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.

$\bullet$ Is it known whether the lattice $(K_{\mathrm{top}(X)},\langle-,-\rangle)$ is a perfect pairing?

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?

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.

$\bullet$ Is it known whether the lattice $(K_{\mathrm{top}(X)},\langle-,-\rangle)$ is unimodular?

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.

$\bullet$ Is it known whether the lattice $(K_{\mathrm{top}(X)},\langle-,-\rangle)$ is a perfect pairing?