Rational tilt discriminant and numerical support from ordinary sheaf BG

Since ch₀,ch₁,ch₂ have bounded denominators (1,1,2), the three numerical coordinates at fixed rational β lie in a discrete lattice $L\subset R^3$. The values of s on Bβ are nonnegative and discrete. Ordinary BG implies the boundary-slope inequality needed by the preceding HN helper: for $\mu (F)=\beta$, $C_1(F)=\beta$ C₀(F) and $\mathrm{\Delta }(F)\ge 0$ give $C_2(F)\le {\beta }^2{C}_0(F)\mathrm{/}2$. Thus that helper applies for every $a>0$, and its HN and finite-part assertions are available without any tilt discriminant assumption.

If $E\in C\beta$, its ordinary cohomology is F[1] and T, where F is zero or a ${\mu }_H-semistable$ torsion-free sheaf of slope β and T has dimension at most one. Therefore

$\mathrm{\Delta }(E)=\mathrm{\Delta }(F)+2$ C₀(F) $C_2(T)\ge 0$.

Here $C_2(T)\ge 0$ is the effective one-dimensional cycle degree. If $F=0$ this simply gives $\mathrm{\Delta }(E)=0$. Also $Z_a(E)=0$ iff $F=0$ and T is zero-dimensional, by strict $a>0$ boundary positivity. If E is finite-slope semistable, E has no nonzero Cβ subobject: such a subobject has assigned slope $\mathrm{\infty }$. Thus E lies in $B{\beta }^{f}in$, and this fact is independent of a.

On ker $Z_a$, $s=0$ and $t=a$ r/2, so $\mathrm{\Delta }=-a$ r²; it is negative definite. Consequently

{ (a,x): $a\in J$, $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }=1$, $\mathrm{\Delta }(x)\ge 0$ }

is compact and disjoint from $Z_a(x)=0$. The positive minimum of $\mathrm{\mid }{Z}_a(x)\mathrm{\mid }$ proves a uniform bound $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }\le K_J\mathrm{\mid }{Z}_a(x)\mathrm{\mid }$ for every $\mathrm{\Delta }(x)\ge 0$. This is just finite-dimensional compactness, with no stability-theoretic deformation theorem.

For a fixed finite slope ℓ, put $t=\mathrm{\ell }s+a$ r/2. Then

$\mathrm{\Delta }=(1+{\mathrm{\ell }}^2\mathrm{/}a)s^2-a(r+\mathrm{\ell }s\mathrm{/}a{)}^2$.

In the half-plane $s>0$, $\mathrm{\Delta }\ge 0$ is the convex cone defined by

|sqrt(a)(r+ℓs/a)| $\le$ sqrt(1+ℓ²/a) s.

It is closed under sums. In particular, equal-slope positive-denominator objects with nonnegative discriminant have a sum of nonnegative discriminant.

Induct on the positive discrete value s(E). The zero case was proved above. Fix E with $s(E)=d>0$ and suppose the result holds for all smaller positive s. If E is semistable at $a_0>0$, then for every subobject $F\subset E$ with $0<s(F)<d$,

Re $Z_{a_0}(F)\ge$ [s(F)/d] Re $Z_{a_0}(E)$.

There is no null subobject. A subobject with $s(F)=d$ cannot destabilize E at any a: its quotient lies in Cβ and has t−a $r\mathrm{/}2\ge 0$, whence ${\lambda }_a(F)\le {\lambda }_a(E)$. Thus every possible destabilizing HN first factor at a nearby a has $0<s(F)<d$. It is a semistable object, so induction gives $\mathrm{\Delta }(F)\ge 0$ and the uniform support estimate on a compact interval J about a₀.

Write $R=\mathrm{Re}{Z}_a(F)$, $R_0=\mathrm{Re}{Z}_{a_0}(F)$, and $\delta =\mathrm{\mid }a-a_0\mathrm{\mid }$. There is a fixed $B\ge 0$ with $R_0\ge -B$, because $0<s(F)<d$ and the preceding semistability inequality holds at a₀. Since $Z_a-Z_{a_0}=(a-a_0)r\mathrm{/}2$,

R $\ge$ −B − (δ/2)|r(F)| $\ge$ −B − (δ $K_J\mathrm{/}2)\mathrm{\mid }{Z}_a(F)\mathrm{\mid }$.

As $\mathrm{\mid }{Z}_a(F)\mathrm{\mid }\le \mathrm{\mid }R\mathrm{\mid }+d$, choosing δ $K_J\mathrm{/}2<1\mathrm{/}2$ gives a uniform lower bound on negative R. The destabilizing inequality gives the upper bound

R $<$ [s(F)/d] Re $Z_a(E)$,

which is also uniform near a₀. Thus R, s(F), $\mathrm{\mid }{Z}_a(F)\mathrm{\mid }$, and then ||(r,s,t)(F)|| are all bounded. The lattice L contains only finitely many possible triples in this bounded set. The same proof includes a semistable equal-slope first factor at a₀, replacing the strict upper inequality by equality. Only the numerical classes of subobjects that actually occur are considered; no assertion about boundedness of their isomorphism classes is needed.

For later use, assume additionally $\mathrm{\Delta }(E)<0$. None of those finitely many subobject classes can satisfy equality at a₀. Indeed, if F is an equal-slope subobject of the semistable E with $0<s(F)<d$, F is semistable of the same finite slope, but $Q=E\mathrm{/}F$ may have null subobjects. Take its maximal Cβ subobject T, which exists by noetherianity. Its preimage P in E has $s(P)=s(F)$ and $(t-a_{0}r\mathrm{/}2)(P)=(t-a_{0}r\mathrm{/}2)(F)+(t-a_{0}r\mathrm{/}2)(T)$. The boundary positivity from step 4.1 makes the last term nonnegative; semistability of E and equality for F force it to be zero. Hence T is zero-dimensional and has all three coordinates zero. The quotient $Qbar=Q\mathrm{/}T$ has no null subobject and is finite-slope semistable of the same slope: any positive-denominator destabilizing subobject lifts first through T and then through F to a destabilizing subobject of E, by additivity and the equal slope of F. Induction therefore gives $\mathrm{\Delta }(F),\mathrm{\Delta }(Qbar)\ge 0$. Since Q and Qbar have identical numerical triples, this is $\mathrm{\Delta }(F),\mathrm{\Delta }(E\mathrm{/}F)\ge 0$; the fixed-ray convexity above gives $\mathrm{\Delta }(E)\ge 0$, a contradiction. There are therefore finitely many strict inequalities at a₀. Shrinking the a-neighborhood preserves all of them. The preceding finite-class bound shows that no new destabilizer outside this finite set appears there. Hence a negative-discriminant E semistable at a₀ remains semistable in a neighborhood of a₀.

The set of parameters $a>0$ where a fixed finite-part E is semistable is closed: it is the intersection over its fixed set of subobjects of the closed inequalities ${\lambda }_a(F)\le {\lambda }_a(E)$; null subobjects never occur and same-denominator inequalities hold automatically. Thus, for $\mathrm{\Delta }(E)<0$, this set is both open and closed in the connected interval $(0,\mathrm{\infty })$. If it is nonempty, it equals $(0,\mathrm{\infty })$. This replaces the usual first-wall assertion by a proved open-and-closed argument.

Suppose E is finite-slope semistable for all arbitrarily large a; put $F=H^{-1}(E)\in F\beta$ and $T=H^0(E)\in T\beta$. All sheaf HN filtrations here are already constructed in the first-heart helper.

If $F\mathrm{\ne }0$, F has ${\mu }_{\max }(F)<\beta$: otherwise its top $\mu =\beta$ HN subsheaf shifted by one would be a Cβ subobject of E. Thus $s(F[1])>0$. The first-heart exact sequence is $0\to F[1]\to E\to T\to 0$. If $s(T)>0$, compare the coefficients of a in the slopes of F[1] and E. Write $R_F=C_0(F)>0$, $R_T=C_0(T)\ge 0$, $d_F=s(F[1])>0$, $d_T=s(T)>0$. Their coefficients are respectively $R_F\mathrm{/}(2d_F)$ and $(R_F-R_T)\mathrm{/}(2(d_F+d_T))$; the first is strictly larger. The subobject F[1] therefore destabilizes for large a, impossible. Hence $s(T)=0$. Since $T\in T\beta$, it follows that T has dimension at most one.

If F is not μ-semistable, its highest-slope HN subsheaf F₁ has $\mu (F_1)>\mu (F)$, both strictly below β. In the heart $F_1[1]\subset F[1]\subset E$. Its positive a-coefficient 1/[2(β−μ(F₁))] is strictly larger than that of E, 1/[2(β−μ(F))]. This too destabilizes for large a. Thus F is μ-semistable. By ordinary BG and $C_2(T)\ge 0$,

$\mathrm{\Delta }(E)=\mathrm{\Delta }(F)+2C_0(F)C_2(T)\ge 0$.

If $F=0$, $E=T$ is a coherent sheaf in Tβ. If it has rank zero, $\mathrm{\Delta }(E)=C_1(E{)}^2\ge 0$, with no purity assertion needed. If it has positive rank, a nonzero torsion subsheaf of dimension at most one would be a null subobject, forbidden. A two-dimensional torsion subsheaf has rank zero and positive denominator, so its slope is independent of a whereas E's slope tends to $-\mathrm{\infty }$; it therefore destabilizes for large a. E is torsion-free. All of its HN slopes exceed β. If it is not μ-semistable, its highest HN subsheaf E₁ has $\mu (E_1)>\mu (E)>\beta$, and its coefficient −1/[2(μ(E₁)−β)] is strictly larger than E's coefficient −1/[2(μ(E)−β)]. Again this is a large-a destabilizer. Thus E is μ-semistable and ordinary BG gives $\mathrm{\Delta }(E)\ge 0$.

If the induction object E were semistable at some a₀ and had $\mathrm{\Delta }(E)<0$, the argument beginning at step 13.1 would make it semistable for every $a>0$. the argument beginning at step 23.1 contradicts its negative discriminant. This proves the induction, hence $\mathrm{\Delta }\ge 0$ for every tilt-semistable object and every $a>0$ at this fixed rational β. the argument beginning at step 1.1 then proves the claimed uniform numerical support estimate.

The proof uses the first-heart/HN packet and ordinary sheaf BG, but it does not use a preexisting tilt discriminant inequality, a wall-existence theorem, a deformation theorem, effective restriction, a BN inequality, or the threefold BMT inequality. It does not yet prove β-varying heart deformation, ordinary sheaf BG, effective restriction, or the zero-discriminant locally-free/projectively-flat classification. In particular the latter classification must not be replaced by an unjustified assertion that $\mathrm{\Delta }=0$ already implies the required ch₃ bound.