Exact semistable transport along numerical wall lines across slope hearts

For an ordinary μ-semistable torsion-free sheaf G of slope μ, ordinary BG gives

L(G) ≤ C₀(G) [ μ²/2−ℓμ−c ].

The bracket is strictly negative for μ∈(b₋,b₊). The same strict inequality holds for any nonzero torsion-free sheaf all of whose ordinary HN slopes lie in a closed subinterval of (b₋,b₊), by addition over its finite HN filtration.

Write $F=H^{-1}(E),T=H^0(E)$. If F≠0, its highest ordinary HN subsheaf F₁ has slope μ₁≤b. If ${\mu }_1>b_{-}$, then L(F₁)<0 and L(F₁[1])>0. But F₁[1]⊂F[1]⊂E in $B_b$. Its denominator is nonnegative; if positive, $L>0$ says it has slope greater than ℓ; if zero, its assigned slope is infinity. Both contradict finite-slope semistability. Consequently μ_max(F)≤b₋.

If the torsion-free quotient of T is nonzero, its lowest HN quotient $T_{last}$ has slope ${\mu }_{last}>b$. It is a quotient of E in $B_b$. If ${\mu }_{last}<b_{+}$ then $L(T_{last})<0$, so its finite slope is less than ℓ, contradicting semistability of E. Thus μ_min(T/tors T)≥b₊. Torsion has slope infinity and causes no change to this membership conclusion. Therefore $F\in F_u$ and $T\in T_u$ for all u∈(b₋,b₊). The same E belongs to every $B_u$, and L(E)=0 gives its slope ℓ as long as its denominator is positive.

The denominator cannot vanish inside this interval. If C₀(E)=0 it is its fixed positive value. If C₀(E)≠0 and it vanished at u, then L(E)=0 would give C₂(E)=ℓC₁(E)+cC₀(E) and hence Δ(E)=C₀(E)²[u²−2ℓu−2c]<0. At the original rational point the discriminant theorem forbids this. Alternatively, without requiring the original point rational, the just-proved heart-membership forces C₁(E)−uC₀(E)≥0 on the entire open interval; a nonconstant affine function cannot have an interior zero while remaining nonnegative on both sides. This second argument proves positivity without any use of tilt discriminant or rationality.

Fix $b_{-}<u_0<u_1<b_{+}$. In $B_{u_0}$, define G to be the ordinary torsion-free sheaves whose HN slopes all belong to (u₀,u₁]. Define T to be the objects A with ordinary $H^{-1}(A)\in F_{u_0}$ and $H^0(A)\in T_{u_1}$. Then (T,G) is a torsion pair in $B_{u_0}$, and

$B_{u_1}=⟨G[1],T⟩$.

Here is the construction, rather than an imported assertion about hearts. Hom(T,G)=0 follows from $Hom(T_{u_1},G)=0$ by the sheaf slope HN inequality, and $Hom(F_{u_0}[1],G)=0$ by negative Ext degree. For $A\in B_{u_0}$, take the ordinary HN quotient $H^0(A)\to G_A$ consisting of slopes in (u₀,u₁], after its torsion and $\text{slopes}>u_1$ have been removed. Its composition $A\to H^0(A)\to G_A$ is an epimorphism in $B_{u_0}$; its kernel $T_A$ has $H^0(T_A)\in T_{u_1}$ and $H^{-1}(T_A)=H^{-1}(A)$. This gives $0\to T_A\to A\to G_A\to 0$. Conversely the elementary torsion-pair truncation used in the first-heart construction says that the tilted heart has old cohomology H⁻¹ in G and H⁰ in T. Combining this description with ordinary cohomology gives exactly $H^{-1}\in F_{u_1},H^0\in T_{u_1}$, hence $B_{u_1}$. One may check the last equality directly by splitting the ordinary HN filtration of H⁻¹ at u₀: its part of slopes≤u₀ shifted by one lies in T, and its remaining part lies in G[1].

Every nonzero G∈G satisfies L(G)<0 by the argument beginning at step 1.1. Also E belongs to T by its already proved cohomology bounds. Both facts are essential; no assertion that arbitrary objects survive a heart change is being made.

Assume E is semistable in $B_{u_0}$, with L(E)=0, and let A⊂E be a subobject in $B_{u_1}$. Taking $B_{u_0}$-cohomology of 0→A→E→Q→0 gives H⁻¹_old(A)=0, so A∈T, and an exact sequence

0→K→A→E→H⁰_old(Q)→0

in $B_{u_0}$, where $K=H_{old}^{-1}(Q)\in G$. Let I be the old-heart image A→E. Then 0→K→A→I→0 is exact in $B_{u_0}$, and old semistability gives L(I)≤0 whenever I≠0; this also rules out a nonzero old null subobject of E. Consequently L(A)=L(K)+L(I)≤0. For positive new denominator this is precisely the required new slope inequality.

If A had zero new denominator, weak-charge positivity would imply L(A)≥0. Equality above would force $K=0$ (strict negativity on nonzero G), and then $A=I$ is a subobject of E in both hearts. A new-null object has ordinary H⁻¹ of slope u₁ and H⁰ of dimension≤1. Since A∈T has H⁻¹ of $\text{slopes}\le u_0<u_1$, its H⁻¹ must vanish. Thus A is a coherent sheaf of dimension≤1. It is old-null too and cannot be a nonzero subobject of the old finite-slope semistable E. Therefore no nonzero new null subobject exists. This proves semistability in $B_{u_1}$.

Weak-charge positivity used here is elementary at a real parameter: an object of zero denominator consists of a slope-u₁ semistable sheaf shifted by one and a sheaf supported in dimension≤1; ordinary BG and $w(u_1)>u_1^2\mathrm{/}2$ give nonnegative numerator. It does not require real-parameter HN for the tilt slope.

Assume instead that E is semistable in $B_{u_1}$. Let I⊂E be a subobject in $B_{u_0}$, and use its torsion-pair sequence $0\to T_I\to I\to G_I\to 0$ from the argument beginning at step 7.1. Since T is a torsion class and E∈T, its old quotient $E\mathrm{/}{T}_I$ also belongs to T. Thus $0\to T_I\to E\to E\mathrm{/}{T}_I\to 0$ is exact in both hearts. New semistability gives $L(T_I)\le 0$, while $L(G_I)<0$ if $G_I\mathrm{\ne }0$. Hence $L(I)=L(T_I)+L(G_I)\le 0$. All positive-old-denominator subobjects obey the required inequality.

A nonzero old-null subobject I would have L(I)≥0. Equality forces $G_I=0$, so I∈T and is a subobject in both hearts. If its old H⁻¹ is nonzero of slope u₀, then its new denominator is (u₁−u₀)C₀(H⁻¹)>0, and ordinary BG gives L(I)>0 because the shifted sheaf contributes strictly positive L while a one-dimensional H⁰ contributes nonnegative L; this contradicts new semistability. If H⁻¹ vanishes, I is a coherent sheaf of dimension≤1, hence also new-null, again impossible. Thus old-null subobjects are ruled out. E is semistable in $B_{u_0}$.

Start with a same-slope semistable short exact sequence 0→E₁→E→E₂→0 at one point on the line, with all three denominators positive. the argument beginning at step 1.1 places each $E_i$ in all hearts along the open line segment. The fixed derived triangle therefore remains a short exact sequence in each heart. the arguments beginning at steps 11.1 and 16.1 make all three objects semistable of slope ℓ there. In particular the original map remains the same subobject or quotient throughout the segment. No chamber or local-finiteness theorem is needed for this persistence assertion.

This component does not alone assert the existence of a first wall along every arbitrary two-parameter path. The preceding rational fixed-β construction supplies the finite-class argument for vertical paths. Together these are enough to move an existing wall sequence to any other β on that same line in the effective-restriction argument. Large-volume existence of semistability for a given ordinary slope-stable sheaf and the quantitative safe-region inequalities are separate next deductions.