Proof. The rational Chern character identifies the Grothendieck group with the rational Chow group (The Stacks Project Authors 2026, Tags 0FDI, 0FEW, 0FB2 and 0FB5). In Riemann–Roch the Euler pairing is the intersection pairing after applying the invertible operations of dualizing a class and multiplying it by \(\mathop{\mathrm{td}}(X)\). Its radical therefore gives precisely the numerical quotient of the Chow group. In dimensions zero and three this quotient is \(\mathbb Q\); in the other two dimensions it is the divisor–curve pairing in (6).

For a rational ample class \(A\), the Hodge index theorem makes \((D,D')\mapsto ADD'\) nondegenerate on \(N^1(X)_\mathbb Q\); see (Greb et al. 2016, Theorem 6.4 and Corollary 6.5) for its real-ample formulation. The divisor–curve pairing is nondegenerate by numerical equivalence, so multiplication by \(A\) is an isomorphism \(N^1(X)_\mathbb Q\to N_1(X)_\mathbb Q\). Divisors, their products with \(A\), and \(A^3\) consequently span every summand of (6).

Choose an affine slice transverse to the ray of \(H_*\) and a rational simplex in its ample cone whose interior contains that ray. Its \(q=\rho(X)\) vertices are linearly independent. Taking sufficiently large integral multiples makes them very ample without changing their positive cone. This also explains the rank-one case, where a single very ample class suffices.

Pushforward is defined on numerical groups by adjunction with perfect complexes. If \(i_*v=0\) numerically, the projection formula gives \[\chi(i^*W,v)=\chi(W,i_*v)=0\qquad(W\in K_0(Y)).\] Chern characters of line bundles on \(Y\) span its polynomial Chow ring, and their restrictions span (6). Nondegeneracy of the Euler pairing gives \(v=0\). The spanning argument here uses dimension three; it makes no analogous claim in arbitrary dimension. ◻

Proof. We use the product-with-a-curve construction of (Liu 2021, Lemma 5.7 and Remark 5.8), the positive-genus product construction of (Li et al. 2025, Theorem 4.5), and the support induction of (Li et al. 2026, Lemma 6.6). For \(n=1\), the usual slope stability on \(C\) has charge \(-x_{\varnothing}+iw_1x_{\{1\}}\) and the assertion is immediate.

Suppose that the scaled support bound has been proved in dimension \(n-1\). Omitting factor \(r\) splits the coordinates into \(x',x''\), where \(x'\) consists of those containing \(r\), with that index removed. The product construction has charge \[Z_{n-1}(x'')-iw_rZ_{n-1}(x').\] It initially gives support with respect to its own quotient lattice. Every choice of omitted factor gives the same exponential charge and the same phase \(1\) for point sheaves. The uniqueness theorem (Li et al. 2025, Theorem 3.11) therefore identifies the slicings: that theorem permits the two support lattices to differ. We may consequently test all the product support forms on the same semistable objects, before proving support on the full coordinate lattice.

Write \(M_{\widehat r}\) for scaling the remaining coordinates by their products of weights. The scaled coordinates split as \[y'=w_rM_{\widehat r}x',\qquad y''=M_{\widehat r}x''.\] The lower-dimensional charge in scaled coordinates is a fixed linear map \(\widehat Z\). Increase its support constant \(C_{n-1}\) if needed and choose the support form \[ q_{n-1}(v)=C_{n-1}^2|\widehat Z(v)|^2-\|v\|^2. \tag{8}\] It is nonnegative on lower-dimensional semistable classes. Crucially, \[q_{n-1}(v)\le C_{n-1}^2|\widehat Z(v)|^2 \quad\text{for every }v,\] not merely for semistable classes. This is the norm form used in the product support estimate.

For clarity, that estimate has the following coefficient condition. Write \(Z_{n-1}(x')=\alpha+i\beta\) and \(Z_{n-1}(x'')=\gamma+i\delta\). With equal positive product parameters \(s=t=w_r\), the form \[\beta\gamma-\alpha\delta +\eta_r q_{n-1}(M_{\widehat r}x')\] is nonnegative on product-semistable classes when \(0\le\eta_r\le w_r/C_{n-1}^2\) (Liu 2021, Lemma 5.7); this is the substitution in (Li et al. 2026, Lemma 6.6). Choose \(0<\lambda_r<C_{n-1}^{-2}\), independently of \(R\), and put \(\eta_r=\lambda_rw_r\). If \[\widehat Z(y')=\widehat\alpha+i\widehat\beta, \qquad \widehat Z(y'')=\widehat\gamma+i\widehat\delta,\] then multiplication of the product form by \(w_r\) gives \[ q_r(y)=\widehat\beta\widehat\gamma -\widehat\alpha\widehat\delta +\lambda_rq_{n-1}(y'). \tag{9}\] All powers of \(R\) have disappeared. This form is nonnegative on the common semistable objects just constructed.

The total charge in these coordinates is \((\widehat\gamma+\widehat\beta) +i(\widehat\delta-\widehat\alpha)\). On its kernel, \[q_r(y)=-|\widehat Z(y')|^2+\lambda_rq_{n-1}(y') =(\lambda_rC_{n-1}^2-1)|\widehat Z(y')|^2 -\lambda_r\|y'\|^2.\] It is nonpositive there, and vanishes only when \(y'=0\); the kernel condition then also gives \(\widehat Z(y'')=0\). Sum (9) over all omitted factors. If the sum vanishes on the total charge kernel, then \(y'=0\) for every factor. This kills every coordinate with \(I\ne\varnothing\), and the charge kills the remaining top-degree coordinate. The summed form is thus negative definite on the charge kernel and nonnegative on semistables. Compactness on the unit sphere of its nonnegative cone gives \(\|y(u)\|\le C_n|Z_{\mathbf w}(u)|\) for every semistable class \(u\). The forms and all choices were made in the scaled coordinates. In particular they are independent of \(R\) and can be kept uniform on compact sets of positive ratios. This completes the support induction. The geometricity assertion is part of the positive-genus product construction; see also (Li et al. 2026, Theorem 4.1(1)). ◻

Proof. Cheng proves the two-block construction and phase estimate in (Cheng 2026, Proposition 4.3 and Remark 4.4). We apply the same quotient construction block by block, using Lemma 5 to control all coordinates uniformly in the volume.

First take \(R\) and the positive weights \(s_h\) rational. Put \(n=\sum_hd_h\) and give each of the \(d_h\) elliptic factors in block \(h\) the weight \(w_r=2Rs_h\). The lemma supplies the starting stability condition and its scaled support bound. Only the coordinate lattice of the elliptic power is used here; full numerical support will follow on \(Y\) from the explicit pullback calculation below.

First apply (Li et al. 2026, Proposition 5.1) to the stability condition just constructed on the coordinate lattice. It supplies the descent by \((\mathbb Z/2)^n\) to \((\mathbb P^1)^n\) and the Bayer property with \(k=0\) for every effective line bundle \(\mathcal O(a_1,\ldots,a_n)\), with all \(a_r\ge0\). We then take the symmetric quotients one block at a time.

Here is the filtration used at a symmetric step. For \(Q=(\mathbb P^1)^d\to\mathbb P^d\), there is a finite filtration of \(\mathcal O_{Q\times_{\mathbb P^d}Q}\) whose successive quotients are \[p_2^*\mathcal L_j^{-1}\otimes\mathcal O_{\Gamma(g_j)}, \qquad g_j\in S_d,\qquad \mathcal L_j=\mathcal O(a_{j1},\ldots,a_{jd}),\quad a_{jr}\ge0,\] where \(p_2\) is the second projection and \(\Gamma(g_j)\) is the graph of the permutation \(g_j\) (Li et al. 2026, Definition 3.19 and Example 3.20(3)). At a block stage the map is \(f:Q\times M\to\mathbb P^d\times M\), where \(M\) is the product of the other blocks, and its fiber square is \((Q\times_{\mathbb P^d}Q)\times M\). Flat pullback to this product preserves the filtration and its exact sequences. Its graph automorphisms are \(g_j\times\operatorname{id}_M\), and its line bundles remain pulled back from \(Q\).

The finite-flat descent criterion (Li et al. 2026, Proposition 3.21) therefore requires precisely invariance under these permutations and the Bayer properties with \(k=0\) for these effective line bundles. Indeed, for a phase-\(\phi\) object \(F\), the factors of \(f^*f_*F\) are \(\mathcal L_j^{-1}\otimes g_{j*}F\); the two properties put their largest phases at most \(\phi\). On the elliptic product, signs and permutations within each equal-weight block preserve the charge and point phases, so the uniqueness theorem used above gives their invariance. Block permutations normalize the full sign group; their invariance descends through the completed involution quotient. For subsequent block quotients, the required properties pass to the remaining blocks as follows (Li et al. 2026, Lemma 3.9 and Remark 3.14(3)). At each quotient \(f:V\to V'\), pullback computes extremal phases for the induced slicing \(\mathcal S'=f_\#\mathcal S\). If a line bundle \(L\) comes from an unquotiented block, then \[\phi^-_{\mathcal S'}(F\otimes L[k]) =\phi^-_{\mathcal S}(f^*F\otimes f^*L[k]).\] The Bayer property for \(f^*L\) therefore descends to \(L\). An automorphism of another block commutes with \(f\), so its invariance descends by the same pullback description. Thus every remaining quotient has the required Bayer properties and invariance, for any finite number of blocks.

For completeness, let \(g\) be the resulting map from the elliptic power to \(Y\). Pullback satisfies \[g^*\xi_h^{j_h}=2^{j_h}j_h! \sum_{|J_h|=j_h}H_{J_h}, \qquad \deg g=2^n\prod_hd_h!.\] If \(|I\cap\text{block }h|=d_h-j_h\), then the scaled \(I\)-coordinate of \(g^*u\) is \[ \Bigl(\prod_{r\in I}w_r\Bigr) H_I\mathop{\mathrm{ch}}_{n-|I|}(g^*u) =2^n\Bigl(\prod_hj_h!\Bigr)(M_Ru)_{\mathbf j}. \tag{13}\] The fixed multiplicities are \(\prod_h\binom{d_h}{j_h}\). Pullback is thus injective on the complete Chern-character lattice of \(Y\), with norm comparison constants independent of \(R\). Semistability pulls back, and \(Z_{\mathbf w}(g^*u)=\deg(g)U_R(u)\). This proves (11).

For the phase estimate, vary the real tensor parameter on each elliptic factor successively. The one-factor argument in the proof of (Cheng and Feyzbakhsh 2026, Theorem 2.4), used in (Cheng 2026, Remark 4.4), gives its absolute variation divided by \(\pi w_r\). A twist \(e_h\xi_h\) pulls back with coefficient \(2e_h\) on the \(d_h\) factors in block \(h\); its factor \(2\) cancels the factor \(2\) in \(w_r\). Adding the variations gives (12). Finally the deformation and multiplication-isogeny argument in the proof of (Cheng 2026, Proposition 4.3) extends the rational-weight construction continuously to positive real weights. It applies factor by factor: the scaled support forms just constructed remain fixed on compact positive weight sets, the exponential charges vary continuously, and uniqueness on overlapping deformation neighborhoods identifies the lifts. The closed support inequalities and the phase bounds persist. This also extends \(R\) to all real \(R\ge1\). Point stability in this extension follows separately from (Li et al. 2026, Theorem 4.1(1)) on the elliptic power and preservation of geometricity under finite pushforward (Li et al. 2026, Remark 3.11). Concretely, for the finite faithfully flat quotient \(g\), the sheaf \(g^*\mathcal O_y\) has a finite filtration by point sheaves, all stable of phase \(1\). A phase-one subobject and quotient of \(\mathcal O_y\) pull back to objects whose Jordan–Hölder factors are among these points, so both pullbacks are coherent finite-length sheaves. Faithful flatness descends this cohomological concentration. The original subobject and quotient are therefore sheaves, and simplicity of \(\mathcal O_y\) rules out a proper nonzero subobject. Continuity and the charge value \(-1\) normalize the point phase to \(1\). ◻

Proof. This is the point-object argument of (Bridgeland 2008, Lemma 10.1); we include it to specify the dimension and boundary statements being used. Let \(E\) be stable with phase \(0<\phi\le1\) and not a point of phase \(1\). Phase vanishing gives \(\mathop{\mathrm{Hom}}(E,\mathcal O_x[j])=0\) for \(j<0\). For \(j\ge n\), Serre duality identifies this space with \(\mathop{\mathrm{Hom}}(\mathcal O_x,E[n-j])^*\), which is zero by phases. At \(j=n,\phi=1\), use nonisomorphism of the two stable objects. The canonical bundle causes no change, since its restriction to a point is trivial. A minimal free complex at each closed point consequently has nonzero terms only in degrees \([-n+1,0]\). Its bottom cohomology is a subsheaf of a locally free sheaf and hence is torsion-free. At phase \(1\) we also have \(\mathop{\mathrm{Hom}}(E,\mathcal O_x)=0\) for every \(x\), so \(\mathcal H^0(E)=0\). Point sheaves themselves satisfy the stated cohomological bounds. Finite Jordan–Hölder filtrations and extensions give these assertions for the whole heart; in phase \(1\) its zeroth cohomology is an extension of sheaves supported at finitely many points.

A coherent sheaf cannot receive a nonzero morphism from an object whose phase is greater than \(1\), by the cohomological bound just proved and a shift. It cannot map nontrivially to an object of phase at most \(1-n\): such an object is in \(D^{\ge1}\), including the endpoint by the phase-one observation. Applied to its extreme Harder–Narasimhan factors, these vanishings prove \(\mathop{\mathrm{Coh}}(V)\subset\mathcal R(1-n,1]\). Shifting this inclusion and the heart bound proves (15). ◻

Proof. If \(E\) is strictly semistable, take a first stable subobject \(G\) in a finite Jordan–Hölder filtration. Its class cannot be real proportional to \([E]\). Indeed their aligned charges would make \([G]=c[E]\) with \(0<c<1\), whereas primitivity makes an integral class on this line an integral multiple of \([E]\). Choose a phase-window heart of length one centered at their common phase. The triangle \(G\to E\to E/G\) remains a short exact sequence in this heart for small deformations: all three objects stay strictly inside its phase window. An independent small perturbation of the two charge values makes the phase of \(G\) greater than that of \(E\). This contradicts the assumed semistability of \(E\) in the entire neighborhood. ◻

Proof. Take \(V\) bounded in the \(b_h\) coordinates, and restrict \(W\) by \(|\ell_h/s_h-1|<\eta\). In the scaled coordinates (10), expansion of the two exponentials gives \[ \bigl\|(U_t^{B,H}-U_t)\circ M_t^{-1}\bigr\| \le C_1\eta+C_2t^{-1}. \tag{22}\] To see the uniformity, the coefficient of \(u_{\mathbf j}\) in the uncorrected exponential has degree \(n-|\mathbf j|\) in \(t\). Varying the positive coefficients \(s_h\) to \(\ell_h\) changes its ratio to the weight of \(M_t\) by \(O(\eta)\). A codimension-\(d>0\) term of \(e^{-\sum b_h\xi_h}\gamma-1\) lowers that degree by \(d\) and hence contributes \(O(t^{-d})\). There are finitely many monomials, and their coefficients are bounded on the chosen neighborhoods. This proves (22) independently of the object.

Combine it with (11). Choose \(\eta\) first and then \(T\) so that \[D(C_1\eta+C_2/T)<\sin(\pi\epsilon).\] To check the support hypothesis of the deformation theorem explicitly, put \[Q_t(u)=D^2|U_t(u)|^2-\|M_tu\|^2.\] It is nonnegative on the original semistable classes. If \(U'(u)=0\) and \(\|(U'-U_t)M_t^{-1}\|\le\delta<1/D\), then \[Q_t(u)\le(D^2\delta^2-1)\|M_tu\|^2<0\qquad(u\ne0).\] Thus \(Q_t\) is negative definite on every charge kernel along the straight deformation, and throughout the indicated open charge region. The support-property deformation theorem (Bayer 2019, Theorem 1.2) lifts the straight path while preserving \(Q_t\). On its nonnegative cone, \(\|M_tu\|\le D|U_t(u)|\), so the relative charge change is less than \(\sin(\pi\epsilon)\). The quantitative phase estimate (Bayer 2019, Lemma 2.9) then makes the slicing distance less than \(\epsilon\) as long as it is less than \(1/4\). It is zero at the start; continuity along the path and \(\epsilon<1/8\) prevent a first exit from that \(1/4\) neighborhood. This proves the asserted strict distance bound. The charge bound is strict, so for each fixed \(t\) it also holds on an open neighborhood in all ambient numerical charges. This is the space on which we deform, before applying restriction. ◻

Proof. Set \(A=\mathcal O_Y(1,\ldots,1)\). Choose a locally free resolution \(P^\bullet\to i_*\mathcal O_X\) with \(P^0=\mathcal O_Y\) and with each \(P^{-k}\) a finite sum of line bundles \(A^{-b}\) for \(k\ge1\). Such a resolution is obtained by repeatedly generating the successive coherent kernels after sufficiently positive twists. Fix \(N>\dim Y+1\) and truncate after the terms in degrees \(0,\ldots,-(N-1)\). The remaining kernel is shifted by \([N]\), so the cofiltration remainder lies in \(D^{\le-N}(Y)\).

For a defining ambient linear form of a smooth section \(S_h\), first form the cone of the untruncated map \(P^\bullet(-\xi_h)\to P^\bullet\). It resolves \((i\circ j_h)_*\mathcal O_{S_h}\), and its term in degree \(-k\) is \[P^{-k}\oplus P^{1-k}(-\xi_h),\qquad P^1=0.\] Truncate each cone resolution after its terms in degrees \(0,\ldots,-(N-1)\). Its remaining kernel is again shifted by \([N]\), so its cofiltration remainder also lies in \(D^{\le-N}(Y)\). The line bundles in all these terms are independent of the chosen smooth section. Include all their summands, in particular the mixed twists \(A^{-b}\otimes\mathcal O_Y(-\xi_h)\), together with \(A\) and \(\mathcal O_Y(\xi_h)\), in one finite list. This list is fixed before the parameters \(B,H,t\) and the sections are chosen.

Let \(\mathcal R\) be a lift from Lemma 9. Tensoring is an isometry for slicing distance, so (12) gives, after increasing the single threshold \(T\), \[ d(\mathcal R,\mathcal R\otimes L)<1 \tag{23}\] for every line bundle \(L\) in the finite list, uniformly in the claimed sector. The triangle inequality bounds the left side by \(2\epsilon+O(t^{-1})\). The same strict inequalities hold in a full open neighborhood of each lifted charge.

In particular tensoring by \(A^{-1}[1]\) has the Bayer property. (Li et al. 2026, Proposition 3.27) makes every point sheaf on \(Y\) semistable throughout that open neighborhood. Its numerical class is primitive, because \(\chi(\mathcal O_Y,\mathcal O_y)=1\). Lemma 8 upgrades semistability to stability. At the charge (20) its value is \(-1\), and closeness to \(\mathcal R_t\) fixes its phase to be \(1\).

We can now apply Lemma 7 on \(Y\). It supplies (16) with \(n_1=-\dim Y\) and \(n_2=0\). Each resolution term \(L[k]\), \(k\ge1\), has the Bayer property: (23) says that tensoring by \(L\) lowers no extremal phase by as much as one, and the shift \([k]\) restores the required lower bound. The initial \(\mathcal O_Y\) term is the identity. The chosen \(N\) exceeds the truncation bound. Thus the finite-map criterion applies to the closed immersion \(i\) and, when needed, to every \(i\circ j_h\). It supplies slicings with the indicated charges and Harder–Narasimhan filtrations. Definition (17) and uniqueness of those filtrations show that pushforward computes their extremal phases.

Pushforward is exact for these hearts and detects zero objects. A proper destabilizing subobject of a point would therefore give one for the stable point on \(Y\). The induced points are stable of phase \(1\). If \(E\) is semistable on \(X\), the ambient support inequality and the injectivity in Lemma 4 give \[\|[E]\|\le C\|i_*[E]\|\le C'|U_t^{B,H}(i_*E)|.\] This is support on the full Euler numerical group of \(X\). For each auxiliary surface the ambient lattice through pushforward suffices; no injectivity assertion for its full numerical group is needed. Finally the support bound on a finite-rank lattice gives local finiteness: in a sufficiently short phase interval, the projection of a nonzero semistable charge onto its middle ray has a fixed positive lower bound. Additivity of this projection bounds the length of any chain of strict subobjects or quotients of a fixed object in that interval. The induced slicings are thus locally finite, as asserted. ◻

Proof. Lemma 7 gives \(\mathcal R(0,1]\subset D^{[-1,0]}(S)\) and \(\mathop{\mathrm{Coh}}(S)\subset\mathcal R(-1,1]\). The comparison of these two bounded \(t\)-structures makes \(\mathcal R(0,1]\) a tilt of \(\mathop{\mathrm{Coh}}(S)\), with torsion pair \[\mathcal T_0=\mathop{\mathrm{Coh}}(S)\cap\mathcal R(0,1],\qquad \mathcal F_0=\mathop{\mathrm{Coh}}(S)\cap\mathcal R(-1,0].\] For instance their torsion sequence is obtained by truncating a sheaf at phase \(0\); the cohomological range ensures that both pieces are sheaves.

If \(G\in\mathcal T_0\), every quotient sheaf also belongs to \(\mathcal T_0\) and has nonnegative imaginary charge. A nonzero positive-rank quotient cannot have imaginary charge zero: it would lie in \(\mathcal R(1)\), whereas Lemma 7 makes its zeroth cohomology zero-dimensional. Thus all positive-rank quotients of \(G\) have strictly positive slope. Applying this to its last slope Harder–Narasimhan quotient proves \(\mathcal T_0\subset\mathcal T_{\omega,\beta}\). If \(F\in\mathcal F_0\), then \(F[1]\in\mathcal R(0,1]\), so \(F\) is torsion-free by the same lemma. Every subsheaf remains in \(\mathcal F_0\) and has nonpositive imaginary charge. Its maximal-slope factor therefore has slope at most zero, giving \(\mathcal F_0\subset\mathcal F_{\omega,\beta}\).

These inclusions identify the torsion pairs. Indeed the torsion sequence for either pair, applied to an object in the corresponding part of the other, leaves a quotient or subobject in the intersection of its two orthogonal parts, hence zero. This proof includes slope zero; it does not use a generic-parameter argument. ◻

Proof. The range and torsion-freeness follow from Lemma 7. Write \(F=\mathcal H^{-2}(E)\) and \(G=\mathcal H^0(E)\). For each \(h\), choose a general smooth \(S_h\) which is a nonzerodivisor on the finitely many cohomology sheaves and the subobjects and quotients used below. Those sheaves and sequences then restrict without Tor. Such choices are available inside the complete very ample linear system.

By (28), \(E(L_h)|_{S_h}\) has largest phase at most \(2\). The surface tilt (26), applied to its lowest ordinary cohomology, implies \(F(L_h)|_{S_h}\in\mathcal F_h\). Similarly (27) places \(E|_{S_h}\) in \(\mathcal P_h(>0)\), and the highest ordinary cohomology of the surface tilt gives \(G|_{S_h}\in\mathcal T_h\). These conclusions can also be read directly from the aisles: \(\mathcal P_h(\le2)\) has its degree \(-2\) cohomology in \(\mathcal F_h\), while \(\mathcal P_h(>0)\) has its degree \(0\) cohomology in \(\mathcal T_h\).

If \(F\ne0\), restrict its maximal-slope subsheaf \(F'\subset F\). The class \(\mathcal F_h\) is closed under subsheaves, so the shifted twist in the definition of the surface slope gives \[\frac{HL_h(c_1(F')-B\mathop{\mathrm{rk}}F')}{\mathop{\mathrm{rk}}F'} \le-\frac12HL_h^2.\] If \(G\) has a positive-rank last slope quotient \(G''\), quotient-closure of \(\mathcal T_h\) gives \[\frac{HL_h(c_1(G'')-B\mathop{\mathrm{rk}}G'')}{\mathop{\mathrm{rk}}G''} >\frac12HL_h^2.\] For a torsion sheaf \(G\) its minimum slope is \(+\infty\) instead. Multiply the displayed inequalities by the positive \(\ell_h\) and sum. Since \(\sum_h\ell_hL_h=H\), they are exactly (30). Positivity of the weights is what permits this argument for a real ample \(H\). ◻

Proof. Let \(D\in\mathcal T_{H,B}\) be a sheaf with \(m^-_{H,B}(D)>0\). Lemma 7 places all its phases in \((-2,1]\). If a last Harder–Narasimhan factor \(G\) had phase in \((-2,-1]\), then \(G[2]\in\mathcal P(0,1]\). It follows that \(G\in D^{[0,2]}\) and \(m^+_{H,B}(\mathcal H^0(G))\le-c\). The canonical nonzero map \(D\to G\) is tested on its degree-zero cohomology: \[\mathop{\mathrm{Hom}}(D,G)=\mathop{\mathrm{Hom}}(D,\mathcal H^0(G))=0.\] Here the equality follows from \(G\in D^{\ge0}\) and standard truncation; the vanishing is the slope comparison, including the case of torsion \(D\) since the target is torsion-free. This contradiction excludes that phase interval and gives \(D\in\mathcal P(-1,1]\).

Now let \(J\in\mathcal F_{H,B}\), so \(J\) is torsion-free with \(m^+_{H,B}(J)\le0\). The coherent-sheaf range gives \(J[1]\in\mathcal P(-1,2]\). If a first Harder–Narasimhan factor \(G\) had phase in \((1,2]\), then \(G[-1]\in\mathcal P(0,1]\), \(G\in D^{[-3,-1]}\), and \(m^-_{H,B}(\mathcal H^{-1}(G))>c\). The canonical nonzero map \(G\to J[1]\) is tested on its top cohomology: \[\mathop{\mathrm{Hom}}(G,J[1])=\mathop{\mathrm{Hom}}(\mathcal H^{-1}(G),J)=0,\] again by truncation and slope. Thus \(J[1]\in\mathcal P(-1,1]\). Extensions prove (31).

The final assertion is the comparison theorem for two bounded hearts one of which lies in two adjacent degrees of the other. Explicitly, truncate an object of \(\mathcal B\) at phase \(0\) in \(\mathcal P\). The inclusion (31) and its corresponding inclusions of aisles make both truncations objects of \(\mathcal B\); they give its torsion sequence with the two stated intersections. Tilting that sequence recovers \(\mathcal P(0,1]\). ◻

Proof. Let \(E\) belong to the stated intersection. Its ordinary cohomology is \(J=\mathcal H^{-1}(E)\in\mathcal F_{H,B}\) and \(U=\mathcal H^0(E)\in\mathcal T_{H,B}\). Lemma 7 makes \(U\) zero-dimensional. Choose the smooth sections \(S_h\) to avoid its finite support and to restrict \(J\) without Tor. If \(J\ne0\), it is torsion-free of positive rank, and \(E|_{S_h}=J|_{S_h}[1]\). Inequality (27) gives \(\phi^-_{\mathcal P_h}(J|_{S_h})\ge0\). Its phases as a coherent sheaf are also at most \(1\), by Lemma 7. Hence \(\Im Z_h(J|_{S_h})\ge0\) and (25) gives \[\frac{HL_h(c_1(J)-B\mathop{\mathrm{rk}}J)}{\mathop{\mathrm{rk}}J}\ge\frac12HL_h^2 \quad\text{for each }h.\] The positive weighted sum says \(m_{H,B}(J)\ge c>0\), contrary to \(m^+_{H,B}(J)\le0\). Therefore \(J=0\) and \(E=U\). ◻

Proof. Let \(J=\mathcal H^{-1}(E)\) and \(U=\mathcal H^0(E)\). The first torsion pair makes each contribution to \(H^2\mathop{\mathrm{ch}}_1^B(E)=H^2\mathop{\mathrm{ch}}_1^B(U)-H^2\mathop{\mathrm{ch}}_1^B(J)\) nonnegative. If the sum is zero, \(J\), when nonzero, is slope-semistable of slope zero, and \(U\) has dimension at most one. Indeed positive-rank quotients in the positive torsion part have strictly positive degree, and a nonzero effective divisorial cycle has positive intersection with \(H^2\). For real \(H\), the Kähler Bogomolov–Gieseker inequality follows from (Li et al. 2017, Theorem 1.1, Equation (1.6)), applied with zero Higgs field to the reflexive hull \(J^{**}\) when \(J\ne0\). This hull is semistable for the same \(H\): intersecting a subsheaf of \(J^{**}\) with \(J\) preserves its rank and first Chern class. The quotient \(Q=J^{**}/J\) has codimension at least two, and, writing \(r=\mathop{\mathrm{rk}}J\), additivity gives \[c_1(J)^2-2r\mathop{\mathrm{ch}}_2(J) =c_1(J^{**})^2-2r\mathop{\mathrm{ch}}_2(J^{**})+2r\mathop{\mathrm{ch}}_2(Q),\] where \(\mathop{\mathrm{ch}}_2(Q)\) is an effective curve cycle, so the inequality descends to \(J\). The same argument on a surface uses an effective zero-cycle correction; neither application changes the real polarization. Classical Bogomolov–Gieseker and the real-ample Hodge index theorem (Greb et al. 2016, Theorem 6.4), applied to \(H^2\mathop{\mathrm{ch}}_1^B(J)=0\), give \(H\mathop{\mathrm{ch}}_2^B(J)\le0\). The curve cycle of \(U\) is effective. Consequently \[\Im Z_{B,tH}(E)= tH\mathop{\mathrm{ch}}_2^B(U)-tH\mathop{\mathrm{ch}}_2^B(J)+\frac{t^3H^3}{6}\mathop{\mathrm{rk}}J\ge0.\] If \(J\ne0\), the last term is strictly positive. If \(J=0\) and \(U\) has a nonzero one-dimensional part, its curve degree is strictly positive. Thus equality is possible exactly when \(E=U\) is zero-dimensional. ◻

Proof. Compare \((\mathcal T',\mathcal F')\) with the torsion pair \((\mathcal T_0,\mathcal F_0)\) of Lemma 13. For \(D\in\mathcal T'\), take its latter torsion sequence \(0\to D_1\to D\to D_2\to0\). A nonzero \(D_2\in\mathcal F_0\) has \(\Im Z(D_2)\le0\). Its denominator cannot vanish: Lemma 15 would make it zero-dimensional, whereas zero-dimensional sheaves are in \(\mathcal P(1)\) and \(\mathcal F_0\subset\mathcal P(-1,0]\). Thus \(D_2\) has positive denominator and \(\nu^{\mathrm{phys}}_{B,tH}(D_2)\le0\), contradicting the strictly positive slopes of every nonzero quotient of \(D\). Hence \(\mathcal T'\subset\mathcal T_0\).

For \(D\in\mathcal F'\), consider \(D_1\) in the same torsion sequence. A nonzero subobject of \(D\) has second slope at most zero, so \(D_1\) has positive denominator and \(\Im Z(D_1)\le0\). But \(D_1\in\mathcal T_0\subset\mathcal P(0,1]\) makes its imaginary charge nonnegative. It must therefore vanish, and \(D_1\in\mathcal P(1)\). Lemma 14 makes it zero-dimensional, whose second slope is \(+\infty\), a contradiction. Thus \(\mathcal F'\subset\mathcal F_0\). The two inclusions identify the torsion pairs, and their tilts coincide. ◻

Proof. Write \(V=\mathcal H^{-1}(E)\) and \(U=\mathcal H^0(E)\) for ordinary cohomology. The defining torsion pair gives \(d_B(V)\le0\) and \(d_B(U)\ge0\), whence \(d_B(E)=d_B(U)-d_B(V)\ge0\). If it is zero, a nonzero \(V\) is torsion-free and slope-semistable of slope zero, whereas \(U\) has dimension at most one. Bogomolov–Gieseker and Hodge index give \(w_B(V)\le0\); effectiveness of the curve cycle gives \(w_B(U)\ge0\). Thus \[n(E)=w_B(U)-w_B(V)+\tfrac12a^2r(V)\ge0.\] Equality forces \(V=0\) and the curve cycle of \(U\) to vanish, so \(U\) is zero-dimensional. The converse is immediate.

In a short exact sequence \(0\to F\to E\to G\to0\) with \(d(E)>0\), a subobject with \(d(F)=0\) has slope \(+\infty\) while \(d(G)>0\). It is forbidden by semistability. If both denominators are positive, additivity makes \(\nu(E)\) their weighted average, proving the desired equivalences. If \(d(G)=0\), then \(n(G)\ge0\), so \(\nu(F)\le\nu(E)\) and comparison with \(\nu(G)=+\infty\) is automatic. ◻

Proof. The first inequality of (42) gives \(a^2r^2+2rn\le d^2\), and hence \(|r|\le 2|n|/a^2+d/a\). The equality \(w=n+a^2r/2\) bounds \(|w|\). Finally the second inequality gives \[\|c_B\|^2=2d^2-\langle c_B,c_B\rangle \le(2+C_H)d^2-2rw,\] which proves the norm bound. These expressions also prove uniformity on compact \(a\)-intervals. ◻

Proof. Let \(\nu(E)=u\in\mathbb R\). If \(E\) is not stable, there is a proper nonzero subobject \(F\) with \(\nu(F)=\nu(E/F)=u\) and positive denominators. The object \(F\) is semistable by Lemma 17. To obtain a splitting into two semistable objects, we may have to absorb a zero-dimensional subobject of the quotient back into \(F\). Put \(G=E/F\) and take its Harder–Narasimhan filtration. Its last factor is a quotient of \(E\), so, if its slope is finite, that slope is at least \(u\). The other finite slopes are no smaller. There must be a finite factor since \(d(G)>0\). Additivity of \(n-ud\), together with the nonnegativity of \(n\) on infinite-slope factors, now forces all finite slopes to equal \(u\) and all infinite-slope factors to have \(d=n=0\). Thus \(G\) is either semistable or has a zero-dimensional initial subobject \(T\) followed by a semistable quotient of slope \(u\). Replace \(F\) by the preimage of \(T\). It is still a proper equal-slope subobject of \(E\), hence semistable, and its quotient is now semistable as well.

Repeated splitting cannot produce arbitrarily many nonzero factors. For a semistable factor of slope \(u\), Lemma 18 gives \(|r|\le C_a(1+|u|)d\). If \(r\ne0\), integrality of rank therefore bounds \(d\) below by \(1/(C_a(1+|u|))\). If \(r=0\), then \(d=H^2\mathop{\mathrm{ch}}_1/h\) is positive and belongs to \(h^{-1}\mathbb Z\), so \(d\ge1/h\). Each factor has denominator at least the smaller of these two positive constants, and their denominators sum to \(d(E)\). A finite number of splits consequently gives stable factors.

For the extension assertion, intersect a subobject with the first term and take its image in the last term. A nonzero intersection or image has positive denominator and slope at most \(u\) by semistability. Their sum has the same properties, proving the subobject criterion. ◻

Proof. Since \(w_b=ud_b+a^2r/2\), the twist formulas give \(w_c=R^2r/2\) and \(d_c=d_b-ur\). Thus \(\Delta=d_c^2-R^2r^2\ge0\). If \(r\ne0\), the negative alternative \(d_c\le-R|r|\) would give \(d_b=d_c+ur\le-(R-|u|)|r|<0\). It follows that \(d_c\ge R|r|\); for \(r=0\) we have \(d_c=d_b>0\). On the whole open arc, therefore, \[d_{b'}=d_c-xr>0,\qquad n_{a',b'}=-x(d_c-xr).\]

Consider the set of points on the arc where \(E\) is semistable in the corresponding heart. It contains the original point. It is closed within the arc: the formal charge has positive real part, and continuity of the largest and smallest phases makes both converge to its finite phase. In particular, this argument also ensures membership in the limiting heart. It is open within the arc as well. At a semistable point take the finite stable filtration from Lemma 19. Every factor has the same slope there and thus the same \(c,R\) in (44). Openness of stability keeps all these finitely many factors stable in nearby hearts, with their common slope \(-x\). Their fixed triangles are short exact sequences in those hearts, and their extensions are semistable by the preceding lemma. Connectedness of the arc proves the assertion. ◻

Proof. For rational \(\omega=\sqrt3aH\) and rational \(B\), (Bayer et al. 2014, Proposition 5.1.3(b)) gives the triangle and semistability directly: its hypothesis is finite maximal tilt slope, which holds for a finite-slope semistable object. In that triangle \(\widetilde E\) has ordinary cohomology in degrees \(-1,0\), whereas \(T_0[-1]\) is in degree \(1\). Hence \(\widetilde E\) is the stated ordinary truncation; it depends only on \(E\).

First suppose \(E\) is stable at real parameters. By openness it is stable at a sequence of nearby parameters with rational \(\omega,B\). The rational triangles all have the same truncation \(\tau^{\le0}\mathbb D(E)\) and the same sheaf \(\mathcal H^1(\mathbb D(E))\). Their mirror finite phases converge to the phase of slope \(-u\), with the denominator still positive. Continuity of extremal phases therefore gives semistability of this fixed truncation in \(\mathcal B_{H,-B}\).

For a semistable \(E\), use Lemma 19 and induct on its number of stable factors. For an extension \(0\to E_1\to E\to E_2\to0\), duality reverses the triangle. We must show that its degree-zero cohomology in the mirror tilt heart is semistable and that its degree-one cohomology is zero-dimensional. The cohomology sequence is \[0\longrightarrow\widetilde E_2\longrightarrow \mathcal H^0_{\mathcal B_{H,-B}}(\mathbb D(E)) \longrightarrow\widetilde E_1\longrightarrow T_{0,2} \longrightarrow\mathcal H^1_{\mathcal B_{H,-B}}(\mathbb D(E)) \longrightarrow T_{0,1}\longrightarrow0,\] and all other tilt cohomology vanishes. Subobjects and quotients of a zero-dimensional sheaf in this heart are zero-dimensional: additivity first forces both denominators to vanish, then both nonnegative numerators to vanish, and Lemma 17 applies. Consequently the kernel of \(\widetilde E_1\to T_{0,2}\) has the same positive denominator and the same slope as \(\widetilde E_1\); it is semistable by the subobject criterion. The degree-zero term is an extension of it by \(\widetilde E_2\), and is semistable of slope \(-u\). The degree-one term is zero-dimensional. This proves (45), including its truncation description, for the extension.

Finally the shifted dual has characters \((-r,c_B,-\mathop{\mathrm{ch}}_2^B,\mathop{\mathrm{ch}}_3^B)\) at twist \(-B\). The term \(T_0[-1]\) has third character \(-\operatorname{length}(T_0)\), so taking characters of the triangle gives (46). ◻

Proof. Write \[\operatorname{Disc}(G) =2r c_2(G)-(r-1)c_1(G)^2=c_1(G)^2-2r\mathop{\mathrm{ch}}_2(G).\] The characteristic-zero semistable case of Langer’s restriction inequality (Langer 2004b, Theorem 3.1) has zero extremal-slope term. On \(X\) it gives, for raw restricted slopes \(s_i=h\mu_i\), \[\sum_{i<j}r_i r_j(s_i-s_j)^2 \le hH\operatorname{Disc}(G)=h^2r^2\xi_G.\] Since \(\sum r_i\mu_i=r\mu(G)\), the left side is \(h^2r\sum_i r_i(\mu_i-\mu(G))^2\). Division gives (48). If the restriction is semistable, its left side is zero, and the inequality follows directly from \(\xi_G\ge0\). Refinement at a fixed slope leaves the sum unchanged. For a surface restriction, apply the same theorem with its very ample divisor \(mH\). The raw slopes are \(s_i=mh\mu_i\) and the right side is \(m^2h\operatorname{Disc}(G)=m^2h^2r^2\xi_G\); the factors \(m^2h^2r\) cancel in the same way.

We use the corrected very-general formulation of the theorem (Langer 2004a, 1211). In the ample polarizations used here, that addendum also permits general divisors: the stated difficulty concerns the remaining nef polarizations when they are not ample.

For each original factor, the weighted mean of its restricted slopes is unchanged. Subtracting a fixed scalar \(\mu(T)\) from these slopes therefore does not change their variance. The elementary identity \[\sum_i r_i(\mu_i-\mu(T))^2 =r(\mu(G)-\mu(T))^2 +\sum_i r_i(\mu_i-\mu(G))^2\] proves the second-moment assertion. From \(X\) to \(S\), restriction preserves the total \(w_T\); hence \(\sum r_i\delta_{G_i}^T=\sum r_i\mu_T(G_i)^2-2w_T(G)\) changes by exactly the same amount. Sum these identities over the original factors. Their restricted and refined filtrations are exact for a suitable choice of the divisor, as justified next. ◻

Proof. Choose \(m\) so large that \((m-1)H-K_X\) is ample, \(H^1(X,\mathcal O_X((m-1)H))=0\), and \(m\ge\mu(K_S)+1\). The last slope is independent of \(S\), by adjunction \(K_S=(K_X+H)|_S\). Kodaira vanishing on \(S\) (Demailly 2012, VII, Theorem 3.3) gives the first equality in (49), and the restriction sequence on \(X\) gives surjectivity on sections.

Bertini’s theorem supplies a smooth pair meeting transversely. Shrink an analytic neighborhood of that pair so that its closure is compact and all intersections remain smooth. The universal surfaces and curves over this closure are smooth proper families. Local trivializations on a finite cover give uniform bounds for the metrics, their derivatives, coordinate comparisons, and the partitions of unity used below. Surjectivity on sections allows the curve on each chosen surface to vary in an analytic open subset of its complete linear system.

For a fixed coherent sheaf, a general defining section avoids all its associated points, and is thus a nonzerodivisor. Applying this to the quotients in a finite collection of exact sequences preserves their exactness under restriction. To ensure torsion-freeness, embed a torsion-free sheaf into a vector bundle and require the defining section to be a nonzerodivisor on its cokernel. Restriction then embeds the restricted sheaf into the restricted bundle. Such an embedding exists because a sufficiently positive twist of the dual is globally generated and the original sheaf injects into its double dual. Repeating the argument on the surface gives the curve assertion. Applied to the ordinary cohomology sheaves of a bounded complex, the nonzerodivisor condition kills the first Tor terms in the restriction spectral sequence, giving the asserted cohomology identification.

These are finitely many nonempty Zariski-open conditions at each stage. A very-general condition removes at most countably many proper algebraic closed subsets. Each such subset has empty analytic interior, so the Baire theorem ensures a choice in every one of our nonempty analytic neighborhoods. The geometric constants were fixed before this choice and do not depend on the sheaves being restricted. ◻

Proof. The uniform charts and partitions of unity give the Nash inequality \[ \|f\|_2^{2+4/n} \le C\bigl(\|df\|_2^2+\|f\|_2^2\bigr)\|f\|_1^{4/n}. \tag{51}\] For completeness, in a Euclidean chart split the Fourier integral at frequency \(R\): its low-frequency part is bounded by \(CR^n\|f\|_1^2\), and its high-frequency part by \(R^{-2}\|df\|_2^2\). Optimize in \(R\), and use a fixed finite partition of unity. Derivatives of the partition produce the additional \(\|f\|_2^2\) term. All comparison constants are uniform over the chosen families.

Set \(L_{\mathrm{sc}}=\Delta_{\mathrm{sc}}+1\) and \(f_s=e^{-sL_{\mathrm{sc}}}f\). The scalar semigroup is positivity preserving and contracts \(L^1\). For \(y(s)=\|f_s\|_2^2\), (51) and \(y'=-2\langle L_{\mathrm{sc}}f_s,f_s\rangle\) imply \[y'\le-c\|f\|_1^{-4/n}y^{1+2/n}.\] Integration gives \(\|e^{-sL_{\mathrm{sc}}}\|_{1\to2}\le Cs^{-n/4}\). Selfadjointness and composition give \(\|e^{-sL_{\mathrm{sc}}}\|_{1\to\infty}\le Cs^{-n/2}\) for every \(s>0\). Multiplication by \(e^s\) gives the stated estimate for \(k_s\) when \(s\le1\).

For a section-valued heat solution, Kato’s inequality and the maximum principle give \[|e^{-sL}v|(x)\le(e^{-sL_{\mathrm{sc}}}|v|)(x).\] One obtains the differential inequality by differentiating \((|v|^2+\varepsilon^2)^{1/2}\) and then letting \(\varepsilon\downarrow0\); compatibility of the connection with the metric is the only bundle input. Approximate a vector-valued point mass at \(y\) in this inequality. This bounds the operator norm of \(K_s(x,y)\) by the scalar kernel of \(e^{-sL_{\mathrm{sc}}}\), and proves (50). ◻

Proof. On a curve \(G\) is locally free. On a surface its reflexive hull \(W=G^{**}\) is locally free and slope-stable: intersecting a subsheaf of \(W\) with \(G\) preserves rank and first Chern class, since \(W/G\) is zero-dimensional. The Hermitian–Einstein theorem (Donaldson 1985; Uhlenbeck and Yau 1986) therefore gives a metric on \(W\) for which \(i\Lambda F_W\) is the scalar determined by \(\mu(G)\). Its proportionality constant is fixed by the dimension and our volume normalization.

For a holomorphic section \(s\) of \(W\), the Bochner identity gives \[\Delta_{\mathrm{sc}}|s|^2\le C\mu(G)_+|s|^2.\] Scalar heat comparison consequently yields \(|s(x)|^2\le e^{C u\mu(G)_+}(e^{-u\Delta_{\mathrm{sc}}}|s|^2)(x)\). Take \(u=(1+\mu(G)_+)^{-1}\) and apply Lemma 24. The evaluation operator, with the \(L^2\) norm on \(H^0(Y,W)\), satisfies \[ \|\operatorname{ev}_x\|_{\mathrm{op}}^2 \le C\bigl(1+\mu(G)_+\bigr)^{\dim Y}. \tag{53}\] Give \(H^0(Y,J)\subset H^0(Y,W)\) the induced \(L^2\) norm and choose an orthonormal basis \(s_1,\ldots,s_N\). Outside a proper analytic subset, their evaluations lie in a subspace of dimension at most \(q\). Thus \[\sum_{j=1}^N|s_j(x)|^2 =\|\operatorname{ev}_x|_{H^0(J)}\|_{\mathrm{HS}}^2 \le q\|\operatorname{ev}_x\|_{\mathrm{op}}^2.\] Integration proves (52); the volumes are uniformly bounded. The case \(q=0\) means \(J=0\). Dualization and tensoring with a line preserve slope stability, so the same proof gives the last assertion. ◻

Proof. The compact positive operator \(L^{-1/2}PL^{-1/2}\) has at least \(\dim U\) eigenvalues greater than or equal to \(1\): apply min–max to the subspace \(L^{1/2}U\) and (54). Its nonzero eigenvalues, with multiplicity, are those of \(P^{1/2}L^{-1}P^{1/2}\). Define an operator with an additional time variable by \[(\mathcal Bv)(s)=e^{-sL/2}P^{1/2}v, \qquad \mathcal B:L^2(S,\mathcal V)\longrightarrow L^2((0,\infty)\times S,\mathcal V).\] Integration of the heat semigroup gives \(\mathcal B^*\mathcal B=P^{1/2}L^{-1}P^{1/2}\).

Fix \(0<\eta<1\) and split \(\mathcal B=\mathcal B_{\le}+\mathcal B_>\) by inserting the fiberwise spectral projections \(\mathbf1_{sP\le\eta}\) and \(\mathbf1_{sP>\eta}\) immediately after \(P^{1/2}\). Heat contraction gives \[\|\mathcal B_{\le}v\|^2 \le\int_0^\infty\langle P\mathbf1_{sP\le\eta}v,v\rangle\,ds \le\eta\|v\|^2.\] Indeed, each positive eigenvalue \(p\) contributes \(\int_0^{\eta/p}p\,ds=\eta\), and a zero eigenvalue contributes zero.

Let \(K_s\) be the kernel of \(e^{-sL}\). The semigroup identity and Lemma 24, in real dimension four, imply \[\begin{align*} \|\mathcal B_>\|_{\mathrm{HS}}^2 &=\int_0^\infty\int_S \operatorname{tr}\bigl(P\mathbf1_{sP>\eta}K_s(x,x)\bigr)\,dx\,ds\\ &\le C\int_S\int_0^\infty s^{-2}\operatorname{tr}\bigl(P\mathbf1_{sP>\eta}\bigr)\,ds\,dx\\ &=C\eta^{-1}\int_S\operatorname{tr}(P^2). \end{align*}\] The last identity follows eigenvalue by eigenvalue from \(\int_{\eta/p}^\infty s^{-2}p\,ds=p^2/\eta\). All integrands are nonnegative, so Tonelli’s theorem applies.

On the spectral subspace where \(\mathcal B^*\mathcal B\ge1\), the triangle inequality gives \(\|\mathcal B_>v\|\ge(1-\sqrt\eta)\|v\|\). Apply this to an orthonormal basis of that subspace and sum. Taking \(\eta=1/4\) proves (55). ◻

Proof. We prove the bound for one factor \(G\), writing \(r=\mathop{\mathrm{rk}}G\), \(C_1^{\mathrm{num}}=c_1(G)/r\), and \(\rho=\|C_1^{\mathrm{num}}-T\|\). We separate \(\rho\ge1\), where the numerical error already controls rank, from \(\rho<1\), where the curvature argument is needed.

Suppose first that \(\rho\ge1\). The class \(T\) is uniformly bounded once \(A\) is fixed, so \(1+\|C_1^{\mathrm{num}}\|^2\le C_A\rho^2\). Lemma 25, applied to \(G(tH)\), gives \(h^0(G(tH))\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2)\). Put \(W=G^{**}\). Serre duality identifies \(H^2(G(tH))^*\) with \(\mathop{\mathrm{Hom}}(G,K_S(-tH))=H^0(W^*\otimes K_S(-tH))\); the equality follows by extending across the finite support of \(W/G\). The dual version of Lemma 25 gives the same bound for \(h^2\).

Surface Riemann–Roch reads \[\chi(G(tH)) =r\chi(\mathcal O_S)+\frac{hr}{2} \left(\langle C_1^{\mathrm{num}}+tH, C_1^{\mathrm{num}}+tH\rangle-\xi_G\right) -\frac r2(C_1^{\mathrm{num}}+tH)K_S.\] The Hodge norm controls both the intersection form and pairing with \(K_S\). Consequently \(|\chi(G(tH))|\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2+\xi_G)\). Since \(I\) is finite, \(h^1=h^0+h^2-\chi\) is at most \(Cr(\rho^2+\xi_G)\), as required.

Now assume \(\rho<1\). The hull sequence \(0\to G\to W\to Q\to0\) has \(Q\) zero-dimensional and \[ \xi_G-\xi_W=\frac{2\operatorname{length}Q}{hr}, \qquad \operatorname{length}Q\le\frac{hr\xi_G}{2}. \tag{57}\] Here \(W\) is stable, as in the proof of Lemma 25, and \(\xi_W\ge0\) by the classical Bogomolov–Gieseker inequality. The long exact cohomology sequence gives \(h^1(G(tH))\le h^1(W(tH))+\operatorname{length}Q\). It remains to bound \(h^1(W(tH))\) by \(Cr\xi_W\).

Equip \(W\) with its Hermitian–Einstein metric. Write its Chern curvature as \[F_W=F_0+\frac{\operatorname{tr}F_W}{r}\operatorname{id}_W, \qquad \vartheta=\frac{i}{2\pi r}\operatorname{tr}F_W.\] The real \((1,1)\)-form \(\vartheta\) is closed and has constant contraction, hence is harmonic. It represents \(C_1^{\mathrm{num}}\). The Hodge star formula on primitive \((1,1)\)-forms identifies its \(L^2\) norm with a fixed multiple of the divisor norm. Uniform elliptic estimates on our compact smooth families therefore give a uniform bound for the supremum norm of a harmonic representative in terms of this divisor norm. In particular, since \(\rho<1\) and \(|b|\le1\), \[ -\vartheta-t\omega_S\ge(A-M-C_1)\omega_S \qquad(t\le M). \tag{58}\] The fixed constant \(C_1\) bounds the harmonic representatives of \(B_0|_S+bH\) and the unit ball of errors. Here \(\omega_S\) is the induced Kähler form, and the line bundle \(\mathcal O_S(H)\) has curvature \(2\pi\omega_S\). Choose \(A_0\) so that the right side is strictly positive with a fixed lower bound. This choice uses only the upper endpoint \(M\), not the lower endpoint of \(I\).

The trace-free curvature \(F_0\) is primitive. Chern–Weil and the primitive Hodge star formula give \[ \int_S|F_0|_{\mathrm{HS}}^2\le C\frac{\operatorname{Disc}(W)}r =Chr\xi_W. \tag{59}\] Indeed \(\operatorname{Disc}(W)=-r\int_S \operatorname{tr}((iF_0/2\pi)\wedge(iF_0/2\pi))\), and the last integrand is the negative squared norm because \(F_0\) is primitive. The fixed conversion constant depends on the curvature convention, not on \(r\).

By Serre duality and the Dolbeault theorem, \(h^1(W(tH))\) is the dimension of the space \(U\) of harmonic \((2,1)\)-forms with values in \(W^*(-tH)\). On this bundle the Bochner–Kodaira–Nakano identity (Demailly 2012, VII, Corollary 1.3) is \[ \Delta''=\Delta'+[iF,\Lambda]. \tag{60}\] For the central curvature, (58) makes the commutator on \((2,1)\)-forms at least \(\lambda I\) for a fixed \(\lambda>0\). To see the sign directly, diagonalize the central \((1,1)\)-form in a unitary frame. On top holomorphic degree, its commutator is the sum of the positive eigenvalues whose antiholomorphic indices occur; there is one such index here. Denote the remaining selfadjoint curvature endomorphism by \(K_0\). Wedge and contraction act on fixed-dimensional form spaces, so \[ |K_0|_{\mathrm{HS}}\le C|F_0|_{\mathrm{HS}}. \tag{61}\]

We explain explicitly how to retain an elliptic operator in (60). Let \(\nabla\) be the total metric connection on \((2,1)\)-forms with these bundle coefficients. For a pure-type form the terms in \(\partial u+\bar\partial u\) have different types, as do their adjoints. Hence the quadratic form of the connection de Rham Laplacian is the sum of the forms of \(\Delta'\) and \(\Delta''\). On \(U\), the latter vanishes, so the de Rham form equals the \(\Delta'\) form. The Weitzenböck formula, restricted to this type, consequently gives \[ \langle\Delta'u,u\rangle \ge\|\nabla u\|_2^2-C_2\|u\|_2^2+\langle R_0u,u\rangle \qquad(u\in U), \tag{62}\] where \(R_0\) is selfadjoint and \(|R_0|_{\mathrm{HS}}\le C|F_0|_{\mathrm{HS}}\). The bounded term contains the tangent curvature and the central bundle curvature; its bound \(C_2\) is uniform for fixed \(A,I\). The formula follows by anticommuting wedge and contraction in \(d_\nabla\) and \(d_\nabla^*\): the second-order part is \(\nabla^*\nabla\), and the remaining terms contract the tangent and bundle curvature. Only the fixed form dimension enters the contraction constants.

Choose \(0<\varepsilon\le\min\{1,\lambda/(C_2+1)\}\). On \(U\), retain \(\varepsilon\Delta'\) in (60), discard the nonnegative remainder, and use (62). This gives \[\varepsilon\|\nabla u\|_2^2+ (\lambda-\varepsilon C_2)\|u\|_2^2 \le\langle(|K_0|+\varepsilon|R_0|)u,u\rangle.\] Our choice ensures \(\lambda-\varepsilon C_2\ge\varepsilon\). Thus, with spectral absolute values of selfadjoint matrices, \[L=\nabla^*\nabla+1,\qquad P=\varepsilon^{-1}|K_0|+|R_0|, \qquad \langle Lu,u\rangle\le\langle Pu,u\rangle\quad(u\in U).\] Moreover, \[ \operatorname{tr}(P^2) \le2\varepsilon^{-2}|K_0|_{\mathrm{HS}}^2 +2|R_0|_{\mathrm{HS}}^2 \le C|F_0|_{\mathrm{HS}}^2. \tag{63}\] This uses the Hilbert–Schmidt norm of the matrices, without replacing them by their operator norms times the identity.

Lemma 26, followed by (59), now proves \(h^1(W(tH))=\dim U\le Cr\xi_W\). Together with (57), this gives the required bound for the factors with \(\rho<1\) as well. Sum over the factors. For an extension \(0\to G'\to G\to G''\to0\), the cohomology sequence gives \(h^1(G(tH))\le h^1(G'(tH))+h^1(G''(tH))\); induction proves the final filtration assertion. ◻

Proof. For each line bundle in the fixed list \(\mathcal O_X(\pm D),\mathcal O_X(\pm H),K_X^{\pm1}\), choose three successive surjections from sums of negative powers of \(\mathcal O_X(H)\), with locally free kernels. They exist by global generation after twisting; a surjection onto a locally free sheaf has locally free kernel. Spread these finitely many sequences and a regularity bound for \(\mathcal O_X\) to a model, and shrink so that they remain exact on the fibers.

There is consequently an integer \(v_0\), independent of the fiber, such that tensoring by any one line bundle in the list increases by at most \(v_0\) a threshold above which all higher cohomology vanishes. Indeed, tensor its three-step presentation by the original line bundle and use the cohomology sequences. The three dimension shifts end in cohomological degree greater than three. If the finitely many twists in the presentation are bounded by \(v_0\), every intervening sum of lines has vanishing higher cohomology above the shifted threshold. Starting with the fixed bound for \(\mathcal O_X\) proves the claim by induction on the number of tensor factors.

The line \(K_X^{1-p}\otimes L^{-1}\) is a product of at most \(c_0p+c_1\) members of this fixed list, uniformly for \(|b|\leq1\) and for both choices of \(L\). It therefore has no higher cohomology after twisting by \(sH\) whenever \(s\geq c_2p+c_3\). Projection formula gives \[H^i\bigl(G_{p,L}(j-i)\bigr) =H^i\bigl(K_X^{1-p}\otimes L^{-1}(p(j-i))\bigr)=0 \quad(1\leq i\leq3)\] for a fixed sufficiently large \(j\). Castelnuovo–Mumford regularity then gives the evaluation surjection in (66), using a basis of \(H^0(G_{p,L}(j))\), and gives surjectivity of multiplication of these sections at every higher twist.

The cohomology sequence for its kernel gives a uniform regularity bound \(j_2\). For degrees at least two use regularity of \(G_{p,L}\) and of \(\mathcal O_X(-j)\); in degree one use the just-proved multiplication surjectivity. Increasing \(j_2\) makes the kernel generated at \(j_2\). Finally, the higher cohomology of \(K_X^{1-p}\otimes L^{-1}(pj)\) vanishes, so \[N_{p,L}=\chi\bigl(K_X^{1-p}\otimes L^{-1}(pj)\bigr).\] This Euler characteristic is constant under specialization for each of the indicated line bundles. On the complex fiber, Riemann–Roch is a polynomial of degree at most three in exponents bounded by a fixed multiple of \(p\). This proves \(N_{p,L}\leq C_0p^3\). All the choices preceded any choice of a tilt object. ◻

Proof. Finite duality and projection formula give \[ h^0(F^*P_1\otimes L)=\dim\mathop{\mathrm{Hom}}(G_{p,L},P_1). \tag{68}\] Indeed \(F^!P_1=F^*P_1\otimes K_X^{1-p}\), so adjunction applied to \(K_X^{1-p}\otimes L^{-1}\) proves the identity. The semilinearity of absolute Frobenius does not change these dimensions. Apply (66). The obstruction to injectivity of \[\mathop{\mathrm{Hom}}(G_{p,L},P_1)\longrightarrow \mathop{\mathrm{Hom}}(\mathcal O_X(-j)^{N_{p,L}},P_1)\] is the image of \[\mathop{\mathrm{Hom}}(K_{p,L}[1],P_1) =\mathop{\mathrm{Hom}}(K_{p,L},\mathcal H^{-1}(P_1)).\] The equality uses the ordinary cohomological range of \(P_1\). Generation of \(K_{p,L}(j_2)\) and the assumed vanishing make this group zero. The Hom map is therefore injective, and its target has dimension \(N_{p,L}h^0(P_1(j))\leq C_0p^3h^0(P_1(j))\). ◻

Proof. Put \(N=\dim W\) and take the tilt image \(I\) and kernel \(K'\) of the evaluation map \(\mathcal O_X(-tH)^N\to P\) at the side point. We first control the rank, first character, and \(H\)-degree of the second character of \(I\) by \(e\), independently of \(N\). We then use the evaluation kernel, whose rank is close to \(N\), to cancel all but \(O(e)\) of the sections in the evaluation source. Ordinary cohomology gives a sheaf \(K'\) and an exact sequence \[ 0\longrightarrow V_I\longrightarrow K' \longrightarrow\mathcal O_X(-tH)^N\longrightarrow U_I\longrightarrow0, \qquad V_I\subset V_P, \tag{82}\] where \(V_I=\mathcal H^{-1}(I)\), \(U_I=\mathcal H^0(I)\) and \(V_P=\mathcal H^{-1}(P)\). Since \(I\subset P\), \(0\leq d(I)\leq d(P)=O(e)\). The sheaf \(U_I(t)\) is globally generated. Its first Chern class has nonnegative \(H\)-degree, including its divisorial torsion, and \(H^2B_0=0\); hence \[d(U_I)\geq u r(U_I),\qquad d(V_I)\leq0.\] Thus \(r(U_I)=O(e)\). The inclusion into \(V_P\) gives \(r(V_I)=O(e)\) as well.

In the positive decomposition \(I\subset P\subset E\), so \(n(I)\leq c d(I)\). In the negative decomposition \(P/I\) is a quotient of \(E\) and satisfies \(n(P/I)\geq c d(P/I)\), also when its denominator is zero. Additivity and the numerical bounds on \(P\) therefore give \(n(I)\leq O(e)\) in both cases.

Every finite Harder–Narasimhan slope of \(I\) is at least \(-C_1\). Indeed, its last factor receives a nonzero map from one of the lines in the evaluation map. The tilt image of this map is a quotient of that line and a subobject of the last semistable factor; the line comparison forces the claimed lower bound. If the last factor has infinite slope there are no finite factors. Denominator-zero factors have nonnegative numerator. Writing \(d_i,n_i\) for the factors, we obtain \[\sum_i|n_i|\leq n(I)+2C_1d(I)=O(e).\] Lemma 18, applied at the fixed side volume and summed over the filtration, now gives \[\|c_{B_s}(I)\|+|w_{B_s}(I)|=O(e).\] The twist change to \(I(t)\) is bounded. Its rank is \(O(e)\) by (82), so \[ \|c_0(I(t))\|+|w_0(I(t))|=O(e). \tag{83}\]

The sheaf \(K'(t)\) is torsion-free and has rank \(N+O(e)\). Before twisting it belongs to the positive part of the torsion pair, while (82) bounds its maximum slope by \(-t\). Its slopes after twisting therefore lie in \((-u,0]\). Its first and second characters are the negatives of those in (83). For its refined ordinary slope factors this proves \[ \sum_i r_i\mu_i^2+\sum_i r_i\delta_i^0=O(e). \tag{84}\] In detail, the bounded nonpositive slope interval and the total degree \(O(e)\) bound the first sum; subtracting the total \(2w_0\) bounds the second.

The next step is to obtain almost as many sections of this kernel on a surface as the trivial evaluation source has. Its small first character and projected second character control the Riemann–Roch error; the same numerical bounds will control the cohomology lost from the negative term \(V_I\).

Choose a very general \(S\) in the fixed neighborhood and a very general curve \(C_m\in|mH|_S\) in its fixed neighborhood. Impose also the finitely many open conditions that all the sheaf sequences and filtrations used here restrict exactly, with torsion-free restrictions where required. Such choices are possible by Lemma 23. Intersect the refined restriction filtrations of \(K'(t)\) with \(V_I(t)\). The successive intersections are subsheaves of the stable restriction factors and have total rank \(O(e)\). By Lemma 22, the second slope moments on both \(S\) and \(C_m\) are \(O(e)\). Applying Lemma 25 to the intersections, after the additional fixed twist \(m\), yields \[ h^0\bigl(V_I|_S(t+m)\bigr)+ h^0\bigl(V_I|_{C_m}(t+m)\bigr)=O(e). \tag{85}\] The reason this has no \(N\) term is that each bound is a constant times the rank of the intersection plus its rank times \(\mu_{i,+}^2\); its rank is at most \(r_i\), and the second moments in (84) control the sum.

We also have \(h^2(K'|_S(t+m))=O(e)\). A stable factor of \(K'(t)|_S\) with slope \(\mu_i>\mu(K_S)-m\) has no homomorphism to \(K_S(-m)\). For all the other factors, (69) gives \(\mu_i\leq-1\), so their total rank, as well as their second slope moment, is \(O(e)\). Serre duality and Lemma 25 applied to their dual hulls, twisted by \(K_S(-m)\), bound their \(h^2\) by \(O(e)\). Surface Riemann–Roch and (83) give \[\chi\bigl(K'|_S(t+m)\bigr) =r(K')\chi(\mathcal O_S(m))+O(e).\] Indeed the error consists of \(w_0(K'(t))\) and the intersection of \(c_0(K'(t))\) with the fixed class \(mH-K_S/2\); here \(K_S=(K_X+H)|_S\). Consequently \[ h^0\bigl(K'|_S(t+m)\bigr) \geq r(K')h^0(\mathcal O_S(m))-O(e). \tag{86}\]

Restrict the triangle \(K'\to\mathcal O_X(-tH)^N\to I\) and twist by \((t+m)H\). The kernel on \(H^0\) from \(K'|_S(t+m)\) has dimension at most \(h^{-1}(I|_S(t+m))=h^0(V_I|_S(t+m))\). Equations (85) and (86), with \(r(K')=N+O(e)\), therefore give \[\begin{aligned} &\dim\mathop{\mathrm{im}}\!\left( H^0(K'|_S(t+m))\longrightarrow H^0(\mathcal O_S(m))^N \right)\\ &\hspace{25mm}\geq N h^0(\mathcal O_S(m))-O(e). \end{aligned}\] The source of the next map has dimension \(Nh^0(\mathcal O_S(m))\). Subtracting the displayed lower bound shows that the image of \[H^0(\mathcal O_S(m))^N\longrightarrow\mathbb H^0(I|_S(t+m))\] has dimension \(O(e)\). This is the cancellation which removes the unrestricted dimension \(N\) from the estimate.

It remains to remove the auxiliary twist \(m\). Multiplication by the section defining \(C_m\) gives a map from \(\mathbb H^0(I|_S(t))\) to this latter group. Its kernel has dimension at most \(h^{-1}(I|_{C_m}(t+m))\), which is again \(O(e)\) by (85). The image of \(H^0(\mathcal O_S)^N\) in \(\mathbb H^0(I|_S(t))\) consequently has dimension \(O(e)\). Composing with \(I\to P\) proves the assertion for \(W\). All equalities involving \(h^{-1}\) use derived restriction with nonzerodivisors on the ordinary cohomology sheaves; no exactness of a tilt-image operation under restriction was assumed. ◻

Proof. For the positive decomposition, apply Lemma 31 to all of \(H^0(P(t))\) and the triangle \(P(t-1)\to P(t)\to P|_S(t)\). The increase in \(h^0\) is at most the dimension of the restriction image, at most \(C_2e\). Telescope over the fixed set of integers \(t_0<t\leq j\), starting with (81).

For the negative decomposition, apply the lemma to the image of \(H^0(E(t))\) in \(H^0(P(t))\). The kernel of \[\mathbb H^0(E|_S(t))\longrightarrow\mathbb H^0(P|_S(t))\] has dimension at most \(h^1(V_0|_S(t))=O(e)\) by (76) and (78). Thus the restriction image of \(H^0(E(t))\) itself has dimension \(O(e)\). Telescope the restriction triangle for \(E\) from the same vanishing at \(t_0\). Finally every slope of the negative cohomology sheaves in (88), after twisting by \(j_2\), is at most \(b-A+j_2<0\), by (70). They have no sections. ◻

Proof. Embed \(G\) in \(\mathcal O_Y(u)^k\) by choosing generators for a twist of its dual and using the injection \(G\to G^{**}\). Spread this injection and make its cokernel flat, so it remains an injection on the fibers. We describe a finite-type set containing every destabilizing saturated subsheaf on every geometric fiber.

For a saturated subsheaf \(J\) of rank \(s\), \(0<s<r(G)\), its generic Plücker coordinates extend across codimension two to sections of \(\mathcal O_Y(su)\otimes(\det J)^{-1}\). If \(J\) destabilizes, the degree of the zero divisor of a nonzero coordinate is bounded above by a number depending only on \(u,s\) and the degree of \(G\). An effective divisor of degree at most \(d_0\) is contained, with its multiplicities, in the zero divisor of a section of \(\mathcal O_Y(v)\) with \(v\leq\max(1,d_0)\). To see this, choose a finite linear projection to projective space of dimension \(\dim Y\). Each prime component has hypersurface image of degree at most its own degree. Pull back an equation for each such image and multiply with the required multiplicities. The projection is finite because its pullback of \(\mathcal O(1)\) is ample.

Divide the Plücker tuple by one nonzero coordinate, then multiply by this section to clear its pole divisor. Normality extends the resulting tuple to global sections of \(\mathcal O_Y(v)\). The saturated subsheaf \(J\) is the kernel, inside \(G\), of wedging with this tuple, viewed as a map to a fixed sum of \(\mathcal O_Y(u+v)\): the kernels agree at the generic point and are saturated. The finitely many possible values of \(s,v\) give finite-type spaces of tuples after shrinking the base so that their section spaces commute with base change. On each parameter space, flatten the cokernel of the universal wedge map. The kernel then commutes with taking fibers, and its rank and degree are constructible, as read from its Hilbert polynomial. Hence the condition that some tuple gives a kernel of smaller positive rank and larger slope than \(G\) is constructible on the base. Allowing tuples not satisfying the Plücker equations does not cause a problem: a kernel with those numerical properties is itself a destabilizing subsheaf.

This constructible subset excludes the generic point. Otherwise a destabilizer over its algebraic closure would extend to one over \(\mathbb C\). A constructible subset of an integral noetherian scheme whose closure is the whole scheme contains a nonempty open set and hence the generic point. Its closure is therefore proper; removing that closure proves the assertion. ◻

Proof. We use the Cartier-descent argument of (Langer 2022, Theorem 4.1 and Proposition 4.2), keeping the chosen cotangent twist in the slope normalization. Generation of a fixed positive twist of the tangent bundle gives an embedding \(\Omega_Y^1\hookrightarrow\mathcal O_Y(c_Y)^k\) with locally free cokernel. It can be chosen on the model and retained on its fibers. Frobenius is finite flat because the fiber is smooth over a perfect field.

Suppose two consecutive Harder–Narasimhan slopes of \(F^*G\) differ by more than \(c_Y\). For the truncation \(J\) above that gap, the second fundamental map for the canonical connection is \[J\longrightarrow(F^*G/J)\otimes\Omega_Y^1.\] It is \(\mathcal O_Y\)-linear. Its source has minimum slope larger than the maximum slope of its target, the latter bounded using the displayed embedding, so the map is zero. The connection thus preserves \(J\). The canonical connection has zero \(p\)-curvature, as does its restriction to \(J\). Cartier descent therefore descends this inclusion to a subsheaf of the Frobenius twist of \(G\). Its slope is \(\mu(J)/p>\mu(G)\), contradicting semistability. Here the Frobenius twist of the algebraically closed base field is an automorphism and does not change the slope assertion. Thus no adjacent gap is greater than \(c_Y\). There are at most \(r\) factors, so the maximum slope is at most the mean plus \(c_Y(r-1)\), as claimed. ◻

Proof. We first compute along the slope-zero curve of a fixed class. Its positive-denominator branch is \[ d(a)=\sqrt{\Delta+a^2r^2},\qquad b(a)=\frac{d_{B_0}-d(a)}r\quad(r\ne0). \tag{103}\] For rank zero, \(d\) and \(b\) remain constant. If the branch starts at a semistable object, then \(\Delta\ge0\), and its denominator stays positive at every positive \(a\). Indeed \(d(a)\ge a|r|\) when \(r\ne0\), whereas \(d\) is a positive constant when \(r=0\). The identities \(dz_b/db=-w_b\) and \(w_b=a^2r/2\) give \[ D_a'=-\frac a3S_a,\qquad S_a'=-\frac{ar^2}{d(a)^2}S_a. \tag{104}\] For example, when \(r\ne0\) one has \(d'=ar^2/d\), \(b'=-ar/d\), and \(z'=a^3r^2/(2d)\); these give both equalities directly. The rank-zero case follows from the definitions. For a differentiable nonnegative function \(k\), therefore, \[ (D_a-k(a)S_a)'= S_a\left(-\frac a3-k'(a)+k(a)\frac{ar^2}{d(a)^2}\right). \tag{105}\]

There are three comparisons. For the corrected bound below \(A\), take \(k(a)=C+(A^2-a^2)/6\) and move upward toward \(A\). The right side of (105) is \(k(a)ar^2S_a/d(a)^2\ge0\), so a strict violation persists upward. To rule out positive defect at a starting value \(R>R_0\), take \(k(a)=a(R-a)/3\) on \([A,R]\) and move downward. Since \(ar^2/d(a)^2\le1/a\), \[-\frac a3-k'(a)+k(a)\frac{ar^2}{d(a)^2} \le-\frac a3-k'(a)+\frac{k(a)}a=0.\] A strict violation again persists in the direction of movement. Here \(k(R)=0\) and \(k(A)=A(R-A)/3>C\). For the corrected bound in the remaining interval, on any interval \([A,R]\) use \(k(a)=Ca/A\) and move downward. The same upper bound for the last term gives a derivative at most \(-aS_a/3\le0\), and \(k(A)=C\).

Each comparison preserves a strict violation while the class remains stable. We next show how to continue through a change of stability, with a strict decrease of a discrete discriminant at each replacement. For an equal-slope-zero filtration with stable factors \(E_i\), all \(d_i\) are positive and \(w_i=a^2r_i/2\). Additivity and weighted Cauchy–Schwarz give \[ D_a(E)=\sum_iD_a(E_i),\qquad S_a(E)=\sum_i d_i-a^2\frac{(\sum_i r_i)^2}{\sum_i d_i} \ge\sum_iS_a(E_i). \tag{106}\] Thus a strict inequality \(D_a(E)>k(a)S_a(E)\), with \(k(a)\ge0\), passes to at least one stable factor. Such a factorization is finite by Lemma 19.

Start with a violating stable factor and follow its branch (103) toward \(A\) as long as it remains stable. Stability is open. At an endpoint in the positive \(a\)-interval, continuity of extremal phases gives semistability, since the formal denominator remains positive. The strict violation persists at the endpoint: along the movement \(D-kS\) is at least its initial positive value. If the endpoint is \(A\), this contradicts (101). Otherwise the endpoint is nonstable, since stability there would extend the interval, and we split again and select a violating stable factor.

At every such genuine wall, the discriminant of each factor is strictly smaller than that of its parent. Indeed, \[ \Delta(E)=\sum_i\Delta(E_i)+ 2\sum_{i<j}(d_i d_j-a^2r_i r_j). \tag{107}\] Every term is nonnegative because \(d_i\ge a|r_i|\). If all cross terms vanished, all \(r_i\) would be nonzero with the same sign and \(d_i=a|r_i|\); hence all factors would have \(\Delta(E_i)=0\) and proportional triples \((r_i,d_i,w_i)\). Twisting preserves this proportionality. Openness keeps the finitely many stable factors stable near the wall, so their fixed extension triangles make \(E\) an extension of objects with equal finite slope immediately before the wall as well. The filtration has at least two factors, so this contradicts the preceding stability of \(E\). Some cross term in (107) is therefore positive, and every factor has smaller discriminant.

Because \(B_0\) is rational, formula (41) puts all these discriminants in \(N^{-1}\mathbb Z\) for one positive integer \(N\) depending only on \(X,H,B_0\). They are nonnegative on the semistable factors. Strict descent consequently permits only finitely many wall splits. This also excludes an accumulating sequence of walls with changing factors: every change consumes at least \(1/N\) of discriminant, and every final stable interval either reaches \(A\) or has an interior semistable endpoint at which another such split is necessary. There is no denominator-zero endpoint inside the interval, by (103). The proposed violation must therefore reach \(A\), giving a contradiction in each of the three comparisons.

The first comparison proves \(D_a\le(C+(A^2-a^2)/6)S_a\) for \(0<a\le A\). The second proves \(D_a\le0\) for \(a>R_0\). The third proves \(D_a\le(Ca/A)S_a\) for \(A\le a\le R_0\). Finally use \(S_a\le d_b\) and (102). ◻

Proof of Theorem 2. First use the very-ample and transverse-twist reductions of Section 4. Proposition 28 supplies \(A>0\) and \(C\ge0\), chosen independently of \(E\) and \(b\), such that \[D_A(E)\le C\bigl(d_B(E)-A|r(E)|\bigr).\] At slope zero put \(e=d_B-A|r|\). Since \(d_B\ge A|r|\), \[S_A=\frac{(d_B-A|r|)(d_B+A|r|)}{d_B},\qquad e\le S_A\le2e.\] The hypothesis of Proposition 35 follows. Multiplying its conclusion by \(h\) gives \[\mathop{\mathrm{ch}}_3^B(E)\le\frac{a^2}{6}H^2\mathop{\mathrm{ch}}_1^B(E) +k_0H^2\mathop{\mathrm{ch}}_1^B(E),\] and gives the same inequality without the correction for \(a>R_0\). Undo the polarization rescaling as described in Section 4. This gives constants \(k\ge0\) and \(R_*>0\) for the original \(X,H,B_0\), uniform in both \(E\) and \(b\).

Enlarge \(k\) to a rational nonnegative number. The curve class \(\Gamma=kH^2\in N_1(X)_{\mathbb Q}\) then satisfies \(\Gamma\cdot H=kh\ge0\), and the correction is exactly \(\Gamma\cdot\mathop{\mathrm{ch}}_1^B(E)\). This proves the corrected statement with its required rational class, as well as the independent uncorrected large-volume threshold. No canonical-bundle hypothesis has entered either argument. ◻

Proof of Corollary 3. Take the threshold of Theorem 2 with \(B_0=0\). Set \(b=\beta\) and \(a=\sqrt{2\alpha-b^2}\). The numerator in (108) is \(h(w_b-a^2r/2)\) and its denominator is \(hd_b\), so its slope is exactly \(\nu_{a,b}\). Expanding (109) with (34)–(35) gives \[ \frac{Q_{\alpha,b}(v(E))}{h^2} =\frac{a^2\Delta(E)}2+2w_b(E)^2-3d_b(E)z_b(E). \tag{110}\] If \(d_b(E)=0\), this is nonnegative by (42); it includes zero-dimensional objects.

Otherwise put \(u=\nu_{a,b}(E)\), \(c=b+u\), and \(R=\sqrt{a^2+u^2}\). Lemma 20 makes \(E\) semistable of slope zero at \((R,c)\), with \(d_c>0\). Using \(w_b=ud_b+a^2r/2\), \[d_c=d_b-ur,\qquad z_c=z_b-uw_b+\tfrac12u^2d_b-\tfrac16u^3r\] in (110) gives the exact identity \[ \frac{Q_{\alpha,b}(v(E))}{h^2} =-3d_b(E)\left(z_c(E)-\frac{R^2d_c(E)}6\right). \tag{111}\] Since \(\alpha>f(b)\) implies \(R\ge a>R_*\), the expression in parentheses is nonpositive by the uncorrected tail. This proves the claim. The passage uses the entire semistability arc and the inequality \(R\ge a\); an all-parameter equivalence alone would not justify a conclusion on this truncated region. ◻

Proof. Write \(\ell=H|_D\), \(s=\ell^2=D^3>0\), \(\rho=\mathop{\mathrm{rk}}(F)\), and \(C=B_0|_D\). Put \[u=\mathop{\mathrm{ch}}_1^C(F),\qquad v=\mathop{\mathrm{ch}}_2^C(F),\qquad \mu=\frac{\ell u}{\rho s},\qquad \delta=\mu^2-\frac{2v}{\rho s}.\] Every exact sequence of sheaves on \(D\) pushes forward to an exact sequence of torsion sheaves in the first-tilt heart. Their tilt slopes differ from their \(C\)-twisted normalized surface slopes by the common constant \(1/2-b\). Twisting shifts all surface slopes equally, so tilt semistability of \(i_*F\) implies slope semistability of \(F\) for \(\ell\). The classical surface Bogomolov–Gieseker inequality (Bayer et al. 2014, Theorem 3.1.4) and the Hodge index theorem give \[u^2-2\rho v\ge0, \qquad (\ell u)^2\ge s u^2, \qquad \delta\ge0.\] The first expression is unchanged by twisting the Chern character by \(C\), so these conclusions hold for the specified \(B_0\).

The normal line bundle has first Chern class \(-\ell\), and hence \(\mathop{\mathrm{td}}(N_{D/X})^{-1}=1+\ell/2+\ell^2/6\). Grothendieck–Riemann–Roch now gives \[H^2\mathop{\mathrm{ch}}_1^B(i_*F)=\rho s, \qquad H\mathop{\mathrm{ch}}_2^B(i_*F)=\rho s(\mu+1/2-b).\] Since the threefold rank of \(i_*F\) is zero, its zero-slope condition is \(b=\mu+1/2\). The degree-three term of the same formula yields \[\frac{\mathop{\mathrm{ch}}_3^B(i_*F)}{\rho s} =\frac{v}{\rho s}+\frac\mu2+\frac16 -b(\mu+1/2)+\frac{b^2}{2} =\frac1{24}-\frac\delta2\le\frac1{24}.\] This proves (115); the two strong bounds follow by subtracting \(a^2/6\). ◻