5 problems

Let $k$ be the coefficient field, let $d_1,d_2,d_3$ be nonnegative integers, and write $x=(x_1,\ldots,x_{d_1})$, $y=(y_1,\ldots,y_{d_2})$, and $z=(z_1,\ldots,z_{d_3})$. For a tuple…

Let $(x,y,z)$ be the standard coordinates of the vector space $k^d=k^{d_1}\times k^{d_2}\times k^{d_3}$. Let $f\in k[x,y,z]$ satisfy $f(0,0,0)=0$ and … for every…

Let $Z'$ be a smooth scheme with trivial $\mathbb{G}_m$-action, and let $\mathbb{A}_{\operatorname{\mathbb{C}}}^n$ carry a $\mathbb{G}_m$-action with nonnegative weights. Let…

Let $(Q,W)$ be a symmetric quiver with potential, and suppose that $(Q,W)$ admits two cuts $I$ and $I'$ such that $I\cap I'=\emptyset$. Let $\Om_\al$ denote the Donaldson–Thomas in…

Absence of poles conjecture. For any $3d$ Calabi–Yau category with a stability condition and any strict sector $V$, the automorphism $x\mapsto A_{V,q}xA_{V,q}^{-1}$ preserves the s…