Abstract
Let X be a smooth, geometrically connected curve over a perfect field k. Given a connected, reductive group G, we prove that central extensions of G by the sheaf \({\mathbf {K}}_2\) on the big Zariski site of X, studied in Brylinski–Deligne [5], are equivalent to factorization line bundles on the Beilinson–Drinfeld affine Grassmannian \(\text{ Gr}_G\). Our result affirms a conjecture of Gaitsgory–Lysenko [13] and classifies factorization line bundles on \(\text{ Gr}_G\).
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Notes
Any central extension over \({\mathbf {F}}\) extends to one over \(X_1\) for some open \(X_1\subset X\). Any two such extensions to \(X_1\) become canonically isomorphic over some open \(X_2\subset X_1\).
As of now, even the definition of \(\varPhi _G\) appeals to the Brylinski–Deligne classification ( [12, §5.1]).
Aside from this descent technique, which was suggested to us by D. Gaitsgory, our paper lives entirely within classical (i.e., non-derived) algebraic geometry.
See [11, §2] for an introduction to lax prestacks.
i.e., the group of k-points of \(\mathscr {L}_xT\).
Indeed, for every \(\lambda \in \varLambda _T\), suppose \(z\leadsto z^{\kappa (\lambda ,\mu )}\) and \(z\leadsto z^{\kappa '(\lambda ,\mu )}\) define the same map \({\mathbb {G}}_m(k')\rightarrow {\mathbb {G}}_m(k')\) for all field extension \(k\subset k'\). By suitably choosing \(k'\), we can ensure that \((k')^{\times }\) contains an element of infinite order. Thus \(\kappa (\lambda ,\mu )\) agrees with \(\kappa '(\lambda ,\mu )\).
Caution: we do not yet know that \(q_1(\lambda )\) depends quadratically on \(\lambda \).
i.e., the \(k^{\times }\)-action on the two lines intertwines \(k^{\times }\rightarrow k^{\times }\), \(a\leadsto a^2\).
Recall: suppose \(X,Y\in \mathbf {Sch}_{/k}\) are connected schemes of finite type with base points, and X is integral, projective with \(\text{ H}^1(X,\mathscr {O}_X)=0\). Then \(\mathbf {Pic}^e(X)\times \mathbf {Pic}^e(Y)\xrightarrow {\sim }\mathbf {Pic}^e(X\times Y)\) (see [16, Exercise III.12.6]).
Recall that for an I-family of co-characters \(\lambda ^{(I)}=(\lambda _1,\cdots ,\lambda _{|I|})\), there is a closed immersion \(X^I\hookrightarrow \text{ Gr}_{T_1,X^I}\) whose image we call \(X^{(\lambda _1,\cdots ,\lambda _{|I|})}\).
Recall that k is assumed to be algebraically closed.
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Acknowledgements
We thank D. Gaitsgory for suggesting this problem to us, and for many insights that played a substantial role in its solution. We also benefited from discussions with Justin Campbell, Elden Elmanto, Quoc P. Ho, and Xinwen Zhu.
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Tao, J., Zhao, Y. Central extensions by \({\mathbf {K}}_2\) and factorization line bundles. Math. Ann. 381, 769–805 (2021). https://doi.org/10.1007/s00208-021-02154-1
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DOI: https://doi.org/10.1007/s00208-021-02154-1