Rational cohomology of Bun_{G,X} for general smooth projective X

Determine the rational cohomology of any connected component Bun_{G,X}^0 of the moduli stack Bun_{G,X} of principal G-bundles over a connected smooth complex projective variety X of complex dimension n, where G is a semisimple complex algebraic group and dim H^j(X,Q) = d_j for 0 ≤ j ≤ 2n. Specifically, prove that H^*(Bun_{G,X}^0,Q) is isomorphic to the tensor product of ∏_{j=0}^n (Sym V[2j])^{⊗ d_{2j}} and ∏_{j=1}^n (Λ V[2j−1])^{⊗ d_{2j−1}}, where V is the span of the positive-degree Chern classes of the universal principal G-bundle EG over the classifying space BG, and that this isomorphism is induced by the pullback map from BG. In particular, show that H^*(Bun_{G,X}^0,Q) is a direct sum of Hodge–Tate structures.

Background

Earlier in the paper, the authors determine the rational homotopy type of the mapping stack Map(X,BG) and, under the special condition that X has only even-dimensional cells and connected even skeleta, they establish an explicit description of the rational cohomology of connected components of Map(X,BG). This yields Corollary 4.3, which gives a formula for H^*(Bun_{G,X}^0,Q) as a tensor product of symmetric powers of shifted generators when X’s CW-structure has no odd-dimensional cells.

Conjecture 4.4 proposes the generalization beyond the special CW-structure assumptions. It asserts that for arbitrary connected smooth complex projective varieties X (with possibly nontrivial odd cohomology), the rational cohomology of Bun_{G,X}^0 factors as a tensor product involving both symmetric and exterior powers of shifted classes V[2j] and V[2j−1], where V is generated by the Chern classes of the universal principal G-bundle over BG. The conjecture also specifies that the isomorphism should come from the pullback map induced by the classifying space and predicts the Hodge–Tate nature of the resulting cohomology.