nLab complex analytic ∞-groupoid

under construction

Idea

A complex analytic $\infty$-groupoid is an ∞-groupoid equipped with geometric structure in the sense of complex analytic geometry, such that complex analytic spaces constitute a full subcategory of the 0-truncated complex analytic $\infty$-groupoids. Hence a complex analytic $\infty$-groupoid is an (∞,1)-sheaf/∞-stack on the site of complex manifolds (or some of its dense subsites). This is directly analogous to how (∞,1)-sheaves over the site of smooth manifolds may be regarded as smooth ∞-groupoids.

Definition

Definition

Write $CplxMfd$ for the category of complex manifolds, regarded as a site with the standard Grothendieck topology.

Write

$$\mathbb{C}Disc \hookrightarrow SteinSp \hookrightarrow CplxMfd$$

for the full subcategories of Stein spaces and of complex polydiscs, respectively, regarded as sites by equipping them with the induced coverages.

Proposition

The inclusions in def. exhibit dense subsite inclusions.

Definition

Write

$$\mathbb{C}Analytic\infty Grpd \coloneqq Sh_\infty(CplxMfd)$$

for the hypercomplete (∞,1)-sheaf (∞,1)-topos over the sites of complex manifolds def. .

Remark

By the discussion at model structure on simplicial presheaves this means that $\mathbb{C}Analytic\infty Grpd$ is equivalently the simplicial localization of any of the hypercomplete local model structures on simplicial (pre-)sheaves, such as the Joyal model structure on simplicial sheaves.

This is considered in (Hopkins-Quick 12, section 2.1).

Properties

Cohesion

Proposition

The global section geometric morphism

$$\Gamma \;\colon\; \mathbb{C}Analytic\infty Grpd \longrightarrow \infty Grpd$$

exhibits a cohesive (∞,1)-topos.

Proof

We discuss the existence of the extra left adjoint $\Pi$ (the shape modality). This proceeds essentially as in the discussion of the cohesion of Smooth∞Grpd (see there) only that where there one may choose good open covers here we have to choose genuine split hypercovers by polydiscs. The rest of the proof is verbatim as for Smooth∞Grpd.

To start with, since the hypercompletion depends only on the underlying sheaf topos of a site we may represent

$$\mathbb{C}Analytic\infty Grpd \simeq L_{lwhe} sPSh(\mathbb{C}Discs)_{proj, loc}$$

by the simplicial localization of the local model structure on simplicial presheaves over $\mathbb{C}Disc$, localized (by the theorem of descent recognition along hypercovers) at the hypercovers $U_\bullet\to X$ as seen over $CplxMfd$ restricted to $X$ a polydisc (this roundabout way since $\mathbb{C}Disc$ does not carry a Grothendieck topology but just a coverage).

Now before Bousfield localization we have a simplicially enriched Quillen adjunction

$$sPSh(\mathbb{C}Disc)_{proj, loc} \stackrel{\overset{\underset{\to}{\lim}}{\longrightarrow}}{\underset{\Delta}{\longleftarrow}} sSet_{Quillen}$$

which is simplicial-degree wise just the defining adjunction of the colimit functor $\underset{\to}{\lim}$ left adjoint to the constant presheaf functor.

To see that this descends to a Quillen adjunction on the local model structure, by the recognition theorem for simplicial Quillen adjunctions we hence need to check that $\Delta$ preserves fibrant objects with respect to the local model structure, hence that any constant simplicial presheaf $\Delta S$ for $S$ a Kan complex already satisfies descent with respect to hypercovers as seen over $CplxMfd$.

By the theorem of descent recognition along hypercovers $\Delta S$ satisfies descent precisely if for each hypercover $U_\bullet \to X$ of any polydisc $X$. the induced morphism of derived hom-spaces

$$RHom(X, \Delta S) \longrightarrow RHom(U_\bullet, \Delta S)$$

in the global model structure on simplicial presheaves is a weak equivalence. This $RHom$ in turn may be computed as the $sSet$-enriched hom object out of a cofibrant resolution $\hat X \to X$ and $\hat U_\bullet \to U_\bullet$, respectively. By the recognition of cofibrant objects in the projective model structure the representable $X$ is already cofibrant and a sufficient condition for a resolution $\hat U_\bullet \to U_\bullet$ to be cofibrant is that it is a split hypercover.

Since $\mathbb{C}Disc \hookrightarrow CplxMfd$ is a dense subsite, we may always choose such a split hypercover such that $\hat U_\bullet$ consists simplicial-degreewise of coproducts of polydiscs: by the proposition at hypercover – Existence of split refinements

Since the colimit of a representable functor is the point, this means that with such a choice $\underset{\to}{\lim} \hat U_\bullet$ is the simplicial set obtained by replacing in the hypercover by polydiscs each polydisc by a point. Forgetting the complex structure on all manifolds involved, one sees that this is precisely the “etale homotopy type” of $X$ as seen by the site CartSp of Cartesian spaces, by the discussion at Smooth∞Grpd, and this comes out as the ordinary homotopy type of the underlying topological space of $X$. But this is of course contractible.

In conclusion this means that

$$\begin{aligned} RHom(U_\bullet, \Delta S) &\simeq sPSh(\mathbb{C}Disc)(\hat U_\bullet, \Delta S) \\ & \simeq sSet(Sing(X), S) \\ & \simeq sSet(\ast, S) \\ &\simeq S \end{aligned}$$

Therefore the descent condition for $\Delta S$ is satisfied. (This is also the statement of (Hopkins-Quick 12, lemma 2.3, prop. 2.4, lemma 2.5, prop. 2.6)).

From here on the argument for the cohesion of $\mathbb{C}Analytic \infty Grpd$ proceeds as at ∞-cohesive site (which might just as well be adapted to the hyper-discussion here).

Corollary

For $X\in CplxMfd \hookrightarrow \mathbb{C}Analytic\infty Grpd$ a complex manifold, then the shape $\Pi(X)$ is the homotopy type of its underlying topological space.

In (Hopkins-Quick 12)) this is part of prop. 2.6.

Oka principle

Discussion of the Oka principle in terms of $\mathbb{C}Analytic\infty Grpd$ is in (Larusson 01).

Definition

Say that a complex manifold $X$ is an Oka manifold if for every Stein manifold $\Sigma$ the canonical inclusion

$$Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)$$

from the mapping space of holomorphic functions to that of continuous functions (both equipped with the compact-open topology) is a weak homotopy equivalence.

Theorem

This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies descent with respect to finite covers.

(Larusson 01, theorem 2.1)

Remark

By corollary , in terms of cohesion, prop. , definition should (…check…) read

$$\Pi[\Sigma,X] \simeq \flat [\Pi(\Sigma), \Pi(X)] \,.$$

Complex analytic differential generalized cohomology

By prop. $\mathbb{C}Analytic \infty Grpd$ is cohesive and hence by the discussion at differential cohomology hexagon the objects $\hat E \in Stab(\mathbb{C}Analytic \infty Grpd)$ (hence the sheaves of spectra on $\mathbb{C}Mfd$ ) qualify as differential cohomology refinements of the cohomology theories represented by the shapes $E\coloneqq \Pi \hat E \in$ Spectra.

Discussion of such complex analytic differential generalized cohomology is in (Hopkins-Quick 12, section 4),.

Examples

Multiplicative group and holomorphic line $n$-bundles

The multiplicative group is a canonical ∞-group object

$$\mathbb{G}_m \in Grp(\mathbb{C}Analytic\infty Grpd)$$

given as an (∞,1)-presheaf by the assignment

$$\mathbb{G}_m \;\colon\; \Sigma \mapsto \mathcal{O}_\Sigma^\times$$

that sends a Stein manifold to the multiplicative abelian group of non-vanishing holomorphic functions on it.

The delooping $\mathbf{B}\mathbb{G}_m$ is the universal moduli stack for holomorphic line bundles (the Picard stack) and the double delooping $\mathbf{B}^2 \mathbb{G}_m$ that for holomorphic line 2-bundles (the Brauer stack).

Related concepts

• smooth ∞-groupoid

• complex analytic stack

• holomorphic line n-bundle

References

• Finnur Lárusson, Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle (arXiv:math/0101103)

• Jacob Lurie, section 4.4. of Structured Spaces

• Jacob Lurie, sections 11 and 12 of Closed Immersions

• Michael Hopkins, Gereon Quick, Hodge filtered complex bordism, Journal of Topology, 8 1 (2015) (arXiv:1212.2173, doi:10.1112/jtopol/jtu021)