 10 Oct 2025).

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

Feature	Description	Source
Stack type	Artin stack of finite type	(Hashizume et al., 30 Apr 2025)
Stability notion	K-(semi,poly)stability via DF invariant	(Hashizume et al., 30 Apr 2025)
Projectivity	Coarse space is projective via CM line bundle	(Hashizume et al., 30 Apr 2025)
Comparison	Quasi-finite morphism from K-moduli of CY fibrations	(Hashizume et al., 30 Apr 2025)
CM line bundle	Ample on coarse moduli, governs positivity	(Hashizume et al., 30 Apr 2025)
Enumerative applications	Mirror symmetry, wall-crossing, quantum K-theory	(Zhang et al., 2020, Jinzenji et al., 2023)