Stability conditions on smooth projective complex varieties

If $X$ is empty, its zero derived category has the zero slicing and zero charge lattice, satisfying every assertion vacuously. Otherwise put $r=n+1$. Starting with the canonical surjection ${\mathcal{O}}_S\to i_{\ast }{\mathcal{O}}_X$, repeatedly cover its coherent kernel and subsequent kernels by finite direct sums of sufficiently negative twisting lines, using F4. There are finite such covers: from a globally generated coherent twist, take finitely many generating sections, using quasi-compactness of $S$ and finite generation on each of finitely many affine charts. After $r-1$ covers this gives an exact sequence $0\to K\to {\mathcal{O}}_S(-a_{r-1}{)}^{m_{r-1}}\to \cdots \to {\mathcal{O}}_S(-a_1{)}^{m_1}\to {\mathcal{O}}_S\to i_{\ast }{\mathcal{O}}_X\to 0,$ with $a_j\ge 0$ and coherent $K$. This is a finite partial resolution; no unproved assertion that its last kernel is zero or locally free is needed. Its successive kernel triangles give a finite derived filtration with factors ${\mathcal{O}}_S$, ${\mathcal{O}}_S(-a_j)[j]$ for $1\le j\le r-1$, and $K[r]$.

Check the geometric hypotheses of F3 for $F=i_{\ast }$ and $L=\mathbf{L}{i}^{\ast }$. A closed immersion is affine; restriction of scalars on affine charts is exact, preserves coproducts, and is conservative. Coherent cohomology and boundedness of a quasi-coherent complex are equivalent before and after $i_{\ast }$, because a module over $A\mathrm{/}I$ is finite over $A\mathrm{/}I$ exactly when it is finite over $A$. Thus bounded-coherent reflection holds. The algebraic adjunction $(A\mathrm{/}I){\otimes }_A^{\mathbf{L}}-⊣\mathrm{R}\mathrm{e}\mathrm{s}$ glues, using finite complexes, to $\mathbf{L}{i}^{\ast }⊣i_{\ast }$. On $S$, every coherent complex is perfect by F4, so its derived pullback is bounded coherent. The projection formula $i_{\ast }\mathbf{L}{i}^{\ast }E=i_{\ast }{\mathcal{O}}_X{\otimes }^{\mathbf{L}}E$ follows from the same tensor identity on affine charts.

Choose an integer $N\ge 1$ such that $a_j\le jN$ for every $j$, and choose $\sigma$ using F1. Transitivity and invariance of the order under tensor and shifts give $\sigma \otimes \mathcal{O}(a_j)⪯\sigma \otimes \mathcal{O}(jN)\prec \sigma [j]\mathrm{.}$ The first inequality follows by repeated Bayer comparison; the second follows by iterating the restriction comparison $j$ times and tensoring at each step. For $E\in \mathcal{P}(\theta )$, the dual form of this order says ${\varphi }_{\sigma \otimes \mathcal{O}(a_j)}^{-}(E)\ge \theta -j$, hence ${\varphi }_{\sigma }^{-}(E\otimes \mathcal{O}(-a_j)[j])\ge \theta$. To verify that dual form directly, if a bottom HN phase $b<\theta -j$ existed in the left slicing, its bottom truncation would receive a nonzero map from $E$ but would have all $\sigma$-phases at most $b+j<\theta$, contradicting phase-Hom vanishing. This also proves the weak endpoint used here; the available strict comparison is stronger.

The remaining compactness requirement holds on both smooth projective schemes. F4 gives local regular rings of finite dimension and hence finite projective dimension for finite modules. Iterated negative-twist surjections, whose kernels are coherent, then give global finite vector-bundle resolutions as in the proof of F2: after $\dim X$ syzygies the kernel is locally free, and finite-presentation basis lifting makes the local freeness hold on an open neighborhood. Bounded coherent complexes lie in the triangulated closure of these bundles by standard cohomology truncations. For a vector bundle $B$, $\mathrm{Hom}(B,-)$ in the unbounded quasi-coherent derived category is computed by the finite affine Čech complex for $B^{\vee }\otimes -$; a finite affine cover has affine intersections because the scheme is separated. Each affine section functor on quasi-coherent modules commutes with direct sums, as does the finite Čech total complex, and direct sums of modules are exact. Thus vector bundles, and then their finite cones and shifts, are compact. This uses the honest unbounded quasi-coherent model required in F3, not unrestricted Hom classes in an arbitrary unbounded Verdier quotient.

Take $E\in \mathcal{P}(\theta )$ with $0<\theta \le 1$ and derived-tensor the finite filtration in step 1.1 with $E$. The factors are $E$, the $E\otimes \mathcal{O}(-a_j)[j]$, and $E{\otimes }^{\mathbf{L}}K[r]$. The first factor has phase $\theta$, each middle factor has lower HN phase at least $\theta$ by step 2.1, and the last has every phase strictly greater than $1\ge \theta$ by F2 and $r=n+1$. Zero factors may be omitted. Closure under extensions gives $i_{\ast }{\mathcal{O}}_X{\otimes }^{\mathbf{L}}E\in \mathcal{P}([\theta ,\mathrm{\infty }))\mathrm{.}$ Every real phase differs by an integer from one in $(0,1]$, and tensor commutes with shifts, so this inclusion holds for every real $\theta$. Its direction is $\ge \theta$, not $\le \theta$.

Steps 3.1 and 1.2 identify $FL\mathcal{P}(\theta )\subseteq \mathcal{P}([\theta ,\mathrm{\infty }))$, and steps 1.2–2.2 check every other hypothesis of F3. Apply that local supplier. It constructs exactly $\mathcal{Q}(\theta )=\{A:i_{\ast }A\in \mathcal{P}(\theta )\}$, finite HN filtrations, phase-Hom vanishing, and local finiteness in the induced strict exact structure. The charge factors through ${\mathrm{\Lambda }}_X=\mathrm{im}(v{i}_{\ast })$, a subgroup of the finite free lattice ${\mathrm{\Lambda }}_{\sigma }$, hence a finite-rank free group. Restrict the norm from ${\mathrm{\Lambda }}_{\sigma }\otimes \mathbb{R}$. If $A$ is nonzero $\mathcal{Q}$-semistable, $i_{\ast }A$ is nonzero $\mathcal{P}$-semistable, so $\mathrm{\parallel }v(i_{\ast }A)\mathrm{\parallel }\le C_{\sigma }\mathrm{\mid }Z(v(i_{\ast }A))\mathrm{\mid }\mathrm{.}$ This is precisely finite-rank support for $Z_X$; nonzero phase masses and local finiteness are inherited by F3, rather than assumed from the existence of a stability function. Thus the induced condition meets the current library definition in full.