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Looijenga line bundles in complex analytic elliptic cohomology. (English) Zbl 1442.55005
In this paper, the author describes how complex analytic curves arise from the cohomology of some spaces parametrizing principal bundles on orientable genus $$1$$ surfaces.
The construction is as follows: fixed a genus $$1$$ orientable surface $$\Sigma$$ and a topological group $$G$$, one can consider the wreath product group $$\mathcal{W}(G):=\mathrm{Map}(\Sigma,G)\rtimes Diff(\Sigma)$$. Letting $$\mathcal{W}_0(G)$$ be the component of the identity and $$\overline{\mathcal{W}}(G)=\mathcal{W}(G)/\mathcal{W}_0(G)$$, one has that $$\overline{\mathcal{W}}(G)$$ acts naturally on $$B\mathcal{W}_0(G)$$. For the $$G$$ considered in the paper, the cohomology ring $$H^*(B\mathcal{W}_0(G);\mathbb{C})$$ is concentrated in even degree thus one obtains an action of $$(\overline{\mathcal{W}}(G)\times \mathbb{C})^{op}$$ on $$\mathrm{Spec}(H^*(B\mathcal{W}_0(G);\mathbb{C})$$.
Finally, one defines $$\mathcal{X}_G:=[\mathrm{Spec}(H^*(B\mathcal{W}_0(G);\mathbb{C})]_{\mathrm{an}}\setminus \{B_G\}$$, the analytification of the complex variety with $$B_G$$, a closed set of “bad” points removed.
It is shown that if $$\mathcal{X}_e$$ is the space of $$\mathbb{R}$$-linear bijections $$\mathbb{R}^2\to\mathbb{C}$$, then $$\mathcal{X}_G$$ can be identified with the preimage of $$\mathcal{X}_e$$ under the map induced by the terminal morphism $$\pi_G\to e$$.
The objects of interest in this paper are the (stacky) quotients $$\mathcal{M}_G:=\overline{\mathcal{W}}(G)\times \mathbb{C}\backslash\backslash \mathcal{X}_G$$.
In particular the author shows that
$\mathcal{M}_e\approx\mathcal{M}=\text{the moduli stack of (complex analytic) elliptic curves,}$ $\mathcal{M}_{U(1)}\approx\mathcal{E}=\text{the universal elliptic curve over $$\mathcal{M}$$,}$ $\mathcal{M}_{U(1)^d}\approx\mathcal{E}^d=\mathcal{E}\times_\mathcal{M}\cdots\times_\mathcal{M}\mathcal{E}=\text{the $$d$$-fold product of $$\mathcal{E}$$,}$ $\mathcal{M}_{K(\mathbb{Z},2)}\approx\mathbb{G}_m\times\mathcal{M}=\text{the multiplicative group as a trivial bundle of groups over $$\mathcal{M}$$,}$ $\mathcal{M}_{U(1)^d\times_{\phi}K(\mathbb{Z},2)}\approx\mathcal{P}_\phi=\text{the principal $$\mathbb{G}_m$$-bundle associated to $$\mathcal{L}_\phi$$,}$ where $$\mathcal{L}_\phi\to\mathcal{E}^d$$ is the Looijenga line bundle associated to $$\phi\in H^4(BU(1)^d,\mathbb{Z})$$ regarded as a quadratic function $$\phi:H_2(BU(1)^d)=\mathbb{Z}^d\to\mathbb{Z}$$.
The paper is organized as follows:
in Section 2 the author defines the $$\mathcal{M}_G$$ and shows the first three cases of the list above. In Section 3 the author proves the last two and shows that Looijenga-type line bundles emerge naturally in this formulation (Theorem 3.7). In Section 4 it is shown how finite coverings of genus-$$1$$ surfaces correspond to isogenies of elliptic curves. Section 5 is devoted to the derived picture: the author sketches how this setting might give rise to derived elliptic curves following [J. Lurie, Abel Symp. 4, 219–277 (2009; Zbl 1206.55007)] and elliptic cohomology theories of Grojnowski type. Sections 6 to 9 are devoted to the proof of Theorem 3.7 obtained from a more general and coordinate invariant formulation, Theorem 7.6. In the last section, Section 10, the author fixes all the conventions needed in the rest of the paper.

##### MSC:
 55N34 Elliptic cohomology 55N91 Equivariant homology and cohomology in algebraic topology 55R40 Homology of classifying spaces and characteristic classes in algebraic topology
##### Keywords:
elliptic cohomology; Looijenga line bundle
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##### References:
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