Small-slope counterexample with uniform HN bounds on every integral hyperplane restriction

Let X be a smooth integral projective complex threefold, H ample Cartier, $v=H^3$, and $0<\epsilon <1\mathrm{/}3$. Assume ordinary numerical sheaf BG. Suppose BG₃(ε) fails: there is an ordinary H-slope-stable torsion-free sheaf A with $0<\mu (A)\le \epsilon$ and C₂(A)≥−C₁(A)/2. Then there is a reflexive ordinary H-stable E with

$0<\mu (E)\le m:=2\epsilon \mathrm{/}(1-\epsilon )$, C₂(E)≥−C₁(E)/2,

such that, for every integral Cartier divisor S∈|H|, E|S is torsion-free and all its ordinary normalized H|S-slope HN factors satisfy

$0\le {\mu }_{H\mathrm{\mid }S}(F_i)\le 2\epsilon \mathrm{/}(1-3\epsilon )$.

Slope on a singular integral S is defined by Hilbert coefficients, normalized by $H\mathrm{\mid }{S}^2=v$. Thus this statement includes integral normal rational Gorenstein S without asserting any singular-surface BG theorem. The latter is a distinct input to the subsequent surface summation.

Define f on [0,∞) by

$f(x)=-x\mathrm{/}2$ (0≤x≤ε), $f(x)=[(1+\epsilon )\mathrm{/}(2(1-\epsilon ))]x-\epsilon \mathrm{/}(1-\epsilon )$ (ε≤x≤m), $f(x)=x^2\mathrm{/}2$ (x≥m).

It is continuous and f(x)≥−x/2 on [0,m]. Let Sε be the following set of positive-rank numerical objects: E∈DᵇCoh(X), with $C_0(E)>0$, $0<\mu (E)=C_1(E)\mathrm{/}{C}_0(E)\le m$ and $y(E)=C_2(E)\mathrm{/}{C}_0(E)\ge f(\mu (E))$, such that either E∈B₀ is tilt-semistable at some $w>0$, or E[1]∈B₁ is tilt-semistable at some $w>1\mathrm{/}2$. No sheaf assertion is built into this set. It is nonempty: the failed BG₃ sheaf A has y≥f(μ) and is tilt-stable at $b=0$ for large w by the constructed large-volume converse.

The rational tilt discriminant gives Δ(E)≥0 for every E in this set. The number $\mathrm{\Delta }=C_1^2-2C_0{C}_2$ is an integer: C₁ is integral, $C_0=v$ rk is integral and 2C₂ is integral. C₁(E) is a positive integer. Choose an element minimizing the ordered pair (Δ(E),C₁(E)) lexicographically in $N\times N_{>0}$. This is a genuine well-founded minimum; no boundedness of ranks or moduli is required.

The function f(x)/x is nondecreasing on (0,m]: it equals −1/2 on (0,ε] and equals (1+ε)/(2(1−ε))−[ε/(1−ε)]/x on [ε,m]. Consequently, if y≥f(μ) and $0<x\le \mu \le m$, every point on the line segment from (μ,y) toward (0,w₀), $w_0>0$, with coordinate x satisfies $y_x\ge f(x)$. The inequality is strict compared with (x/μ)y when $x<\mu$, because the intercept w₀ is positive.

The function $g(x)=[f(x)-1\mathrm{/}2]\mathrm{/}(1-x)$ is nonincreasing on [0,1): it equals −(1+x)/(2(1−x)) on [0,ε], the constant −(1+ε)/(2(1−ε)) on [ε,m], and −(1+x)/2 on [m,1). Thus if $\mu \le x<1$, the line from (μ,y≥f(μ)) toward (1,w₁), $w_1>1\mathrm{/}2$, has $y_x\ge f(x)$, strictly for $x>\mu$ whenever comparison with the line to (1,1/2) is used. In particular, for x≥m with $x>\mu$ this line lies strictly above x²/2, and cannot contain the numerical point of a tilt-semistable nonzero-rank object, by Δ≥0. At $x=\mu =m$ the only possible point $y=f(m)=m^2\mathrm{/}2$ has $\mathrm{\Delta }=0$.

Suppose a minimal-set object at $b=0$ has a proper equal-finite-phase splitting 0→U→E→V→0 with both denominators positive. Such factors are themselves semistable. Their C₁ coordinates are positive and add to C₁(E); among the positive-rank factors choose F with $0<\mu (F)\le \mu (E)$. To prove existence, add the positive-rank factors: their ranks sum to at least rank E, while their positive C₁ sum to at most C₁(E). Their rank-weighted average slope is therefore at most μ(E). This argument needs only the two factors U,V, not a Jordan–Hölder filtration. Equal phase says Π(F) lies on the line through Π(E) and (0,w). Its slope coordinate lies between 0 and μ(E), so the argument beginning at step 10.1 gives y(F)≥f(μ(F)). Thus F∈Sε.

At $b=1$ use the actual heart object E[1], whose rank is negative. If it has a proper equal-finite-phase split with both denominators positive, sum the negative-rank factors and write $R_i=-C_0(F_i)>0$. Their denominators are $R_i(1-\mu (F_i))$; their total $R_i$ is at least C₀(E), while their total denominator is at most C₀(E)(1−μ(E)). Hence at least one such factor has μ(F)≥μ(E), and every negative-rank factor has $\mu (F)<1$. Replace this chosen factor by F[−1] to obtain positive rank. Equal phase places its point on the line toward (1,w), and the argument beginning at step 10.1 plus Δ≥0 forces $\mu (E)\le \mu (F)<m$, except the harmless possible equality $\mu (F)=\mu (E)=m$. It therefore belongs to Sε too.

The discriminant comparison is rigorous in both cases. On a fixed finite-phase ray, with $s>0$, Δ has Lorentz form as in the discriminant supplier. If x,y are vectors in its nonnegative forward cone, their associated bilinear pairing is nonnegative; it vanishes only for proportional null vectors. Thus Δ(x+y)≥Δ(x)+Δ(y), and for either nonzero positive-denominator summand

$\mathrm{\Delta }(summand)<\mathrm{\Delta }(total)$,

unless total and both summands are proportional and null. The strict case contradicts minimal Δ of Sε. In the exceptional null case, the selected factor has numerical vector t times the heart object's vector with $0<t<1$, because its positive denominator is smaller than the total. After undoing the possible shift, $0<C_1(F)=t$ $C_1(E)<C_1(E)$, contradicting the secondary minimum. No exception remains.

Consequently a minimal object has no proper equal-finite-phase splitting at either vertical line whenever it is semistable there. Numerically zero quotients are not such a splitting and do not enter the rank-average argument; weak strict stability correctly allows them. Also no variation of w can make a minimal object lose semistability: the constructed rational fixed-heart finite-class argument would produce a first equal-finite-phase splitting at the boundary of its semistable interval, and the preceding descent excludes it. Starting from any supplied semistable w, minimality thus gives semistability for every allowed w on that vertical line.

If the chosen object is represented at $b=0$, it is semistable there for arbitrarily large w. The proved large-parameter classification says that its positive-rank class is a torsion-free μ-semistable coherent sheaf: the alternative with nonzero ordinary H⁻¹ and ordinary H⁰ supported in dimension≤1 has negative rank and is impossible. Rank-zero torsion in a positive-rank coherent sheaf is likewise excluded by finite-part membership and large-parameter semistability.

If the chosen object is represented only at $b=1$, its heart representative E[1] is semistable there for arbitrarily large w and has negative rank. The same classification yields ordinary $H^{-1}(E[1])=F$ a μ-semistable torsion-free sheaf and $H^0(E[1])=T$ supported in dimension≤1. The positive class of F has $\mu (F)=\mu (E)$, y(F)≥y(E), and Δ(F)≤Δ(E). Refine F by ordinary μ-stable factors (a finite filtration exists by induction on rank, after saturation). They all have the same slope. Choose a factor F₀ with y(F₀)≥y(F). Then

Δ(F₀)≤[C₀(F₀)/C₀(F)]² Δ(F)≤Δ(F),

using Δ(F)≥0. If F has more than one such factor, this strictly lowers Δ unless $\mathrm{\Delta }(F)=0$, when its smaller positive C₁ lowers the secondary coordinate. If F is itself stable, take $F_0=F$. In every case F₀ is ordinary stable, has y≥f at the same slope, and belongs to Sε through its large-volume $b=0$ stability. Minimality forces equality of the minimizing pair and permits replacement by this ordinary stable sheaf. The same stable-factor argument applied to a $b=0$ μ-semistable sheaf proves it is already stable or can be replaced in this way.

Replace this stable sheaf by its reflexive hull E**. It is μ-stable: a destabilizing smaller-rank subsheaf of E** intersects E in the same rank and first Chern class, contradicting stability of E. The quotient E**/E has dimension≤1 and nonnegative C₂; hence its hull has the same μ, larger y and no larger Δ. It belongs to Sε by $b=0$ large-volume stability, so minimality again allows this replacement without increasing the minimum. Rename the resulting reflexive stable sheaf E. It is a minimum of the SAME original set, not a newly enlarged set.

At $b=0$, E is semistable for all $w>0$ by the descent argument. Since $\mu (E)\le m<1$, E[1] is a finite-part object of B₁. Its truncated dual is E*, which is ordinary stable and lies in $B_{-1}$. The constructed large-volume converse and semistability duality therefore make E[1] semistable at $b=1$ for large w. Its minimality and the argument beginning at step 12.1 make it semistable there for EVERY $w>1\mathrm{/}2$. Tensoring by O(−H) gives that E(−H)[1] is semistable at $b=0$ for every $w>0$: the exact untwisted parameter change is $b^{\mathrm{\prime }}=b-1$, $w^{\mathrm{\prime }}=w-b+1\mathrm{/}2$, so $b=1$ maps to $b^{\mathrm{\prime }}=0$ and $w^{\mathrm{\prime }}=w-1\mathrm{/}2>0$. The source slopes change by subtracting one; no twisted-numerator convention is substituted.

Fix any integral Cartier divisor S∈|H|, with i:S→X. Reflexivity makes E|S torsion-free by the depth argument in the effective-restriction supplier. For every $w>0$ there is the short exact sequence in B₀

0→E→i*(E|S)→E(−H)[1]→0.

Put $\mu =\mu (E)\in (0,m]$, $y=C_2(E)\mathrm{/}{C}_0(E)\ge -\mu \mathrm{/}2$. Its endpoint slopes are

$l(w)=(y-w)\mathrm{/}\mu$, $h(w)=(w-y+\mu -1\mathrm{/}2)\mathrm{/}(1-\mu )$.

The value where they agree is $w_0=y-{\mu }^2+\mu \mathrm{/}2$. If $w_0>0$, both endpoints are semistable of the same slope there; their extension i*(E|S) is semistable. Ambient GRR identifies the slope of any positive-rank sheaf on S as $\nu (i\ast F)={\mu }_{H\mathrm{\mid }S}(F)-1\mathrm{/}2$, exactly as derived in the effective-restriction supplier. Every coherent quotient on S is also a quotient in B₀. Thus E|S is ordinary slope-semistable, and its only HN slope μ belongs to (0,m]⊂[0,2ε/(1−3ε)].

If w₀≤0, then l(w)≤h(w) for all $w>0$. The weak HN slopes of the extension i*(E|S) all lie in [l(w),h(w)]. Indeed a highest-slope semistable HN subobject with $\text{slope}>h(w)$ has zero maps to both semistable endpoints, contradicting the exact sequence; a lowest-slope quotient with $\text{slope}<l(w)$ is excluded dually. This also excludes null HN factors at either extreme. Hence every subobject has average slope≤h(w) and every positive-denominator quotient has average slope≥l(w).

The ordinary slope HN filtration on S exists even when S is singular. Here is the needed construction. A torsion-free coherent G on the integral S injects into its double dual: locally, a finite torsion-free module over a domain embeds in a finite free module after clearing denominators in its fraction-field embedding, so the dual linear forms separate its elements. Eventual global generation of G* twisted by $O_S(mH)$ then gives $G\subset O_S(mH{)}^N$. For any rank-q subsheaf F⊂G, choose q coordinate rows that are independent at the generic point; their projection gives an injection $F\subset O_S(mH{)}^q$. The cokernel has dimension at most one and nonnegative ambient C₂. Therefore C₂(i*F)≤q $C_2(i\ast O_S(mH))$, bounding the normalized slope of every subsheaf by m. These slopes form a discrete bounded-above set, since 2C₂ is integral and 1≤q≤rk G. Choose a maximal-slope subsheaf of largest rank and saturate it: saturation is coherent and changes the slope only upward (by a nonnegative effective-cycle degree), so maximality preserves the same slope and gives a torsion-free quotient. Every quotient slope-HN maximal factor has smaller slope, since otherwise its preimage contradicts maximal slope or largest rank. Induction on rank gives a finite HN filtration. This also proves the required HN existence rather than silently importing a smooth-surface theorem.

Take the highest ordinary HN subsheaf $G_{\max }\subset E\mathrm{\mid }S$ and lowest ordinary HN quotient $G_{\min }$. They are also a subobject and a quotient in B₀ after i*, with positive denominators. They need not be tilt-semistable; only their average numerical slopes are used. Therefore

$l(w)\le {\mu }_{H\mathrm{\mid }S}(G_{\min })-1\mathrm{/}2$ $\le {\mu }_{H\mathrm{\mid }S}(F_i)-1\mathrm{/}2$ $\le {\mu }_{H\mathrm{\mid }S}(G_{\max })-1\mathrm{/}2\le h(w)$.

Letting w decrease to zero gives

${\mu }_{\min }(E\mathrm{\mid }S)\ge$ y/μ+1/2≥0, ${\mu }_{\max }(E\mathrm{\mid }S)\le [-y+\mu \mathrm{/}2]\mathrm{/}(1-\mu )$ ≤μ/(1−μ)≤2ε/(1−3ε).

These are precisely the required uniform HN bounds, with no rank-dependent restriction constant. The endpoint $\mu =m$ is included since $m<1$. This proves the stated certificate.