Some results on fake quadrics

In this paper, we give a criterion to assess the effectiveness and ampleness of divisors on a fake quadric surface $S$, and then we establish a relationship between the cones: $$\mathring{\text{Eff}}(S)=\text{Amp}(S)\subset \text{SAmp}(S)=\text{Mov}(S)\subset \text{Nef}(S)=\text{Eff}(S)=\overline{\text{Amp}(S)}\mathrm{.}$$ In particular, we prove that any fake quadric of odd type does not contain a negative curve. This result is central to our manuscript. As applications, first we give that any fake quadric is a fibration over ${\mathbb{P}}^1;$ Subsequently, we show that no fake quadric can be embedded in ${\mathbb{P}}^4$; Finally, we prove that the fake quadric $S$ possesses the bounded cohomology property. This property is characterized by the existence of a positive constant $c_S$ such that $$h^1({\text{CO}}_S(C))\le c_S{h}^0({\text{CO}}_S(C))$$ for any curve $C\subset S$.