Noether inequality for threefolds with Kodaira dimension two
The Noether inequality $Vol(X)\geq \frac{4}{3}p_g(X)-\frac{10}{3}
$$Ivol(X)\geq \frac{3}{4}p_g(X)-\frac{5}{4}.
This is based on a joint work in progress with Meng Chen, Chen Jiang and Yang Liu.