Date of Award

Spring 2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Goncharov, Alexander

Abstract

Denote by $B\Gamma$ the classifying space of an algebraic group $\Gamma(\mathbb{C})$. In this thesis we will take Milnor's realization of it -- the classifying simplicial space $B_{\Gamma\bullet}$. Consider the weight two motivic complex $ \mathbb{Z}_{\mathcal{M}}(2)$ for a regular algebraic variety $X$, placed in degrees $[1,4]$: $$ B_2(\mathbb{C}(X)) \to \bigwedge^2 \mathbb{C}(X)^* \to \bigoplus\limits_{D\in div(X)}\mathbb{C}(D)^* \to \bigoplus\limits_{D\in cod_2( X)}\mathbb{Z}.$$ Let $G$ be a split semisimple simply connected algebraic group over $\mathbb{Q}$. Then the degree 4 cohomology $H^4(B_{G\bullet}, \mathbb{Z}_{\mathcal{M}}(2))$ is known to be isomorphic to $\mathbb{Z}$. This can be deduced, for example, from the main result of Brylinski and Deligne \cite{BD}, although they did not consider the weight two motivic complex. The goal of this thesis is to explicitly construct a cocycle generating $H^4(B_{\Gamma\bullet}, \mathbb{Z}_{\mathcal{M}}(2))$. Such a cocycle was constructed in the special case $G = SL_n$ by Goncharov in \cite{ECCC}. In the case of general $G$ our construction uses the existence of a cluster $K_2$-variety $\mathcal{A}_{G,S}$ as well as a cluster-Poisson variety $\mathcal{P}_{G,S}$ -- the moduli spaces of local $G$-bundles on a surface $S$ with punctures and marked points on its boundary. The existence of such cluster structures was proven by Goncharov and Shen in \cite{QLS} for any split semi-simple algebraic group $G$. In the case of $G = SL_n$ these cluster varieties were discovered by Fock and Goncharov in \cite{MSLS}. The cluster $K_2$-variety structure on the moduli space of a triples of decorated flags (here surface $S$ is a disk with 3 special points on the boundary) provides us with an element $\omega \in \bigwedge^2 \mathbb{C}(G^3 / G)^*$, which gives rise to an invariant 2-form $\Omega$. Our main construction will extend this element to a nontrivial cocycle. As a part of our construction of an element of $H^4(BG, \mathbb{Z}_{\mathcal{M}}(2))$ we will present an algebraic geometric construction of a generator in the 3-dimensional cohomology for the group $G$$$H^3(G(\mathbb{C}),\mathbb{Z}(2)) \simeq H^4(BG,\mathbb{Z}(2))\text{, where } \mathbb{Z}(k) = (2\pi i)^k \mathbb{Z}.$$ The latter group is isomorphic to $\mathbb{Z}$, as $H^*(BG,\mathbb{C}) \simeq \mathbb{C}[\mathfrak{g}]^G$, and there is a unique up to a scalar degree 2 polynomial invariant under the $G$ action, coming from the Killing form on the root lattices. One of the main results of the thesis is that the element we constructed generates $H^3(G(\mathbb{C}), \mathbb{Z}(2))$. This implies that the constructed cocycle of $H^4(BG, \mathbb{Z}_{\mathcal{M}}(2))$ produces the second motivic Chern class for the universal $G$-bundle over the classifying space $BG$. This motivic class gives rise to the topological Chern class, using the Lie-exponential complex of sheaves, as explained in \cite{EC}. This thesis is based on joint work with Alexander Goncharov.

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