Ordinary zero-denominator BG positivity gives tilt HN, duality and the second heart

Use precisely the ordinary sheaf inequality (BG0) assumed in the Statement: every slope-semistable torsion-free sheaf F with μ(F)=β obeys

Hch2(F)≤(β²/2)v rk(F).

The same condition is used at −β. The classical discriminant Bogomolov–Gieseker theorem implies (BG0); its full proof is an independent general geometry prerequisite, not supplied by duality or noetherianity. No source citation is being used as its local proof. At β=0 this is exactly the zero-slope bound already needed by the delivered surface-to-threefold reduction.

For $w>{\beta }^2\mathrm{/}2$ set $A_{\beta ,w}(E)=Hch2(E)-wv$ rk(E). On $C_{\beta }$, (BG0) and the first-heart construction give $A\ge 0$, with equality only for ordinary zero-dimensional sheaves: the shifted boundary-slope sheaf contributes at least $(w-{\beta }^2\mathrm{/}2)vrk>0$, and a nonzero one-dimensional coherent sheaf contributes its positive effective-cycle degree. Thus $Z_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{t}}=-A+i{d}_{\beta }$ takes the closed upper half-plane, with precisely the zero-dimensional sheaves as its zero objects. Source tilt slope is $\nu =A\mathrm{/}{d}_{\beta }$ with zero denominator assigned infinity.

For rational β, the first heart is noetherian by the first-heart construction and d_β is discrete. These facts prove weak tilt HN existence directly. First take the largest C_β-subobject by noetherian finite sums; all its quotients lie in the Serre C_β, so it is slope-infinity semistable. Its quotient belongs to the finite part. On a finite-part object, a descending chain of subobjects with strictly increasing phases cannot be infinite: the discrete d_β eventually agrees, while its zero-denominator quotients have negative real part or zero charge, making the subobject phase decrease or stay equal. Replacing an unstable object by a larger-phase subobject therefore reaches a semistable subobject. Let a be the supremum of phases of semistable subobjects of E. For θ<a take their largest finite sum T_θ over phases>θ; every quotient of this sum has phase>θ, by semistability of the summands, and phase(T_θ)≤a by the preceding replacement argument. Along θ_n↑a, these sums decrease and their d_β eventually agrees, so their phases are nonincreasing and at most a but greater than θ_n. They must attain a, and then are semistable. The largest sum of phase-a semistable subobjects has a quotient with strictly smaller maximal phase: otherwise a higher-phase preimage gives a higher semistable subobject, or an equal-phase preimage enlarges this largest sum. Iterating terminates by noetherianity, giving the finite ordered HN filtration. Zero-charge quotients do not invalidate this argument: they change neither numerical phase of a positive-denominator object nor the conclusion that an equal-phase preimage enlarges the maximal semistable subobject. This proves HN rather than assuming it from BMT.

Under (BG0), β rational and $w>{\beta }^2\mathrm{/}2$, a nonzero finite-part E is ${\nu }_{\beta ,w}$-semistable if and only if $E^d$ is ${\nu }_{-\beta ,w}$-semistable. Here ${\nu }_{-\beta ,w}(E^d)=-{\nu }_{\beta ,w}(E)$, from the first three coordinate identities; no parameter has been switched to a twisted numerator.

If E is unstable, take a finite-slope last HN quotient B with $\nu (B)<\nu (E)$. It lies in the finite part, and its kernel A does too. the saturation construction gives an injection $B^d\to E^d$ with slope $-\nu (B)>-\nu (E)$, so $E^d$ is unstable. Conversely the same argument applied to $E^d$ makes $(E^d{)}^d=E^{\mathrm{s}\mathrm{a}\mathrm{t}}$ unstable whenever $E^d$ is unstable. E and $E^{\mathrm{s}\mathrm{a}\mathrm{t}}$ have the same finite-part semistability: a destabilizing subobject of E also embeds into $E^{\mathrm{s}\mathrm{a}\mathrm{t}}$; a destabilizing subobject $U\subset E^{\mathrm{s}\mathrm{a}\mathrm{t}}$ intersects E in a subobject with the same first three numerical coordinates, since U/(U∩E) is zero-dimensional. This intersection is nonzero, because $E^{\mathrm{s}\mathrm{a}\mathrm{t}}$ has no zero-dimensional subobject. It therefore destabilizes E. This proves the equivalence. The result is semistability reversal; an assertion of strict stability without accounting for numerically zero quotients is deliberately not substituted.

Keep β rational, and choose α² positive rational. Put w=(α²+β²)/2 and use the conventional slope λ=ν_(β,w)−β. Let (T',F') be the torsion pair from its finite HN filtration, cut at zero, and form the bounded second heart A=<F'[1],T'> by the direct torsion-pair truncation argument of the first-heart construction. Define the discrete additive nonnegative quantity

f(E)=Hch2^β(E)−(α²/2)v rk(E)

on A. In the first heart $f=d_{\beta }\lambda$ for positive denominator; on the zero-denominator part it equals $A_{\beta ,w}$, strictly positive unless zero-dimensional. It follows that every $Q\in A$ with $f(Q)=0$ has $H_B^{-1}(Q)$ either zero or λ-semistable of slope zero, and $H_B^0(Q)$ zero-dimensional. Conversely these conditions give $f=0$. This proves the needed zero-imaginary classification.

Consider $A_1\subset A_2\subset \dots \subset E$ in A. Discreteness makes $f(A_i)$ eventually constant, so $Q_i=A_{i+1}\mathrm{/}{A}_i$ have $f=0$. Write $F_i=H_B^{-1}(A_i),T_i=H_B^0(A_i)$. The $F_i$ form an increasing chain in the fixed $F_E$ in the noetherian first heart, so stabilize. For large i the B-cohomology sequence gives an injection $G_i=H_B^{-1}(Q_i)\to T_i$ and an exact sequence from $T_i$ to $T_{i+1}$ with zero-dimensional cokernel. $G_i$ has $\lambda =0$, hence $d_{\beta }(G_i)>0$ unless zero. Consequently $d_{\beta }(T_{i+1})=d_{\beta }(T_i)-d_{\beta }(G_i)$. These nonnegative discrete values cannot decrease forever, so $G_i=0$ eventually and $Q_i$ themselves are zero-dimensional.

Put $E_i=E\mathrm{/}{A}_i$ in $A,U_i=H_B^{-1}(E_i),V_i=H_B^0(E_i)$. The same sequence gives increasing $U_i\to U_{i+1}$ with zero-dimensional cokernel. Each $U_i\in F^{\mathrm{\prime }}$ lies in $B_{\beta }^{\mathrm{f}\mathrm{i}\mathrm{n}}$: it has no slope-infinity subobject. Fix one index i0. By the saturation construction all later $U_i$ embed over $U_{i0}$ into its finite saturation $U_{i0}^{\mathrm{s}\mathrm{a}\mathrm{t}}$. Their increasing chain terminates in the noetherian first heart. The $V_i$ are successive quotients of a fixed first-heart object, so stabilize too. The cohomology sequence then forces $Q_i=0$, proving that A is noetherian. This constructs the second-heart theorem from the actual finite hull, avoiding the external polynomial-heart finite-length theorem. The only numerical premise was the explicitly stated ordinary (BG0) needed for the weak slope and HN construction.