Gamma-BMT inequality and curve Brill–Noether number

Let $X$ be a smooth projective complex threefold and $H$ an ample Cartier divisor. For $i\ge 0$, ${\mathrm{CH}}_i(X{)}_{\mathbb{Q}}$ is the rational vector space of dimension-$i$ algebraic cycles modulo rational equivalence, and ${\mathrm{CH}}_i(X{)}_{\mathbb{R}}={\mathrm{CH}}_i(X{)}_{\mathbb{Q}}{\otimes }_{\mathbb{Q}}\mathbb{R}$. For this smooth threefold, ${\mathrm{CH}}^j(X{)}_{\mathbb{Q}}={\mathrm{CH}}_{3-j}(X{)}_{\mathbb{Q}}$; smoothness gives the Chow intersection product, and the degree map sends a zero-cycle to its degree. Let $K_0(X)$ be the Grothendieck group of coherent sheaves and ${\tau }_X:K_0(X)\to {\mathrm{CH}}_{\ast }(X{)}_{\mathbb{Q}}$ the source Grothendieck–Riemann–Roch transformation. Since $X$ is smooth, its Todd class $\mathrm{td}(X)$ is the Todd class of its tangent bundle, and the source Chern character is $\mathrm{ch}(E)=\mathrm{td}(X{)}^{-1}\cap {\tau }_X([E])$. For a bounded coherent complex, $[E]={\sum }_j(-1{)}^j[{\mathcal{H}}^j(E)]$ in $K_0(X)$. Let $E\in D^b(\mathrm{Coh}X)$, the bounded derived category of coherent sheaves. Write ${\mathrm{ch}}_i(E)$ for the codimension-$i$ component, set ${\mathrm{ch}}^{\beta }(E)=e^{-\beta H}\mathrm{ch}(E)$ for $\beta \in \mathbb{R}$, and denote its codimension-$i$ component by ${\mathrm{ch}}_i^{\beta }(E)$. Products with powers of $H$ are Chow intersection numbers followed by degree. For a real divisor class $D$, $\mathrm{\Gamma }\cdot D$ is the degree of its intersection with $\mathrm{\Gamma }\in {\mathrm{CH}}_1(X{)}_{\mathbb{R}}$; in particular $H^3>0$. Set ${\mathrm{\Delta }}_H(E)=({\mathrm{ch}}_1(E)\cdot H^2{)}^2-2{\mathrm{ch}}_0(E)H^3({\mathrm{ch}}_2(E)\cdot H)\mathrm{.}$

For a torsion-free coherent sheaf $F$, its $H$-slope is ${\mu }_H(F)={\mathrm{ch}}_1(F)\cdot H^2\mathrm{/}({\mathrm{ch}}_0(F)H^3)$; a nonzero torsion sheaf has slope $+\mathrm{\infty }$. Use the source Harder–Narasimhan convention to define ${\mathcal{T}}_{\beta }$ as sheaves all of whose slope factors have slope $>\beta$ and ${\mathcal{F}}_{\beta }$ as torsion-free sheaves all of whose slope factors have slope $\le \beta$. The source tilted heart is the extension-closed subcategory ${\mathrm{Coh}}^{\beta }(X)=⟨{\mathcal{F}}_{\beta }[1],{\mathcal{T}}_{\beta }{⟩}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ of $D^b(\mathrm{Coh}X)$. For $w\in \mathbb{R}$, define ${\nu }_{\beta ,w}(E)=\frac{{\mathrm{ch}}_2(E)\cdot H-w{\mathrm{ch}}_0(E)H^3}{{\mathrm{ch}}_1^{\beta }(E)\cdot H^2}$ when the denominator is positive, and $+\mathrm{\infty }$ when it is zero. A nonzero object of this heart is ${\nu }_{\beta ,w}$-semistable if every nonzero proper subobject in the heart has ${\nu }_{\beta ,w}$-slope at most its slope. The numerator uses the untwisted Chern characters. With $w=({\alpha }^2+{\beta }^2)\mathrm{/}2$, subtracting the constant $\beta$ from this slope gives the conventional twisted tilt slope $({\mathrm{ch}}_2^{\beta }(E)\cdot H-{\alpha }^2{\mathrm{ch}}_0(E)H^3\mathrm{/}2)\mathrm{/}({\mathrm{ch}}_1^{\beta }(E)\cdot H^2)$, so it has the same semistable objects. Only the source range of $\beta ,w$ in the inequality below is used.

For a real one-cycle class $\mathrm{\Gamma }\in {\mathrm{CH}}_1(X{)}_{\mathbb{R}}$ with $\mathrm{\Gamma }\cdot H\ge 0$, the source $\mathrm{\Gamma }$-BMT inequality is the assertion that for every $w>\frac{{\beta }^2}{2}+\frac{(\beta -\lfloor \beta \rfloor )(1-\beta +\lfloor \beta \rfloor )}{2}$ and every ${\nu }_{\beta ,w}$-semistable $E\in {\mathrm{Coh}}^{\beta }(X)$, $0\le Q_{\beta ,w}^{\mathrm{\Gamma }}(E):=(2w-{\beta }^2)({\mathrm{\Delta }}_H(E)+\frac{3\mathrm{\Gamma }\cdot H}{H^3}({\mathrm{ch}}_0^{\beta }(E)H^3{)}^2)+2({\mathrm{ch}}_2^{\beta }(E)\cdot H)(2{\mathrm{ch}}_2^{\beta }(E)\cdot H-3(\mathrm{\Gamma }\cdot H){\mathrm{ch}}_0^{\beta }(E))-6({\mathrm{ch}}_1^{\beta }(E)\cdot H^2)({\mathrm{ch}}_3^{\beta }(E)-\mathrm{\Gamma }\cdot {\mathrm{ch}}_1^{\beta }(E))\mathrm{.}$ This definition fixes the source normalization; it does not assert the inequality for any $\mathrm{\Gamma }$.

If $C$ is an integral projective curve, its arithmetic genus is $g=1-\chi ({\mathcal{O}}_C)$. For a coherent sheaf $E$ on $C$, torsion-free means that no nonzero subsheaf is supported on finitely many points. A positive-rank torsion-free coherent sheaf $E$ has rank equal to its generic rank, degree $\mathrm{deg}(E)=\chi (E)-\mathrm{rk}(E)\chi ({\mathcal{O}}_C)$, and slope $\mathrm{deg}(E)\mathrm{/}\mathrm{rk}(E)$ when its rank is positive. Such a torsion-free sheaf is slope-stable when every nonzero proper subsheaf of positive rank has strictly smaller slope. On a smooth curve, torsion-free coherent sheaves are vector bundles. The source Brill–Noether number is ${\mathrm{BN}}_C={\lim }_{t\downarrow 0}\sup \{\frac{h^0(E)}{\mathrm{rk}(E)}:E\text{ is a positive-rank slope-stable torsion-free sheaf on }C,\frac{\mathrm{deg}(E)}{\mathrm{rk}(E)}\in (g-1-t,g-1+t)\},$ with the limit and supremum convention of the source. In fact every displayed window is nonempty. The integral complex curve has a nonempty smooth locus; choose a smooth closed point $p$. The invertible sheaf ${\mathcal{O}}_C((g-1)p)$ has rank one and degree $g-1$: adding or subtracting a copy of the smooth Cartier point changes its Euler characteristic by one. A nonzero proper subsheaf of positive rank in this line has a nonzero finite-length quotient, so its degree is strictly smaller. The line is therefore slope-stable and belongs to every window with $t>0$. Shrinking the windows decreases their nonnegative suprema, so the displayed limit is their infimum in $[0,+\mathrm{\infty }]$; no empty supremum is needed. For a smooth divisor $S\in \mathrm{\mid }H\mathrm{\mid }$, $\mathrm{\mid }H{\mathrm{\mid }}_S\mathrm{\mid }$ denotes the complete linear system of the restricted line bundle ${\mathcal{O}}_S(H)$, and $\chi ({\mathcal{O}}_X(H))={\sum }_i(-1{)}^i{\dim }_{\mathbb{C}}{H}^i(X,{\mathcal{O}}_X(H))$.

Derived category of an abelian category