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: $$\mathring{\Eff}(S)=\Amp(S)\subset \SAmp(S)=\Mov(S) \subset \Nef(S)=\Eff(S)=\overline{\Amp(S)}.$$ 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 $$h^1(\CO_S(C))\leq c_S h^0(\CO_S(C))$$ for any curve $C \subset S$.