# A theorem about the symplectic geometry of projective bundles

I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta Mathematica, volume 3, pages 601–642(1997).

Theorem: Let $$E_i \rightarrow M_i$$ be Hermitian vector bundles, $$i = 1, 2$$. Let $$\mathbb{P}(E_i)$$ be the projectivization of $$E_i$$. Let $$f : \mathbb{P}( E_1) → \mathbb{P} (E_2)$$ be a fiber-like symplectomorphism, covering a map $$\varphi : M_1 \rightarrow M_2$$. Then $$\varphi^{*}(c_k(E_2)) = c_k(E_1)$$ for $$k \geq 2$$.

Here is where I get confused: taking a simple case where $$M_1=M_2=\,\mathbb{CP}^2$$. Suppose that $$E_{1} = \mathcal{O} \oplus \mathcal{O}$$. $$E_{2} = \mathcal{O}(1) \oplus \mathcal{O}(1)$$. Then, if I am not mistaken, there should be a fibre-wise symplectomorphism $$\mathbb{P}(E_1) \rightarrow \mathbb{P}(E_2)$$ since they are both isomorphic to the $$3$$-fold $$\mathbb{CP}^2 \times \mathbb{CP}^1$$. But there is no diffeomorphism of $$\mathbb{CP}^2$$ mapping $$c_{2}(\mathcal{O} \oplus \mathcal{O}) = 0$$ to $$c_{2}(\mathcal{O}(1) \oplus \mathcal{O}(1)) = H^2$$, where $$H$$ is the hyperplane class (it is not hard to see that a fibre preserving symplectomorphism must induce a diffeomorphism of the base).

I am thinking possibly that this theorem is for a specific choice of symplectic form?

I would be grateful if somebody could clear up my confusion. I am by no means questioning this theorem, I just want to understand the statement correctly. thanks.

• Read carefully the statement of the Theorem. The symplectomorphism is between $\mathbb{E}_1$ and $\mathbb{E}_2$, not $\mathbb{P}(\mathbb{E_1})$ and $\mathbb{P}(\mathbb{E_2})$.
– abx
Mar 9 at 4:50
• Sorry, I misquoted the theorem. The symplectomorphism in the paper is between the projective bundles. Edited. Mar 9 at 5:10

It seems that by "$$c_k(E_i)$$ for $$k\geq 2$$" Reznikov really means the characteristic classes of the $$PU(n)$$ principal bundle obtained by quotienting the $$U(n)$$ bundle underlying $$E_i$$ by its center $$U(1)$$. This is the algebra generated by classes $$\tilde{c}_k, 2\leq k \leq n$$ of degree $$2k$$ which are sort of projections of $$c_k$$ orthogonally to $$\mathbb Z[c_1]$$''.
More concretely, I think the theorem works if you replace $$c_k$$ by $$\tilde c_k$$, where $$1 + \sum_{k=2}^n \tilde c_k(E) t^k = \left [\mathrm{det}\left( \mathrm I_n + \frac{i t}{2\pi} \left( \Omega - \frac{1}{n} \mathrm{Trace}(\Omega) \mathrm{I_n}\right) \right)\right]~,$$ where $$\Omega$$ is the curvature form of some Hermitian connection.