Forcing Axioms and the Definabilty of the Nonstationary Ideal on ${\omega }_1$

We show that under $\text{BMM}$ and "there exists a Woodin cardinal$\mathrm{"}$, the nonstationary ideal on ${\omega }_1$ can not be defined by a ${\mathrm{\Sigma }}_1$ formula with parameter $A\subset {\omega }_1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$-axiom. We further show that there are universes where $\text{BPFA}$ holds and $\text{NS}$ is ${\mathrm{\Sigma }}_1({\omega }_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\text{NS}$ is saturated, ${\mathrm{\Sigma }}_1({\omega }_1)$-definable and $\text{MA}$ holds.