19 problems

The maximal third Chern character conjecture. If $c=-1$ and $d=\frac12+\frac C5\ll0$, then

Let $\mathbb{F}_e$ be a Hirzebruch surface and let $H_m$ be an arbitrary polarization. Let $\mathbf{v}\in K(\mathbb{F}_e)$ be a character such that there are $H_{\lceil m\rceil}$-p…

Let $X$ be a smooth projective surface and let $H$ be an ample line bundle. Fix a rank $r>0$ and a first Chern class $c$, and let $M_{X,H}(r,c,\Delta)$ be the corresponding moduli…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process. Set … Assume $M\geq 0$. First-step cup-pro…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process, including the subspaces $E_r$, $E'_r$,…

Let $M_{d,\chi}$ be the moduli space under consideration, and let the tautological generators in $A^*(M_{d,\chi})$ be those defined in. A relation among these generators has degree…

Let $(r,c,d)\in\mathbb{Z}\oplus\mathbb{Z}\oplus\tfrac{1}{2}\mathbb{Z}$, and suppose that there are Gieseker-semistable sheaves $E$ on $\mathbb{P}^3$ with … Write $E(r,c,d)$ for the…

Let $G$ be a smooth algebraic group and let $X$ be a left $G$-variety over $\operatorname{Spec} K$. Let $\mathsf{Ner}(\operatorname{\mathbf{SfResl}}_G(X)^{\operatorname{op}})$ be t…

Let $X=D53F^1\timesD53F^1$ and let $M_{H_m}({\bf v})$ be the moduli space associated with a character ${\bf v}=(r,\nu,\Delta)\in K(X)$, where $r\geq 2$ and $\Delta\geq\frac{1}{2}$.…

Let $X$ be a projective complex surface with geometric genus $p_g=0$, and let $\mathcal A=\operatorname{Coh}(X)$. For an ample line bundle on $X$, let $\tau$ denote the resulting G…

Let $M_{(0,d,\chi)}$ be the moduli space of Gieseker semistable sheaves on $\mathbb P^2$ with Chern character $(0,d,\chi)$, and write its intersection Betti numbers as…

Let $M(r,c,d,e)$ denote the moduli space of Gieseker-semistable sheaves on ${\mathbb{P}}^3$ with Chern character $(r,c,d,e)$. Let $r \in {\mathbb{Z}}_{\geq 0}$,…

Let $\pi:\mathrm{M}(d,1)\to |\mathcal{O}_{\mathbb{P}^{2}}(d)|$ be the Fitting map, and let … Here $g$ denotes the dimension of a general fiber of $\pi$. Donaldson-number formula co…

Let $X$ be a smooth projective variety, and let $\tau\in B(X)_{\mathbb Q}$. Let $\sigma'$ and $\sigma”$ be bounded stability parameters for the fixed topological type $\tau$. Unifo…

Let $m$ be a positive integer. For $i=1$ or $2$, let $X_i$ be a projective K3 surface, let $v_i=(r_i,c_i,a_i)$ be a primitive positive Mukai vector, and let $H_i$ be a polarization…

Let $V^{\bullet}$ be the sheaf constructed in the paper from closed differential forms and related quotients. Semisimplicity conjecture. The sheaf $V^{\bullet}$ is semisimple. The…

Let $X$ be a smooth proper $k$-variety, and let $C_{k(X)}$ denote the projective-generator sheaf associated with the function field $k(X)$. For a smooth $k$-variety $Y$, write…

Let $k$ be the base field, and let ${\mathcal F}$ be a simple ${\mathbb A}^1$-invariant sheaf. Write $\Omega^{\bullet}_{k(Y)|k}$ for the graded differential forms of the function f…

Let ${\mathbb B}^{\bullet}$ be the fully faithful functor from pure $k$-motives to semisimple sheaves of finite length of finite-dimensional graded ${\mathbb Q}$-vector spaces, wit…