Categorical structures of Kuranishi spaces with $L_{\infty}[1]$-algebras

Abstract: We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of points in the zero locus of the Kuranishi section. We show that these objects collectively form a category into which the category of smooth manifolds naturally embeds. Some notions in \cite{FOOO1} are modified to achieve the desired categorical structures; for instance, the tangent bundle condition for chart embeddings is replaced by a quasi-isomorphism condition for the $L_{\infty}[1]$-structures.

• The paper constructs local L∞[1]-algebras from Kuranishi charts, presymplectic foliations, Koszul complexes, and coisotropic embeddings to encode deformation data independently of auxiliary choices.
• It replaces the traditional FOOO tangent-bundle condition with a quasi-isomorphism requirement, proving that suitable FOOO embeddings induce L∞-Kuranishi chart embeddings and that expansions become isomorphic.
• It establishes a category Kur of L∞-Kuranishi spaces whose morphisms are defined up to homotopy, whose coordinate-change cocycles simplify to base-map conditions, and which contains smooth manifolds as a subcategory.

Motivation: choice-dependence in Kuranishi theory

Kuranishi structures, introduced by Fukaya and Ono to construct virtual fundamental chains on moduli spaces of pseudoholomorphic curves, carry an intrinsic dependence on auxiliary data: the obstruction bundle $E$, the ambient manifold $U$, and the finite group action $\Gamma$. The paper's motivating example is the "expansion" $\mathcal{U} \times V$ of a chart by a finite-dimensional vector space $V$, for which $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$, yet the FOOO notion of chart isomorphism in [FOOO1] is too restrictive to identify the two. The author's stated objective is therefore twofold: to reformulate Kuranishi theory so that ambient choices become homotopically trivial, and to obtain an honest category of Kuranishi spaces containing the category of smooth manifolds. The mechanism is to attach, to each chart and each zero point of the Kuranishi section, a locally defined $L_{\infty}[1]$-algebra, in the spirit of $L_{\infty}$-spaces as developed by Alexandrov–Kontsevich–Schwarz–Zaboronsky, Costello, Behrend–Liao–Xu, Tu, and others.

$L_{\infty}$-Kuranishi charts

An $L_{\infty}$-Kuranishi chart on a compact metrizable space $U$0 is a tuple $U$1 in which the base $U$2 is equipped with a closed two-form $U$3 admitting a stratification $U$4 by the loci where $U$5 has constant rank. By results of Kim–Oh, such stratifications exist for a residual set of closed two-forms and are Whitney stratifications. Around each zero point $U$6, the stratification yields a presymplectic neighborhood $U$7 with regular foliation $U$8, and a local $U$9-algebra

$\Gamma$0

the direct sum of the Koszul complex of $\Gamma$1 (with differential $\Gamma$2 and no higher operations) and the augmented foliation de Rham complex. The latter carries an $\Gamma$3-structure obtained via Gotay's coisotropic embedding of $\Gamma$4 into $\Gamma$5 and the V-algebra formalism of Voronov and Cattaneo–Schätz; the augmentation is constructed recursively using the Poincaré lemma for foliations of Miranda–Solha. The resulting augmented complexes are acyclic, and the construction is independent of the choice of $\Gamma$6 and of the splitting $\Gamma$7 up to $\Gamma$8-isomorphism.

Chart morphisms and the quasi-isomorphism condition

A chart morphism over $\Gamma$9 is a pair $\mathcal{U} \times V$0 with $\mathcal{U} \times V$1 a $\mathcal{U} \times V$2-equivariant smooth map and $\mathcal{U} \times V$3 a family of $\mathcal{U} \times V$4-morphisms $\mathcal{U} \times V$5, required to factor through the completion $\mathcal{U} \times V$6 at the image of $\mathcal{U} \times V$7. The completion is taken with respect to the ideal of functions vanishing on $\mathcal{U} \times V$8, and the paper proves that the natural map $\mathcal{U} \times V$9 is itself an $V$0-morphism and that acyclicity is preserved under completion when the components of $V$1 lie in $V$2.

The central definitional move replaces the FOOO tangent bundle condition — the isomorphism $V$3 — by the requirement that $V$4 be a quasi-isomorphism. The main comparison result states that, under two additional conditions (vanishing of the complementary part of $V$5 on $V$6, and the tangent bundle condition holding on all of $V$7 rather than at the zero locus alone), every FOOO embedding determines an $V$8-Kuranishi chart embedding. The proof proceeds by constructing an explicit $V$9-morphism $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$0 via pullbacks along a retraction $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$1, a symplectic embedding $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$2 of cotangent bundles, and the bundle embedding $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$3, and then showing quasi-isomorphicity through a double-complex argument: the Koszul quotient decomposes into columns whose acyclicity follows from the regularity of the sequence $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$4 of complementary components of $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$5, which in turn follows from the tangent bundle condition via a zero-divisor argument. The Whitehead theorem for strict $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$6-algebras over a field then supplies the homotopy inverse needed in the definition of coordinate changes.

Two structural consequences follow immediately. First, expansions $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$7 become isomorphic to $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$8 in the new category, since the projection $(s \times \mathrm{id}_V)^{-1}(0) \simeq s^{-1}(0)$9 induces an $L_{\infty}[1]$0-quasi-isomorphism — resolving the motivating example. Second, the cocycle condition for coordinate changes is imposed only on the base maps; the $L_{\infty}[1]$1-component is automatically compatible up to homotopy because all relevant morphisms are quasi-isomorphisms between acyclic complexes, hence homotopic by the homotopy model theory developed in the companion paper [Kim2].

The category of $L_{\infty}[1]$2-Kuranishi spaces

An $L_{\infty}[1]$3-Kuranishi atlas assigns to each point of $L_{\infty}[1]$4 a chart with contractible base, together with coordinate changes that are embeddings of charts, satisfying a cocycle condition on base maps only. Two atlases are declared equivalent if, after restriction to open subatlases and expansion by Euclidean factors, they agree; the author verifies this is an equivalence relation using the contractibility of chart bases and the homotopy theory of [Kim2]. An $L_{\infty}[1]$5-Kuranishi space is then an equivalence class $L_{\infty}[1]$6.

Morphisms are equivalence classes of pre-morphisms: tuples $L_{\infty}[1]$7 where $L_{\infty}[1]$8 is continuous and the chart morphisms are compatible with coordinate changes, the $L_{\infty}[1]$9-compatibility being required only up to $L_{\infty}$0-homotopy. Equivalence of pre-morphisms is again defined through expansions and surjective extensions of the base maps, with the $L_{\infty}$1-condition expressed as homotopy commutativity of a diagram of local algebras. The paper proves that composition, defined by composing the base maps, the chart maps, and the $L_{\infty}$2-morphisms, is well defined and associative, with identities given by the evident pre-morphisms. The resulting category $L_{\infty}$3 has $L_{\infty}$4-Kuranishi spaces as objects and these equivalence classes as morphisms.

The main theorem asserts that $L_{\infty}$5 is a category naturally admitting the category of smooth manifolds $L_{\infty}$6 as a subcategory. Smooth manifolds are realized by charts with zero obstruction bundle, zero two-form, and trivial isotropy, so that the local algebra $L_{\infty}$7 reduces to the augmented de Rham complex, a mere chain complex; the embedding functor sends smooth maps to pre-morphisms whose $L_{\infty}$8-components are pullbacks of forms, and the compatibility diagrams commute strictly in this case. The author also notes a subcategory $L_{\infty}$9 of spaces without group actions, through which the functor factors.

Limitations and open questions

Several assumptions and deferrals should be noted. The theory requires the base of each chart to be contractible and the atlas to have uniformly bounded dimension; the stratification of the closed two-form $L_{\infty}$0 relies on a genericity result, so charts with non-generic $L_{\infty}$1 fall outside the framework as stated. The FOOO comparison theorem requires the two supplementary conditions on the embedding (complementary vanishing and the tangent bundle condition on all of $L_{\infty}$2), which the author justifies for moduli spaces of pseudoholomorphic maps but which are genuine restrictions. The higher cocycle condition on the $L_{\infty}$3-components of coordinate changes is not established here; it is deferred to [Kim3], where it is shown to hold after suitable choices of homotopy data. Finally, the anticipated homotopy-theoretic properties of $L_{\infty}$4 (e.g., whether it underlies an $L_{\infty}$5-category) are the subject of the forthcoming [Kim4], and the application to moduli spaces of pseudoholomorphic disks, under a stratification hypothesis on base manifolds, appears in [Kim1]. Whether the equivalence relation on atlases and pre-morphisms can be simplified, and whether the category embeds into known models of derived geometry, remain open.

Conclusion

The paper reformulates Kuranishi chart theory by attaching presymplectic foliation data and locally defined $L_{\infty}$6-algebras to charts, replacing the FOOO tangent bundle condition with a quasi-isomorphism condition that provably generalizes the former. This yields a category $L_{\infty}$7 of $L_{\infty}$8-Kuranishi spaces in which ambient expansions such as $L_{\infty}$9 are canonically identified, the cocycle condition simplifies to a statement about base maps, and the category of smooth manifolds embeds naturally. The construction rests on V-algebra techniques, Gotay's coisotropic embedding, and the homotopy theory of strict $L_{\infty}$0-algebras, and it provides the categorical foundation on which the author's subsequent work on higher cocycle conditions, homotopical properties, and applications to Floer-theoretic moduli spaces builds.