Complex Analytic Infinity-Prestacks

• Complex analytic infinity-prestacks are ∞-functors on the Stein ∞-site that extend classical complex geometry via higher stack theory and robust descent conditions.
• They incorporate analytic Grothendieck topologies, mapping stack equivalences, and GAGA-type results while integrating homotopical algebra and derived geometry.
• This framework enables concrete index computations and cocycle-level HRR identities, bridging algebraic and analytic approaches in modern geometry.

A complex analytic infinity-prestack is an $\infty$-functor from the category of Stein manifolds, equipped with an analytic Grothendieck topology, to the $\infty$-category of spaces or, in a concrete model, a simplicial presheaf on the Stein site with the local projective model structure. This framework generalizes classical complex-analytic geometry and higher stack theory to the $\infty$-categorical setting, supporting robust descent, mapping stack, and GAGA-type results, and integrating techniques from homotopical algebra, derived geometry, and index theory (Porta et al., 2014, Glass et al., 18 Jan 2026).

1. The Stein ∞-Site and Definition of Analytic ∞-Prestacks

Let $\mathrm{Stein}_\mathbb{C}$ denote the category of complex Stein spaces with analytic open coverings, regarded as an $\infty$-site by equipping its nerve with the Grothendieck topology an. An analytic $\infty$-prestack is a functor

$$F : \mathrm{Stein}_\mathbb{C}^{\mathrm{op}} \longrightarrow \mathcal{S}$$

valued in the $\infty$-category of spaces ($\mathcal{S}$). In the model-categorical approach, one may use the site $St$ of Stein coordinate charts, forming simplicial presheaves $F: St^{\mathrm{op}} \rightarrow \mathrm{sSet}$ endowed with the local projective model structure (Porta et al., 2014, Glass et al., 18 Jan 2026). Fibrant objects in the left Bousfield localization at hypercovers model analytic $\infty$-stacks.

A prestack $F$ satisfies descent (is an $\infty$-stack) if, for each analytic covering $\{U_i\to U\}$,

$$F(U) \longrightarrow \mathrm{holim}_\Delta\,F(U_\bullet)$$

is an equivalence, where $U_\bullet$ is the Čech nerve associated to the covering.

2. Sheaf Conditions, Lisse-Étale Sites, and Geometricity

For an analytic $\infty$-stack $X$ (i.e., an $n$-geometric sheaf on $\mathrm{Stein}_\mathbb{C}$ by analogy with HAG-II), the lisse-étale site $(X_{\lisse\et},\,\et)$ consists of pairs $(U, u)$, with $U\in\mathrm{Stein}_\mathbb{C}$ and a smooth $u:U\to X$. Coverings are families $\{(U_i,u\circ\varphi_i)\}$ where $\varphi_i:U_i\to U$ is an analytic open cover. Sheaves of spaces on $X_{\lisse\et}$ are functors satisfying (derived) descent for all such covers.

Geometricity for analytic stacks is axiomatized as follows: $F$ is $n$-geometric if the diagonal $F\to F\times F$ is $(n-1)$-representable, and there exists a smooth atlas $U=\coprod_i U_i\to F$ with each $U_i$ Stein and $(n-1)$-representable. Artin-type criteria (integrability of formal completions, effectivity of infinitesimal thickenings, Schlessinger conditions) further ensure finite-level geometricity (Porta et al., 2014).

3. Analytification Functor and Mapping Stacks

The analytification process passes from affine schemes to analytic geometry: $$(-)^{\mathrm{an}}: \mathrm{Aff}_{/\mathbb{C}}^{\mathrm{op}} \longrightarrow \mathrm{Stein}_\mathbb{C}^{\mathrm{op}},\qquad \mathrm{Spec}\,A \mapsto \mathrm{Sp}(A)_{\mathrm{an}}$$ and extends by left Kan-extension to prestacks. For $\infty$-stacks,

$(-)^{\mathrm{an}}: \mathrm{St}_{\et}(\mathrm{Aff}_{/\mathbb{C}}) \to \mathrm{St}_{\an}(\mathrm{Stein}_\mathbb{C})$

is defined.

A crucial property is the equivalence for mapping stacks: given algebraic stacks $X$ and $Y$,

$$\mathrm{Map}(X,Y)^{\mathrm{an}} \simeq \mathrm{Map}(X^{\mathrm{an}}, Y^{\mathrm{an}})$$

so analytic mapping stacks realize the correct geometric $n$-categorical behavior. Classifying stacks $$BG: U \mapsto \{\text{principal }G\text{-bundles on }U\}$$ are $1$-geometric analytic $\infty$-stacks for any complex Lie or analytic group $G$ (Porta et al., 2014).

4. Coherence, GAGA Theorems, and Direct Images

For an analytic $\infty$-stack $X$, one defines the derived $\infty$-category of coherent sheaves $Coh^+(X)\subset D(\mathcal{O}_X)$ by checking coherence on a smooth atlas. The $\infty$-version of Grauert's theorem states: if $f:X\to Y$ is a proper morphism of analytic $\infty$-stacks, then

$f_*: Coh^+(X)\longrightarrow Coh^+(Y)$

preserves coherence. The proof employs descent over a smooth surjection $p:P\to X$ (with $P$ a representable analytic space proper over $Y$), spectral sequences, and the nerve $P^\bullet\to X$ (Porta et al., 2014).

A Serre GAGA-type result holds: if $X$ is proper over $\mathbb{C}$ or a $k$-affinoid base, analytification induces an equivalence

$Coh(X) \simeq Coh(X^{\mathrm{an}})$

for 1-categories of coherent sheaves, confirming the extension of underived GAGA results to higher analytic stacks. For proper morphisms of algebraic $\infty$-stacks $f:X\to Y$, analytification intertwines derived direct images: $$(Rf_*F)^{\mathrm{an}} \xrightarrow{\sim} Rf^{\mathrm{an}}_*(F^{\mathrm{an}})$$ as equivalences in $Coh^+(Y^{\mathrm{an}})$ (Porta et al., 2014).

5. Cech-Cocycle and Index Theorems: Hirzebruch-Riemann-Roch Perspective

Recent developments provide a cocycle-level Hirzebruch–Riemann–Roch (HRR) identity for complex analytic $\infty$-prestacks (Glass et al., 18 Jan 2026). The model uses simplicial presheaves on the Stein chart site, with mapping spaces as derived Hom-spaces. Basic objects include the representable presheaf $yV(W)=\mathrm{Hom}_{St}(W,V)$, the chart prestack $C$, the quotient prestack $[M/G]$, and prestacks of differential forms.

Prestacks of holomorphic $k$-forms $\Omega^k$ and off-diagonal forms $E$ are constructed, and related via the Hartogs extension

$\mathrm{Hart}: E \to \Omega^n$

and the Bochner-Martinelli Čech parametrix

$\mathrm{BM}: C \to E$

mapping charts to universal forms. Invariant polynomials $T: (gl(n,\mathbb{C}))^{\otimes k} \to \mathbb{C}$ yield Chern–Weil forms, concretely realized in the Todd class and parametrized by

$C\,\mathrm{Todd}_k: C \to \Omega^k.$

The fundamental HRR identity is the equality of two natural transformations in the $\infty$-category of simplicial presheaves: $$\mathrm{Hart} \circ \mathrm{BM} = C\,\mathrm{Todd}_n: C \to \Omega^n.$$ On mapping spaces,

$$\mathbb{R}\mathrm{Hom}(F,C) \rightrightarrows \mathbb{R}\mathrm{Hom}(F,\Omega^n)$$

the cocycle-level push-forwards and Chern–Todd maps coincide, enforcing the HRR formula in Čech–Dolbeault cohomology (Glass et al., 18 Jan 2026).

6. Representative Examples and Computational Models

• Mapping Stacks: For analytic $\infty$-stacks $Z,X$, the mapping stack $U \mapsto \mathrm{Map}(U\times Z, X)$ is analytic under reasonable finiteness assumptions.
• Classifying Stacks: For a complex Lie group $G$, the classifying stack $BG$ is $1$-geometric.
• Equivariant and Orbifold Models: For $F=[M/G]$ with $M$ a complex manifold and $G$ a discrete group of biholomorphisms, Čech-cocycles and group-cohomology-valued cocycles capture equivariant index formulas.
• Local/Orbifold Todd Class: For $V\subset\mathbb{C}^n$ and $G\leq\mathrm{Aut}(V)$ discrete, $F=[V/G]$ yields the analytic torsion or orbifold Todd class $\tau_G\in Z^n(G,\Omega^n(V))$, via both Bochner–Martinelli and Chern–Weil procedures (Glass et al., 18 Jan 2026).

7. Connections, Theoretical Significance, and Extensions

The theory of complex analytic $\infty$-prestacks unifies and generalizes classical analyses of sheaves, cohomology, and stack-theoretic descent in the context of complex geometry. It establishes a direct correspondence between algebraic and analytic contexts via the analytification functor and GAGA theorems, provides a robust geometric criterion for stack representability and geometricity, and enables cocycle-refined index theorems, including the HRR identities for higher analytic stacks.

Fundamental results by Porta and Yu (Porta et al., 2014) form the analytic side parallel to the foundations of higher topos theory and derived algebraic geometry (Lurie, HTT/HA). Extensions to push-forwards, direct images, and the comparison of analytic and algebraic categories are central to ongoing research, as are explicit cocycle formulas, equivariant and orbifold extensions, and deeper explorations in non-archimedean analytic settings.

References:

• M. Porta, T. Yu, "Higher analytic stacks and GAGA theorems" (Porta et al., 2014)
• J. Glass, T. Tradler, M. Zeinalian, "Hirzebruch-Riemann-Roch for complex analytic infinity-prestacks" (Glass et al., 18 Jan 2026)