We prove the Bloch conjecture : c 2 (E) ∈ H 4 D (X, Z(2)) is torsion for holomorphic rank two vector bundles E with an integrable connection over a complex projective variety X. We prove also the rationality of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds. We give a sharp higher-dimensional Milnor inequality for the volume regulator of all representations to P SO(1, n) of fundamental groups of compact n-dimensional hyperbolic manifolds, announced in our earlier paper.

We prove that if a pair of weights (u, v) satisfies a sharp A p -bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map L p (v) into L p (u) with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the A 2 conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weaktype inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele.2010 Mathematics Subject Classification. 42B20, 42B35, 47A30.

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