Effective restriction to integral divisors from constructed tilt and duality theory

Let E be an ordinary slope-stable torsion-free sheaf of rank $R>0$, let $b<\mu (E)$ be rational, and put $d=C_1(E)-b{C}_0(E)>0$. E belongs to $B_b$. We prove it is weak-tilt-stable for all sufficiently large $a={\alpha }^2$. Here weak stability uses $\nu (F)<\nu (E\mathrm{/}F)$ for nonzero proper subobjects, not the stronger and inappropriate condition $\nu (F)<\nu (E)$ when the quotient is numerically zero.

There is no null subobject of E in $B_b$. In fact a $B_b$ subobject A⊂E is a coherent sheaf: taking ordinary cohomology shows $H^{-1}(A)=0$. If $s(A)=0$ it is supported in dimension at most one, so cannot inject as a sheaf into the torsion-free E. For any subobject A, write $Q=E\mathrm{/}A$ and $K=H^{-1}(Q)$. The ordinary sequence is

0→K→A→E→H⁰(Q)→0.

Thus A is torsion-free, $K\in F_b$, and its image G⊂E is nonzero (otherwise $A=K$ would lie in both $T_b$ and $F_b$). Ordinary slope stability gives $\mu (G)\le \mu (E)$, with equality possible only for a full-rank image whose quotient has codimension at least two. Since $\mu (K)\le b<\mu (E)$ when K≠0, one obtains $\mu (A)\le \mu (E)$. If $0<s(A)<d$, equality is impossible: it would force $K=0$, rank $G=R$ and $c_1(G)=c_1(E)$, hence $s(A)=d$. Therefore for relevant proper positive-denominator subobjects

$C_0(A)\mathrm{/}s(A)>C_0(E)\mathrm{/}d=x_E>0$.

The difference is uniformly separated from zero. Indeed $C_0(A)d-C_0(E)s(A)=C_0(A)C_1(E)-C_0(E)C_1(A)$ is a positive integral multiple of $v=H^3$, while $0<s(A)<d$. Hence

$x_A\ge x_E+v\mathrm{/}{d}^2$.

If E is unstable at any sufficiently large a, its first HN factor A is semistable, has $0<s(A)<d$, and $\mathrm{\Delta }(A)\ge 0$ by the already constructed tilt discriminant theorem. The same conclusion holds for a proper finite-slope equal-phase factor of a semistable E. From $\mathrm{\Delta }(A)\ge 0$ and $C_0(A)>0$,

${\lambda }_a(A)\le 1\mathrm{/}(2x_A)-a{x}_A\mathrm{/}2\le 1\mathrm{/}(2x_E)-a(x_E+v\mathrm{/}{d}^2)\mathrm{/}2$.

But ${\lambda }_a(E)=t(E)\mathrm{/}d-a{x}_E\mathrm{/}2$. For large a this makes ${\lambda }_a(A)<{\lambda }_a(E)$, a contradiction. Null quotients have slope infinity and do not violate weak strict stability. Thus E is weak-tilt-stable for all sufficiently large a. This proof uses the tilt discriminant after it was derived by the one-way large-parameter classification, so it is not circular.

For a slope-stable sheaf E define its strict subsheaf gap

δ_L(E)=μ(E)−sup{ μ(G)<μ(E) : 0≠G⊂E is a coherent subsheaf of positive rank }.

The supremum is attained and δ_L(E)>0: ranks are at most R, integral degrees are discrete, and degrees of subsheaves are bounded above. The set is nonempty, for example E(−mH)⊂E for a nonzero section of O(mH). Full-rank subsheaves are INCLUDED; this matters in the dual application.

Let $P=(\mu (E),C_2(E)\mathrm{/}{C}_0(E))$ and $\mathrm{\Delta }(E)\ge 0$. Let L be a line through P with parabola roots $x<y\le \mu (E)$, and suppose

μ(E)−y < δ_L(E), R(μ(E)−y) < y−x.

Then E is tilt-semistable at every point of L∩U. It suffices first to take a rational b∈(x,y). At large a it is stable by the argument beginning at step 1.1. The rational fixed-b local finite-class argument gives a first finite wall if stability is lost before L. To spell out the wall existence being consumed: the tilt-discriminant support bound holds for every semistable class; at any finite positive a the inequalities in the local finite-class proof bound all potential semistable destabilizer classes in a compact parameter neighborhood. This proves local wall finiteness for this fixed object and fixed heart. Starting at a large stable parameter, the first parameter of loss is consequently an equal-phase semistable exact sequence with both denominators strictly between 0 and d. A numerically proportional equality class has a parameter-independent slope equality and cannot be a first loss of stability.

Suppose such a wall lies above L. If $\mathrm{\Delta }(E)>0$, its line through P has parabola roots $x^{\mathrm{\prime }}<x<y<y^{\mathrm{\prime }}<\mu (E)$. Exact numerical-line transport supplies the same semistable sequence $0\to A\to E\to Q\to 0$ in every heart for $b\in (x^{\mathrm{\prime }},y^{\mathrm{\prime }})$. A is a coherent sheaf. Write $K=H^{-1}(Q)$ and $T=H^0(Q)$. From the cohomological endpoint bounds in the transport theorem,

μ_min(A)≥y', μ_min(T/tors T)≥y', μ_max(K)≤x'.

If K has positive rank k, then the ordinary cohomology sequence gives

C₁(E)≥v[ y'(R+k)−x'k ],

because rank A+rank $T=R+k$ and torsion contributes nonnegative divisor degree. Consequently $R(\mu (E)-y^{\mathrm{\prime }})\ge k(y^{\mathrm{\prime }}-x^{\mathrm{\prime }})\ge y^{\mathrm{\prime }}-x^{\mathrm{\prime }}$, contradicting $R(\mu (E)-y)<y-x$. Hence $K=0$ and A is a genuine subsheaf of E. It has $\mu (A)\ge y^{\mathrm{\prime }}>y$. Also $\mu (A)<\mu (E)$: ordinary slope stability proves this for smaller rank, and in equal rank equality would make Q supported in codimension at least two, contrary to its positive denominator. Thus

0<μ(E)−μ(A)<μ(E)−y<δ_L(E),

a contradiction. If Δ(E)=0, any equal-phase semistable factors have Δ≥0 and lie in the same forward Lorentz cone. A sum on its null boundary forces both numerical vectors proportional to E, so they cannot produce a first loss of stability; this covers the y=μ(E) case too. Therefore E stays semistable down to L at this rational b. The numerical-line transport theorem then gives all real b∈(x,y).

The weak inequalities μ(E)−y≤δ_L(E) and R(μ(E)−y)≤y−x also suffice for semistability. At a fixed b on L, raise w slightly while keeping b fixed; the line from this raised point to P has smaller μ(E)−y and larger y−x, so the strict case applies. Semistability is closed in w in the fixed heart, being an intersection of closed subobject inequalities. Let w decrease to its desired value. No two-parameter deformation theorem is used.

Assume E is reflexive. Its dual E* is ordinary slope-stable of slope −μ(E). To verify this, for any positive-rank subsheaf G⊂E* of $\mathrm{rank}<R$, dualization gives a generic quotient $E\to G\ast$ whose image I is a genuine lower-rank torsion-free quotient of E. Stability gives $\mu (I)>\mu (E)$, while the inclusion I⊂G* gives $\mu (G\ast )\ge \mu (I)$. Hence $\mu (G)<-\mu (E)$. After saturation this rules out every destabilizing subsheaf. For a full-rank subsheaf its degree is at most that of E*, so it cannot destabilize either.

For $b>\mu (E)$, E[1] lies in the finite part of $B_b$. There are no maps $G[1]\to E[1]$ from a boundary-slope semistable sheaf G because $\mu (G)=b>\mu (E)$. There are no maps from coherent K supported in codimension at least two: these are $Ex{t}^1(K,E)=0$. The latter follows from the reflexive kernel presentation $0\to E\to V\to W\to 0$ with V locally free and W torsion-free: $\mathrm{Hom}(K,W)=0$ and $Ex{t}^1(K,V)=0$ by depth at least two along the support of K; the regular-sequence/coherent-duality proof of the preceding structural helper supplies exactly this Ext vanishing. A general null object is an extension of these two types.

The constructed truncated derived dual of E[1] is E*: reflexive sheaves on a smooth threefold have only their ordinary dual in cohomological degree zero and a possible zero-dimensional Ext¹ in degree one, which τ≤0 discards. The derived-dual semistability theorem therefore identifies semistability of E[1] at (b,w) with semistability of E* at (−b,w), with opposite source slope. Thus the left safe-region theorem for E* proves the right branch, using the strictly defined dual subsheaf gap δ_L(E*). This handles full-rank modifications as well as lower-rank quotients; no improper use of a proper-quotient gap is made.

Let E be an ordinary ${\mu }_H$-stable reflexive sheaf of rank $R>0$ on the smooth projective threefold X. Put

D=Δ(E)/C₀(E)²≥0, δ=min(δ_L(E),δ_L(E*))>0.

Let k be a positive integer and let Y∈|kH| be any integral Cartier divisor. Suppose

k ≥ (R+2)/sqrt(R+1) · sqrt(D), k/2+sqrt(k²/4−D) ≥ D/δ.

The square root is real under these conditions, and the actual required open intersection is automatic: if $D>0$, the first coefficient is strictly greater than 2, while if $D=0$, $k>0$. Then E|Y is slope-semistable on the integral polarized divisor (Y,H|Y). No smoothness of Y is assumed. Equivalently, with $z=\sqrt{k^2\mathrm{/}4-D}$ and $e=k\mathrm{/}2-z$,

R e≤2z, e≤δ.

The equivalence uses e(k/2+z)=D and the elementary rearrangement R(k/2−z)≤2z iff k²≥(R+2)²D/(R+1).

The line through Π(E) and Π(E(−kH)) has slope ℓ=μ(E)−k/2 and parabola roots

x=μ(E)−k/2−z, y=μ(E)−k/2+z.

Their width is 2z; $\mu (E)-y=e$ and $x-\mu (E(-kH))=e$. the argument beginning at step 11.1 makes E semistable everywhere on this line. Applying the reflected left safe region to the ordinary stable dual E*(kH), with its unchanged gap ${\delta }_L(E\ast )$, makes E(−kH)[1] semistable there too, by the argument beginning at step 25.1. Their source slopes both equal ℓ. The non-strict boundary case follows by raising w at a fixed interior b and using the two strict safe regions separately, then letting w decrease; the two slopes need coincide only at the limiting point.

Since E is torsion-free and Y is Cartier, multiplication by its defining section is injective and gives the exact derived triangle $E\to i\ast (E\mathrm{\mid }Y)\to E(-kH)[1]$. At an interior point on the line the three objects lie in $B_b$, so it is a short exact sequence there. The extension of the two equal-slope semistable objects is semistable: otherwise a higher-slope subobject has zero maps to both factors, by the elementary subobject/quotient inequalities. Thus i*(E|Y) is tilt-semistable of slope ℓ.

The restriction E|Y is torsion-free on the integral divisor. Indeed E has depth at least min(2,dim $O_{X,p}$) at every point because it is reflexive, and the defining nonzerodivisor reduces depth by one. At every nongeneric point of Y this gives depth at least one, so there is no associated point away from the generic point; a torsion submodule would have such an associated point. This also excludes zero-dimensional torsion at the singular points of E or Y.

For a coherent quotient $E\mathrm{\mid }Y\to Q$, sheaves i*Q and its kernel are torsion in X, so the quotient is an epimorphism in $B_b$. A positive-rank quotient on Y has $C_1(i_{\ast }Q)=kv{\mathrm{rk}}_Y(Q)>0$. Tilt semistability yields

C₂(iQ)/C₁(iQ) ≥ C₂(i*(E|Y))/C₁(i*(E|Y)).

This is precisely the Hilbert-polynomial slope inequality on Y, including singular integral Y. Explicitly, ambient GRR gives ${\alpha }_2(i\ast Q)=C_1(i\ast Q)$ and ${\alpha }_1(i_{\ast }Q)=C_2(i_{\ast }Q)-K_{X}H{\mathrm{ch}}_1(i_{\ast }Q)\mathrm{/}2$. Subtract ${\mathrm{rk}}_Y(Q)$ times the same coefficient for $O_Y$, whose Chern character is $1-e^{(-kH)}$, to get

deg_{H|Y}(Q)=C₂(iQ)+rk_Y(Q) k²v/2, μ_{H|Y}(Q)=kv [ C₂(iQ)/C₁(i*Q)+k/2 ].

The positive factor and constant are independent of Q. Rank-zero quotients have assigned infinite slope and impose no restriction. Hence E|Y is slope-semistable, proving the full stated effective restriction theorem.

For $R\ge 2$, both ${\delta }_L(E)$ and ${\delta }_L(E\ast )$ are at least 1/[v R(R−1)]. For a subsheaf of rank $q<R$, a strict slope difference has numerator a nonzero integer in denominator $vRq\le vR(R-1)$. For $q=R$, its degree difference is a positive integer, giving the stronger 1/(vR); rank-one sheaves have only this full-rank case. Thus one may eliminate the unknown gaps by the universal explicit lower bound. This recovers a genuine uniform effective restriction consequence without assuming that the dual subsheaf gap equals a proper-quotient gap.