Matroids and the integral hodge conjecture for abelian varieties(待完善)

$(X,\Theta)$ 是一个 PPAV, $\phi_{\Theta}: X \cong \hat{X}$.

The minimial class for curves $= \frac{\Theta^{g-1}}{(g-1)!} \in H^{2g-2} (X,\mathbb{Z})$.

Fact: For very general $X$, the Hodge classes of curve type are generated by minimal class.

$X$ very general PPAV of $\dim =g \ge 4$, 那么任意余维数为 $g-1$ 的代数 cycle 有上同调类

$$
[z] = m \frac{[\Theta]^{g-1}}{(g-1)!}, m\ \text{even}
$$

$X$ stably rational $\iff$ $\exists r$ 使得 $X \times \mathbb{P}^r$ rational.

$Y$ rationaly connected $3$ fold, 那么 $Y$ admits a decomposition of diagonal of

The decomposition (*) is orthogonal with $\Theta_{C_t}$. 这等价于说

$$
(f_*^\vee H_1(X_t, \mathbb{R})) = \text{ker}(f_*: H_1(J(C_t), \mathbb{R}) \to H_1(X_t, \mathbb{R})).
$$