Homogeneous representations preserve slope semistability in characteristic zero

We first prove the nullcone and separation facts for $S{L}_r$, and for the block-diagonal reductive groups used below. A vector v is unstable if zero belongs to the Zariski closure of its orbit. An orbit is open in its closure: its image is constructible, contains an open subset of the closure, and translates of that open subset cover the orbit. Choose an integral curve through zero with generic point in the orbit, by successive generic affine linear slices of the closure. Normalize the curve at the chosen point. Its generic orbit point lifts to a group element after a finite extension of its DVR fraction field: the fibre of the orbit map is a nonempty finite-type scheme, hence has a closed point over a finite extension. Complete and ramify the DVR if necessary. Thus g(t)v tends to zero with g(t) in the group over C((t)). Smith reduction gives $g(t)=P(t)\mathrm{diag}(t^{a_i})Q(t)$, with P,Q invertible over Ct. For $S{L}_r$ the sum of $a_i$ is zero; determinant units may be moved between P,Q so they have determinant one. Since the left unit preserves vanishing at zero, every nonpositive-weight coordinate of Q(0)v is zero. Therefore the one-parameter subgroup $\lambda (t)=\mathrm{diag}(t^{a_i})$ sends Q(0)v to zero, and its conjugate sends v to zero. The converse is immediate. For products of GL blocks with determinant-character constraints, apply Smith separately to each block. The valuations obey the same character constraints. Unit factors can be corrected inside the formal torus, taking roots of units after a finite extension, so the resulting subgroup and constant factor lie in the connected character kernel. This proves the same criterion for those groups.

If zero does NOT belong to the orbit closure, there is a polynomial f vanishing on that closure with $f(0)=1$, by the definition of its Zariski-closed ideal. Average f over the maximal compact unitary subgroup. The result is a polynomial, still one at zero and zero on the orbit, invariant under that compact group. It is invariant under the complex group: the complex span of its compact Lie algebra is the full complex Lie algebra, so infinitesimal invariance extends to the connected group; average also over any finite component group. Thus 1−average(f) is an invariant polynomial vanishing at zero and nonzero at v. Some positive homogeneous component is likewise nonzero. Conversely a homogeneous invariant polynomial nonzero at v rules out zero in the orbit closure. On polynomials of any fixed bounded degree the averaging operator is a complex linear map on a finite-dimensional space, and lies in the finite complex span of the compact group action matrices (that linear span is closed). It therefore preserves every invariant subspace after ANY field extension containing C, and commutes with that scalar extension. Thus the same separation argument holds over an algebraically closed extension, using this algebraic Reynolds operator rather than an undefined integral of field-valued functions. This proves invariant separation for $S{L}_r$ and the block character kernels of step1.1.

Here is the optimal-weight construction. A one-parameter subgroup of $S{L}_r$ diagonalizes with integer weights $a_i$ summing to zero. Give it norm $(\sum a_i^2{)}^{1\mathrm{/}2}$, invariant under conjugation and permutation. For a vector v write m(v,lambda) for the least lambda-weight of its nonzero components. It is unstable precisely when some m is positive. Fix one diagonal torus. The representation has finitely many weights, and hence only finitely many possible supports of gv in that weight set. For a fixed support S with zero outside its convex hull, the maximum of ${\min }_{b\in S}<b,a>\mathrm{/}\mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }$ is the distance from zero to that convex hull. Its maximizing ray is the ray through the closest point: projection onto the compact convex polytope gives $<b,\beta >\ge \mathrm{\mid }\mathrm{\mid }\beta \mathrm{\mid }{\mathrm{\mid }}^2$ for every b, with equality on the minimal face. The closest point has rational coordinates, by solving the rational linear equations of that face. Thus its ray contains an integral subgroup. Maximizing over the finitely many realized supports gives an actual optimal subgroup, with positive ratio m/||lambda||, whenever v is unstable. This uses no finiteness theorem for invariant rings.

Its weighted flag is unique, up to simultaneous positive rescaling of all weights. To prove this, the subgroup P(lambda) preserving its descending weight flag preserves the least weight of every vector in every representation: the induced action on each associated graded weight space is invertible and no lower weights are introduced. Changing a splitting of this weighted flag is conjugation by P(lambda), so it does not change m(v,lambda) or the norm. Any TWO filtrations of a finite vector space admit a simultaneous splitting: choose bases in the successive quotients of their intersections, or use ordinary Gaussian elimination on the two flags. Choose such a splitting for two optimal weighted flags. The corresponding conjugates lambda and mu now commute, and both are still optimal for the SAME vector v. For their sum the least weight satisfies $m(v,\lambda +\mu )\ge m(v,\lambda )+m(v,\mu )$. Unless their diagonal real weight vectors are positive proportional, strict convexity gives $\mathrm{\mid }\mathrm{\mid }\lambda +\mu \mathrm{\mid }\mathrm{\mid }<\mathrm{\mid }\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }+\mathrm{\mid }\mathrm{\mid }\mu \mathrm{\mid }\mathrm{\mid }$, contradicting optimality. Opposite proportional weights cannot both have positive least weight. Consequently the two weighted flags coincide. This proves the canonical optimal flag directly. The argument applies over any algebraically closed characteristic-zero extension.

For v defined over a characteristic-zero field K containing C, that flag descends to K. Its weights may be normalized as primitive integral weights. Every automorphism of the algebraic closure over K preserves the optimization problem, so uniqueness preserves each of its subspaces. A Galois-invariant subspace of ${\bar{K}}^r$ descends to K, by linear descent of its defining equations (equivalently its Grassmann coordinates). A K-basis adapted to that flag gives an optimal subgroup defined over K; changing its splitting again preserves the least weight by step4.1. Thus the construction applies at the generic point of X, with $K=C(X)$, and gives saturated coherent subsheaves of E by intersecting those generic subspaces with E. They and their quotients are bundles on a big open set.

We need the following explicit polynomial on the leading weight space. Let the optimal weights be $a_1>\mathrm{.}\mathrm{.}\mathrm{.}>a_l$ on blocks of dimensions $r_j$, sum $r_j{a}_j=0$, and let $b=m(v,\lambda )>0$. Denote the leading vector in the weight-b space $W_b$ by v0. The block Levi centralizer has character ${\chi }_{\lambda }(g)={\prod }_{j}det(g_j{)}^{a_j}$, whose pairing with lambda is $\mathrm{\mid }\mathrm{\mid }\lambda \mathrm{\mid }{\mathrm{\mid }}^2$. Inside its determinant-one part take the connected kernel $G_{\perp }$ of ${\chi }_{\lambda }$. The vector v0 is semistable for $G_{\perp }$. Otherwise step1.1 supplies a destabilizing subgroup mu in that kernel. It commutes with lambda and has $<\lambda ,\mu \ge 0$. For sufficiently small positive rational t, the least weight of v under lambda+t mu comes from the leading weight-b components, so it is b+t $m(v0,\mu )>b$. Its norm is $\mathrm{\mid }\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }+O(t^2)$. This improves the optimal ratio, a contradiction. Multiplying by a common denominator makes it an integral subgroup.

Step2.1 gives a homogeneous $G_{\perp }$-invariant polynomial P of degree $d>0$ on $W_b$ with P(v0) nonzero. It is a semi-invariant for the whole block Levi, after a power if needed. To see both its character and its sign explicitly, the quotient by $G_{\perp }$ is generated up to finite isogeny by the scalar torus and lambda. Suppose the original representation W has scalar weight u. Every degree-d polynomial on $W_b$ has scalar transformation $t^{ud}$ and lambda transformation $t^{bd}$. Hence $P(gx)=det(g{)}^{ud\mathrm{/}r}{\chi }_{\lambda }(g{)}^q$ P(x), with $q=bd\mathrm{/}\mathrm{\mid }\mathrm{\mid }\lambda \mathrm{\mid }{\mathrm{\mid }}^2>0$. Taking a power clears both denominators and any finite component character. The polynomial can be chosen with constant complex coefficients: in each degree invariance and the character conditions are linear equations on a finite-dimensional polynomial space defined over C. Their solution space after a field extension is its scalar extension; if evaluation is nonzero after extension, one constant-basis polynomial already has nonzero evaluation. This justifies using that same polynomial in bundle charts, rather than a generic rational polynomial with uncontrolled poles.

Apply these facts to a saturated proper positive-rank subsheaf F of rho(E), of rank s, and its generic Plücker line $\mathrm{\ell }=detF$ in $W={\bigwedge }^s(V)$. This representation has scalar weight $u=ws$. First suppose its generic vector is $S{L}_r$-semistable. A homogeneous $S{L}_r$-invariant polynomial of degree d nonzero on ell transforms under $G{L}_r$ by $de{t}^{wsd\mathrm{/}r}$: write a matrix as a scalar times a determinant-one matrix; the finite scalar ambiguity disappears because the polynomial is invariant under the center of $S{L}_r$. Clearing denominators by a power if necessary, evaluation gives a nonzero morphism of line bundles $(detF{)}^d\to$ $(detE{)}^{wsd\mathrm{/}r}$ on a big open set, and hence globally. The resulting effective zero divisor gives ${\mathrm{deg}}_H(F)\le sw{\mu }_H(E)$.

In the unstable case, use the canonical flag $0=E0$ subset E1 subset ... subset $El=E$ from step5.1, ordered by the strictly decreasing weights $a_j$. Its character line is $C_{\lambda }={\prod }_{j}det(E_j\mathrm{/}{E}_{j-1}{)}^{a_j}$. Semistability of E gives ${\mathrm{deg}}_H{C}_{\lambda }={\sum }_{j<l}(a_j-a_{j+1})[{\mathrm{deg}}_H{E}_j-\mathrm{rank}(E_j){\mu }_H(E)]\le 0$. Here the term involving deg E cancels because sum $r_j{a}_j=0$. In particular the direction/sign of the character is fixed by the descending, not ascending, flag.

On the big bundle open set the flag induces a filtration of W(E). The Plücker line is contained in its weight-at-least-b piece: the quotient is torsion-free and the containment holds generically. Its projection into the weight-b associated graded is a nonzero regular morphism, not a rational map with a pole. The unipotent radical of the flag parabolic acts trivially on this associated graded weight piece, so the polynomial semi-invariant has the required transformation also under all adapted-frame transitions. Evaluate the constant-coefficient polynomial of step7.1 on that projection. It gives a nonzero line morphism $(detF{)}^d\to$ $(detE{)}^{wsd\mathrm{/}r}$ tensor $C_{\lambda }^q,q>0$. It extends across codimension two by the line-bundle Hartogs property on smooth X. Thus $d{\mathrm{deg}}_{H}F\le (wsd\mathrm{/}r){\mathrm{deg}}_H$ E+q ${\mathrm{deg}}_H{C}_{\lambda }\le (wsd\mathrm{/}r){\mathrm{deg}}_{H}E$. This gives exactly the same required slope inequality as in step8.1. No restriction of E to a hypersurface or curve has been used.

Finally the slope of rho(E) itself is $w{\mu }_H(E)$. The differential of det rho is a conjugation-invariant linear functional on $g{l}_r$, hence a multiple of trace (it vanishes on commutators). Evaluating on the identity gives that multiple dim(V)w/r, so $c1(\rho (E))=(\dim (V)w\mathrm{/}r)c1(E)$. Equivalently diagonalize formal Chern roots and count the weights. Steps8.1 and10.1 therefore prove semistability. ${\mathrm{Sym}}^m$ has scalar weight m; $\mathrm{End}{\mathrm{Sym}}^m$ has scalar weight zero and is self-dual by its trace pairing. This proves the exact stated special cases in every dimension.

We prove the product-group assertion explicitly. Replace $S{L}_r$ throughout by $S{L}_r$ times $S{L}_s$, and use the sum of the two Euclidean trace-zero weight norms. Smith reduction and invariant averaging in steps1.1–2.1 were already blockwise. The finite-state polytope argument takes place in the direct sum of the two trace-zero spaces. For uniqueness simultaneously split the two flags in EACH factor; strict convexity of the summed norm gives the same canonical pair of flags. For an unstable Plucker line of a rank-k subsheaf, let $\lambda =({\lambda }_A,{\lambda }_B)$, with least weight $b>0$. The leading vector is semistable for the kernel of ${\chi }_{\lambda }={\chi }_{{\lambda }_A}{\chi }_{{\lambda }_B}$ in the product Levi determinant-one subgroup: a destabilizing perpendicular subgroup would improve lambda+t mu exactly as in step6.1. The polynomial of degree d then transforms by $(detA{)}^{ukd\mathrm{/}r}(detB{)}^{vkd\mathrm{/}s}{\chi }_{\lambda }^q$, where $q=bd\mathrm{/}\mathrm{\mid }\mathrm{\mid }\lambda \mathrm{\mid }{\mathrm{\mid }}^2>0$; take a power to clear denominators. Its degree is bounded above by k d $(u{\mu }_H(A)+v{\mu }_H(B))$, because deg ${\chi }_{\lambda }$ is the SUM of the two nonpositive weighted subflag degrees, by semistability of A and of B separately. In the semistable Plucker case the same invariant polynomial has only the two determinant factors and gives the identical bound. This proves the subobject inequalities. The trace calculation of step11.1 in each Lie-algebra factor computes the asserted total slope. Taking $\rho =\text{standard}$ tensor standard gives $(u,v)=(1,1)$, and $\rho =\text{dual}$ standard tensor standard gives Hom with $(u,v)=(-1,1)$. Finally for torsion-free A,B, their non-locally-free sets have codimension at least two. The generic canonical flags, determinant polynomials and effective-divisor inequalities used above all live in codimension one and extend exactly as in step10.1. Saturating a subsheaf and intersecting with a different codimension-two extension preserve its rank and first Chern class. Thus semistability and the proved inequalities are unchanged on passing to the double dual, to a torsion-free tensor extension, or to the Hom extension. No semistability of a restricted sheaf is used in this extension.