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Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with

  1. $H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$.

  2. $H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module for all $0\leq m\leq n$.

Let $K_{\mathrm{top}}(X)$ be the topological $K$-theory, the Mukai vector $$v\colon K_{\mathrm{top}}(X)\to \bigoplus H^*(X,\mathbb{Q}),\ \ E\mapsto \mathrm{ch}(E)\sqrt{\mathrm{td}(X)}$$ is an injection, and tensoring over $\mathbb{Q}$, we have $K_{\mathrm{top}}(X)_{\mathbb{Q}}\cong \bigoplus H^*(X,\mathbb{Q})$.

The lattice $K_{\mathrm{top}}(X)$ is equipped with the Euler pairing $\langle E,F\rangle=\chi(E^\vee\otimes F)$, it is not necesssarily symmetric.

  • Is it known whether the lattice $(K_{\mathrm{top}(X)},\langle-,-\rangle)$ is a perfect pairing?
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  • $\begingroup$ The vanishing conditions seem to make the Atiyah–Hirzebruch spectral sequence degenerates at $E_2$. Since you mentioned the Poincaré duality, it might also be interesting to compare this with the Atiyah duality (one could find a proof in a note by Rezk). In particular, (almost) complex manifolds admit a $\operatorname{Spin}^c$-structure, therefore are $\operatorname{KU}$-orientable. $\endgroup$
    – Z. M
    Commented Mar 5, 2022 at 23:46
  • $\begingroup$ see Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, around formula (12) $\endgroup$
    – user482036
    Commented May 23, 2022 at 7:19

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