A Gepner stability condition on every smooth quintic threefold

Proof. The Weak Lefschetz Theorem (in the integral form recalled in [6, Theorem 3]) and Poincaré duality give H^{2j}(X,\Q)=\Q H^j for 0\le j\le3; the integral version also gives H^2(X,\Z)=\Z H. Thus the even Chern character is precisely

The tangent sequence gives c(T_X)=(1+H)^5/(1+5H), so that c_1(T_X)=0, c_2(T_X)=10H^2, and

Hirzebruch–Riemann–Roch, obtained by taking the proper map to a point in [4, Section 7], now gives (2.4), and in particular

The matrix of the bilinear form (2.4) is

Moreover the four vectors

span \Q^4: their row matrix has determinant one. It follows that the kernel of v on K_0(X) is exactly the radical of the Euler pairing. Indeed one implication follows from Riemann–Roch, and the other follows by testing against these four spanning vectors and using the nondegeneracy of J. Hence v identifies the numerical group with its image, which spans \Q^4.

For any class of K_0(X), integrality of Chern classes and

give d\in\tfrac1{10}\Z and e\in\tfrac1{30}\Z; integral Weak Lefschetz gives c\in\Z. The image is therefore a subgroup of the discrete lattice in (2.3). Since it contains the four spanning line-bundle classes, it is itself a full lattice. In particular its real span is the full space \R^4.

Tensoring by \OO_X(1) sends

The twist T_{\OO_X} then subtracts \chi(E(1)) from the rank and leaves the other three coordinates unchanged. Formula (2.6) therefore gives M as displayed. Expanding \det(zI-M) gives (2.5); in particular its four eigenvalues are the nonidentity fifth roots of unity, each with multiplicity one.□

Proof. A closed point has v(\OO_x)=(0,0,0,1/5), so the normalization fixes the coefficient of e in Z/5 to be -1. Write Z/5=Ar+Bc+Cd-e. Reading the e, d, and c columns of ZM=\lambda Z, in that order, gives

The identity

then gives C=-1/2+i t_0 and B=t_0^2-5/6+i t_0/2. Substitution in the remaining, rank-column equation reduces it to 1+\lambda+\lambda^2+\lambda^3+\lambda^4=0. This proves both the eigenvalue identity and uniqueness with the stated normalization. Taking imaginary parts and using

proves (2.9).

Finally, the real coefficient matrix of (d,e) in (\Re Z,\Im Z) is

Thus v\mapsto(r,c,\Re Z(v),\Im Z(v)) is an isomorphism of real vector spaces. Boundedness of its inverse in finite dimension proves (2.10).□

Proof. We use Xu’s Theorem 1.3 [16]. Its hypotheses are precisely a smooth complex quintic threefold with its hyperplane polarization and a torsion-free slope-semistable sheaf of slope in [-1,1]; its coordinates \mu_H and \xi_H are our c/r and d/r. On [0,1/2] its upper bound g is

The first and third lines are at most -3x/25; the second is also at most -3x/25 because 5/13<25/62; and the last is at most x/2-7/25. Thus g\leq f on this half interval. The full upper bound g in Xu’s theorem and the maximum f both satisfy

For f, this reflection exchanges the first and third affine pieces and fixes the second. This proves the comparison on [0,1], and the theorem uses |\mu| on the negative strip.

For completeness, the symmetry is also compatible with the sheaf operations used here. Twisting by \OO_X(n) has the displayed effect on (x,y) and preserves slope semistability. If F is not reflexive, passing to F^{**} preserves slope semistability and r,c, and only increases d. For a reflexive sheaf on a smooth threefold the higher sheaf Ext terms of its derived dual are supported in dimension zero, so its ordinary dual has characters (r,-c,d) through degree two and is slope-semistable. The operations of dualizing and twisting by \OO_X(1) therefore give (x,y)\mapsto(1-x,y+1/2-x) without changing the validity of an upper bound. Finally the three defining lines are nonpositive on [0,1/2], and their maximum at 1/2 is -3/100.□

Proof. We give the rational first-tilt argument; compare [3, proof of Lemma 3.2.4]. Consider epimorphisms E_0\twoheadrightarrow E_1\twoheadrightarrow E_2\twoheadrightarrow\cdots in \mathcal B_b, with kernels K_i for the individual arrows. The nonnegative discrete numbers p_b(E_i) eventually stabilize, so p_b(K_i)=0. The ordinary cohomology sheaves H^0(E_i) form a sequence of epimorphisms and eventually stabilize, since \Coh(X) is noetherian. The preceding description of K_i gives r(K_i)\leq0, hence the integers r(E_i) are nondecreasing. They are bounded above by the now fixed rank of H^0(E_i), so eventually r(K_i)=0. Each remaining K_i is then an ordinary sheaf of dimension at most one.

The cohomology sequence now reads

These torsion-free sheaves agree in codimension one. Their injections extend to isomorphisms of their reflexive hulls, so they form an increasing chain of coherent subsheaves of a fixed reflexive sheaf. That chain stabilizes, forcing K_i=0. Thus every such sequence of epimorphisms stabilizes, as required.□

Proof. High objects are closed under extensions: if one end term has positive N_b^\eta, so does the extension, and otherwise both have p_b=0. It follows, by taking the induced filtration on a quotient, that \mathcal T_b^\eta is closed under quotients and extensions.

If an object has a high subobject, choose such a nonzero subobject A with p_b(A) minimal. When p_b(A)=0, all its quotients have p_b=0. Otherwise N_b^\eta(A)>0, and a low quotient Q would have p_b(Q)>0 and N_b^\eta(Q)\leq0. Its kernel would then be a high subobject with strictly smaller p_b, a contradiction. Thus A\in\mathcal T_b^\eta in either case.

By Lemma 3.2, every object has a maximal \mathcal T_b^\eta subobject; sums of such subobjects again belong to \mathcal T_b^\eta, so maximality has its usual torsion meaning. The quotient has no high subobject, since such a subobject would contain a nonzero member of \mathcal T_b^\eta by the preceding paragraph, and its inverse image would enlarge the maximal torsion subobject. The quotient therefore lies in \mathcal F_b^\eta. Finally a nonzero image of a map from \mathcal T_b^\eta to \mathcal F_b^\eta would be both high and low. This proves the torsion pair, and (3.2) gives boundedness.

Write an object of the new heart as an extension of T by F[1], where T\in\mathcal T_b^\eta and F\in\mathcal F_b^\eta. Then N_b^\eta(T)\geq0 and N_b^\eta(F)\leq0, proving nonnegativity. If N_b^\eta(E)=0, both terms have N_b^\eta=0. Since T itself is high when nonzero, this forces p_b(T)=0, hence T is zero-dimensional. If F\ne0, then p_b(F)>0 and p_b(E)=-p_b(F)<0. Thus [E]=0 would force F=0; it would then force T=0, since e(T)=\operatorname{length}(T)/5. This proves the last assertion.□

Proof. Let \mathcal M be the ordinary torsion-free sheaves all of whose slopes lie in (-1/2,1/2], and let

To see the torsion pair (\mathcal L,\mathcal M) in \mathcal B_-, take E\in\mathcal B_- and let M_E be the quotient of H^0(E) obtained by removing its torsion and its slope factors above 1/2. Then M_E\in\mathcal M. The kernel of E\to M_E in \mathcal B_- has degree-minus-one cohomology H^{-1}(E) and degree-zero cohomology in \mathcal C_{>1/2}, so it lies in \mathcal L. The slope vanishing and the ordinary cohomological degrees give \Hom(\mathcal L,\mathcal M)=0. Tilting this torsion pair gives \mathcal B_+=\langle\mathcal M[1],\mathcal L\rangle; in \mathcal B_+ the corresponding torsion pair is (\mathcal M[1],\mathcal L). In particular W_-[1]\subset W_+\subset W_-, and W_+[1]\subset W_-[1]\subset\mathcal U_-^\theta. It suffices to prove

Indeed an object E\in W_- has maps to a member of \mathcal F_-^\theta exactly through H^0_{\mathcal B_-}(E); vanishing of all such maps puts this cohomology object in \mathcal T_-^\theta. Equation (3.8) therefore puts \mathcal T_+^\alpha in \mathcal U_-^\theta, and the preceding inclusion handles the negative \mathcal B_+ degrees.

Suppose there is a nonzero map E\to G with E\in\mathcal T_+^\alpha and G\in\mathcal F_-^\theta. Its source decomposes in \mathcal B_+ as 0\to M_E[1]\to E\to E_0\to0, with E_0\in\mathcal L. Because \Hom(\mathcal M[1],\mathcal B_-)=0, the map factors through E_0, which is still in \mathcal T_+^\alpha. The (\mathcal L,\mathcal M) decomposition of G in \mathcal B_- then factors it through a subobject G_0\in\mathcal L of G, since \Hom(\mathcal L,\mathcal M)=0.

Take the kernel K and the nonzero image Q of E_0\to G_0 in \mathcal B_-. Let K_{\mathcal L} be the \mathcal L torsion part of K, and put U'=E_0/K_{\mathcal L} in \mathcal B_-. The quotient closure of \mathcal L gives Q,U'\in\mathcal L. All three terms of K_{\mathcal L}\to E_0\to U' belong to both hearts, so its triangle is short exact in \mathcal B_+ as well. Thus U' remains a quotient of E in \mathcal B_+, and we have

We use the sequence (3.9) only as a short exact sequence in \mathcal B_-; its term S need not belong to \mathcal B_+.

We now use the slope envelope to exclude factors close to the two slope boundaries. The resulting rank bounds will contradict the opposite signs of the two cut forms on U' and Q.

Directly from the three affine functions defining f,

The perturbation error on a slope \pm s is at most (1+s_0)\delta, so (3.5) makes the relevant sheaf value of N_- on [-s_0,-1/2], or of N_+ on [1/2,s_0], strictly negative.

Write A=H^{-1}(Q). If A\ne0, its top slope factor A_1 is a saturated subsheaf and A/A_1\in\mathcal C_{\leq-1/2}. Consequently A_1[1] is a subobject of A[1], and then of Q, in \mathcal B_-. If \mu(A_1)\in[-s_0,-1/2], the preceding negative sheaf value makes A_1[1] high, contradicting Q\in\mathcal F_-^\theta. Thus \mu^+(A)<-s_0.

For U', put A'=H^{-1}(U') and D'=H^0(U'). If D' has positive rank, remove its ordinary torsion subsheaf and take its bottom positive-rank slope quotient G'. The kernel of D'\to G' consists of that torsion and the other slope factors, all of slope >1/2; this sequence is therefore exact in \mathcal B_+. The ordinary cohomology sequence also makes D' a \mathcal B_+ quotient of U', so G' is such a quotient. A slope \mu(G')\in(1/2,s_0] would give p_+(G')>0, N_+^\alpha(G')<0, contradicting U'\in\mathcal T_+^\alpha. Every positive-rank slope of D' therefore exceeds s_0.

For any L\in\mathcal L, its two ordinary cohomology sheaves give

Divisorial torsion contributes nonnegative c, and torsion in lower dimension contributes zero. The stronger slope conclusions just proved similarly give

More explicitly, with D=H^0(Q) the estimates are

They include torsion cohomology sheaves: their ranks are zero and their contributions to c are nonnegative. Also d(S)\leq0, by applying Lemma 3.1 to the ordinary slope factors of S.

Set C=c(Q)+c(U'). By (3.11), C\geq0. If C=0, then c(Q)=r(Q)=0, so p_-(Q)=0, impossible for a nonzero member of \mathcal F_-^\theta. Hence C>0; and (3.12) together with (3.9) gives C\geq s_0r(S). By additivity,

Here the absolute value of the last four terms is at most 3\delta C by (3.11); this applies even when a derived rank is negative. The strict sign is (3.6). But N_-^\theta(Q)\leq0\leq N_+^\alpha(U'), a contradiction. This proves (3.8) and the aisle inclusion.□

Proof. The object \OO_X\in\mathcal B_- has the smallest positive value p_-=1/2 and N_-^\theta=u+\theta_0>0. It has no nonzero subobject of p_-=0: a p_-=0 object is an extension of a sheaf of dimension at most one by F[1] of slope -1/2, and both have zero Hom to \OO_X. Any nonzero low quotient of \OO_X would have p_-=1/2, leaving a kernel of p_-=0; that kernel must vanish, contradicting that \OO_X is high. Hence \OO_X\in\mathcal T_-^\theta.

Similarly \OO_X[1]\in\mathcal B_+ has p_+=1/2 and N_+^\alpha=-u-\alpha_0<0. A p_+=0 object has no map to it: for its slope-1/2 part this is \Hom(F[1],\OO_X[1])=\Hom(F,\OO_X)=0, and for its dimension-at-most-one part T it follows from

Here we use Serre duality and K_X\simeq\OO_X. Every nonzero subobject A of \OO_X[1] therefore has p_+(A)=1/2, and its quotient Q has p_+(Q)=0. Hence N_+^\alpha(A)=N_+^\alpha(\OO_X[1])-N_+^\alpha(Q)<0. This proves \OO_X[1]\in\mathcal F_+^\alpha, and thus \OO_X[2]\in\mathcal A_+^\alpha.

For B\in\mathcal U_+^\alpha and j\geq2, Serre duality and the heart orthogonality give

since B[j-1]\in\mathcal U_+^\alpha[1]. Lemma 3.4 puts B in \mathcal U_-^\theta. In the twist triangle

the third term is a finite direct sum of copies of \OO_X[1-j] with j\leq1, because bounded complexes of finite-dimensional vector spaces split into their cohomology. All these terms lie in \mathcal U_-^\theta. Extension closure therefore gives the claimed twist inclusion.□

Proof. Tensoring by \OO_X(1) takes \mathcal B_- to \mathcal B_+ and preserves the p test. Its effect on the other test is

Thus it takes the second torsion pair and aisle to those with this parameter \alpha. If \eta\in\Omega, then \alpha\in B_\delta; also \theta\in\Omega implies \theta\in B_\delta. Lemma 3.5 now gives

The remaining assertions are Lemma 3.3 and (2.9).

The parameter shear was used once, in proving this inclusion for the original family. In particular, applying any power \Phi^i to (3.14) gives \Phi^{i+1}U_\eta\subset\Phi^iU_\theta with the same \eta,\theta\in\Omega; it performs no further shear and requires no further shrinkage. Negative powers are allowed by Lemma 2.1.□

Proof. For \alpha,\beta\in\Omega, consider

Every object of this intersection belongs to the hearts of D_0^\beta and D_3^\alpha. More strongly, for every \gamma\in\Omega and i\in\{1,2\}, (4.2) gives

The first inclusion gives aisle membership, and the second gives the orthogonal membership. Thus the same object lies in the heart of D_i^\gamma for every \gamma in the original fixed square \Omega. This square is independent of the endpoints \alpha,\beta and of the object.

Fix a Euclidean norm on \Knum(X)_\R. Suppose there were nonzero objects E_n\in\mathcal I_{\alpha_n,\beta_n} with \alpha_n,\beta_n\longrightarrow0. The endpoint heart has no nonzero zero-class object, so v_n=[E_n]/\norm{[E_n]} is defined. After a subsequence it tends to a unit vector v. Heart positivity at the two endpoints, and at the unperturbed middle hearts, gives in the limit

These four inequalities force Z(v)=0. Indeed, if Z(v)\ne0, choose its argument \vartheta\in[0,2\pi). The first three inequalities force \vartheta\in[4\pi/5,\pi], whereas the fourth forces \vartheta\in[0,\pi/5]\cup[6\pi/5,2\pi), which is impossible.

Now fix any \gamma\in\Omega. The middle-heart inequality at index one holds for every n, so the same convergent subsequence satisfies

Since \gamma was arbitrary, this holds simultaneously for all \gamma\in\Omega; there is no further subsequence or shrinking of the middle parameter set. Write w=\Phi_*^{-1}v. The eigencharge identity and (2.9) give Z(w)=N_0(w)=0. Varying \gamma_0,\gamma_1 with both signs in \Omega now gives r(w)=c(w)=0. The real-linear independence of the d,e coefficients of Z gives d(w)=e(w)=0. This contradicts \norm{v}=1.

Consequently some square \Omega_1 about zero has \mathcal I_{\alpha,\beta}=0 for every pair \alpha,\beta\in\Omega_1: otherwise choose counterexamples with both parameters within distance 1/n of zero. To deduce (4.3), let E\in D_3^\alpha and truncate in the bounded t-structure with aisle D_0^\beta:

Here E\in D_0^\beta, and A[1]\in D_0^\beta[2]=D_5^\beta\subset D_3^\alpha. Extension closure therefore puts the zeroth cohomology in D_3^\alpha\cap(D_0^\beta[1])^\perp=0. Hence E\in D_0^\beta[1].

Every integer is 2m+5j. Two such representations differ by (m,j)\mapsto(m+5t,j-2t), so \Phi^5\simeq[2] proves that (4.4) is well defined. Since V_1^\alpha=\Phi^3U_\alpha[-1]\subset U_\beta=V_0^\beta, transport by \Phi^m[j] proves the first assertion of (4.5) for all k. Its other assertions follow directly from the definition.□

Proof. Extension closure follows from (4.7). If j>k, then \mathcal S_j\subset V_j\subset V_{k+1} and \mathcal S_k\subset V_{k+1}^\perp, proving the Hom vanishing. Boundedness and V_{k+5}=V_k[1] put any object E in V_l\cap V_h^\perp for some integers l<h. Truncation at V_{l+1} gives

Because E,A[1]\in V_l, we have B\in V_l, hence B\in\mathcal S_l. Also B[-1]\in V_h^\perp: for F\in V_h, F[1]\in V_h\subset V_{l+1} gives \Hom(F,B[-1])=0. The rotated triangle thus shows A\in V_h^\perp. Repeat with A\in V_{l+1}\cap V_h^\perp. At index h the remaining object belongs to V_h\cap V_h^\perp and is zero. Reading the resulting triangles from the first subobject to the last quotient gives the stated decreasing order; zero factors are omitted.

If E\in\mathcal H_k, apply the same construction with l=k and h=k+5. The first triangle has all its terms in \mathcal H_k, so it is a short exact sequence there, with A\in\mathcal T_k and B\in\mathcal S_k. The preceding Hom vanishing proves the torsion-pair axiom. Repeating on A proves the displayed four-block description. For completeness, a subobject G of an object of \mathcal S_k has \Hom(\mathcal T_k,G)=0, by composing with the inclusion, so its torsion part vanishes. Thus G\in\mathcal S_k.□

Proof. For k=2m+5j, the functor \Phi^{-m}[-j] takes \mathcal H_k to \mathcal A_0 and multiplies its charge by e^{-i\pi k/5}. Positivity therefore gives \Im(e^{-i\pi k/5}Z(E))\ge0 on \mathcal H_k. An object E\in\mathcal S_k belongs both to \mathcal H_k and to

The two tests are consequently

Their intersection is the closed sector in (4.9), together with the origin.

Choose \varepsilon>0 such that the closed square [-\varepsilon,\varepsilon]^2 is contained in \Omega_1. For every \eta in this square, independent-parameter nesting gives

Thus E belongs to the heart of V_{k-1}^\eta for every such \eta: the second inclusion supplies exactly its required right orthogonality. Write k-1=2m+5j, and set w=[\Phi^{-m}E[-j]]. This is the class of an actual object of \mathcal A_\eta for every parameter under consideration, including the shift in this formula. Hence

Choosing the two signs independently proves

The invertible real coordinate map w\mapsto(r(w),c(w),\Re Z(w),\Im Z(w)), established in (2.10), now bounds \norm{w} by a constant times |Z(E)|. Explicitly, d(w)=\Im Z(w)/(5t_0)-c(w)/2-ur(w), and then the real part of (2.8) recovers e(w).

We can choose m\in\{0,1,2,3,4\} according to k-1 modulo five. There are therefore only five norm-conversion operators \Phi_*^m; the arbitrary shift j only changes the sign of a numerical class. Taking the maximum of their operator norms gives one C for every k. If Z(E)=0, the estimate gives w=0. The object \Phi^{-m}E[-j] belongs to \mathcal A_0, so Lemma 3.3 forces \Phi^{-m}E[-j]=0 and hence E=0. This excludes the origin for a nonzero block object and completes the proof.□

Proof. Let \ell>0 be a lower bound for the norms of nonzero elements of the full numerical lattice, supplied by Lemma 2.2, and define

For every nonzero block object, Proposition 4.3 implies

This projection is additive in short exact sequences. Hence every filtration of a fixed E with nonzero block factors has length at most q_k(Z(E))/d_*.

If F\subset E is saturated in the block, then both F and E/F are block objects. Their projections are nonnegative, so

Only finitely many lattice classes, and therefore only finitely many charges and phases, can occur for such F. Choose a nonzero saturated subobject F\subset E of maximal phase, and among those of maximal |Z(F)|. These maxima concern finite sets of numerical values; no finiteness of the set of subobjects is needed.

This F is block-semistable. Indeed, a saturated subobject of F is saturated in E, since the resulting quotient of E is an extension of two objects of \mathcal S_k. If G\subset E/F is a nonzero saturated subobject, its preimage \widetilde G\subset E is also saturated and is an extension of G by F. Charges in a sector of width less than \pi have the elementary strict see-saw property: the argument of their sum is strictly between their arguments when those arguments differ, and their magnitudes add when they agree. Thus \varphi_k(G)>\varphi_k(F) contradicts maximal phase of F by using \widetilde G; equality contradicts maximal magnitude. Every such G therefore has phase strictly less than \varphi_k(F).

Apply the same selection to the nonzero quotient E/F, if there is one, and continue. Preimages give a filtration in the original heart with all factors and intermediate quotients in the block. The phases strictly decrease by the preceding argument, and (4.11) bounds the number of nonzero factors, so the process terminates.□

Proof. Suppose G\subset E violates the stated inequality. Decompose E/G by (4.8) and take the preimage of its torsion part:

The object \overline G\subset E lies in \mathcal S_k by subobject closure, and is saturated because E/\overline G=F'. The four-block filtration of T has charge rays with phases in [b,a+1]. If g=\varphi_k(G), then

Thus Z(G) and every added ray lie in the sector with phase interval [g,a+1] of width strictly less than one. Their sum is nonzero and has phase in that same interval. Since \overline G\in\mathcal S_k, its phase in [a,b] must therefore be at least g. In particular, \varphi_k(\overline G)>\varphi_k(E), a contradiction. The strict inequality g>a, supplied by the alleged destabilization, is what excludes an antipodal endpoint in this argument. If T=0, the contradiction already comes from G itself being saturated.

For a nonzero map E\to F, take its image I in \mathcal H_k. Since I\subset F, we have I\in\mathcal S_k. The kernel in E is therefore saturated. If that kernel is zero, \varphi_k(I)=\varphi_k(E); otherwise semistability and the see-saw property give \varphi_k(I)\ge\varphi_k(E). The unsaturated test applied to I\subset F gives \varphi_k(I)\le\varphi_k(F). These inequalities exclude a nonzero map when the source phase is strictly larger.□

Proof. First consider the generators. If two are in different blocks and the source has strictly larger phase, its block index is larger: for j<k one has (j+1)/5\le k/5. Thus their Hom group vanishes by Lemma 4.2. The case of a common block is Lemma 4.5. Induction along extensions in each variable now gives

The categories are additive, since finite direct sums are split extensions. The charge of an extension of phase-\varphi generators is a sum of positive multiples of e^{i\pi\varphi} and hence is again positive on that ray. Likewise the triangle inequality gives

for any such extension filtration. This proves (4.13).

Refine the decreasing block filtration of any object by Lemma 4.4, splicing the resulting triangles by the octahedral axiom. The phases are weakly decreasing globally and strictly decreasing inside each block. Adjacent equal phases can only occur at a shared endpoint; merging each run of equal-phase factors into its extension gives a finite filtration with strictly decreasing phases and factors in the corresponding \mathcal P(\varphi). This also explains why both blocks at a shared endpoint must enter the same phase category.

By (4.5), the functors [1] and \Phi take (\mathcal H_k,\mathcal S_k) to (\mathcal H_{k+5},\mathcal S_{k+5}) and (\mathcal H_{k+2},\mathcal S_{k+2}), respectively. These equivalences take exact sequences in the hearts to exact sequences, preserve the condition of being saturated in a block, and change the assigned phase by 1 and 2/5. They therefore preserve block-semistability with these phase increments. Applying them and their inverses to the extension closures proves both equalities in (4.12). The phase-cut argument that follows proves closure under direct summands without presupposing a slicing heart.□

Proof. Denote either pair in (4.14) by (\mathcal U,\mathcal L). Strict phase separation gives \Hom(\mathcal U,\mathcal L)=0 by extension induction. Equation (4.12) gives \mathcal U[1]\subset\mathcal U and \mathcal L[-1]\subset\mathcal L. Cutting the finite decreasing filtration at c and splicing triangles gives, for every E, a triangle

All factors of phase exactly c are assigned to the same side, according to the selected convention. These are the t-structure axioms in the aisle/right-orthogonal convention, and we can verify the orthogonal characterizations directly before using any summand closure. If E\in{}^\perp\mathcal L, applying \Hom(-,L_E) to (4.15) shows that \Hom(L_E,L_E)=0, since its neighboring terms \Hom(U_E[1],L_E) and \Hom(E,L_E) vanish. Thus L_E=0 and E\in\mathcal U. Similarly, if E\in\mathcal U^\perp, apply \Hom(U_E,-) and use \Hom(U_E,L_E[-1])=\Hom(U_E,E)=0 to obtain U_E=0. Hence

They are consequently closed under direct summands, and (4.15) is a genuine truncation triangle. Finite upper and lower bounds on the phases of each object’s filtration, and the shift law, prove boundedness.

We justify the interval description explicitly. A nonempty decreasing filtration with nonzero phase factors cannot have zero total object. Indeed, the identity of its first factor maps nontrivially into the total object: at each subsequent triangle this map remains nonzero because \Hom(F_1,F_j[-1])=0. Now group a chosen filtration of an object at the two endpoints of an interval. If the object lies in the upper and lower cut categories, the preceding orthogonal characterizations force the exterior truncation objects to be zero. By the observation just made, their groups of factors are empty. The remaining factors lie in the interval. The reverse inclusion follows from extension closure. This works with either choice at either endpoint, as well as for rays and singletons. Intersecting the aisle with the right orthogonal of its shift therefore gives precisely the two displayed hearts. Finally, intersections of orthogonals are closed under summands, which proves the last assertion.□

Proof. The empty interval gives the zero category. Otherwise let a\le b be its endpoints, with b-a<1. If a is excluded, use the heart \mathcal H=\mathcal P((a,a+1]); if it is included, use \mathcal H=\mathcal P([a,a+1)). Inside this heart, the cut at b gives a torsion pair (\mathcal T,\mathcal F) with \mathcal F=\mathcal P(I). Equality at b is assigned to \mathcal F exactly when b\in I. For example, for I=(a,b) it is

For I=(a,b] the corresponding torsion class is \mathcal P((b,a+1]). The lower-closed variants use the other heart above and the same upper-endpoint rule. Hom orthogonality follows from the phase inequality, and the cut triangles have all terms in \mathcal H, so they give the required short exact decompositions.

We prove the quasi-abelian assertion directly for a torsion-free class \mathcal F in an abelian heart \mathcal H; this standard fact is also recorded in [13, Lemma 3.1]. It is closed under subobjects and extensions. For a morphism f:A\to B in \mathcal F, put K=\ker_{\mathcal H}f, J=\im_{\mathcal H}f, and Q=\operatorname{coker}_{\mathcal H}f. The kernel in \mathcal F is K, and the cokernel is Q/t(Q), where t(Q) denotes the torsion part of Q. Indeed, maps from Q to an object of \mathcal F annihilate t(Q); this proves the cokernel’s universal property. Both K and J lie in \mathcal F. Consequently the internal coimage of f is J, whereas its internal image is the preimage of t(Q) under B\to Q.

Recall that a morphism is strict if its canonical coimage-to-image map is an isomorphism. The preceding formulas show that a strict monomorphism in \mathcal F is exactly a monomorphism in \mathcal H whose cokernel belongs to \mathcal F. A strict epimorphism in \mathcal F is exactly an epimorphism in \mathcal H between its objects: for an internal epimorphism the internal image is B, and strictness says precisely that J=B. This statement is about strict epimorphisms, not all epimorphisms of the subcategory.

The pushout in \mathcal H of a strict monomorphism A\hookrightarrow B along A\to A' fits into

Both outside terms are in \mathcal F, so B'\in\mathcal F and the new monomorphism is strict. The pullback in \mathcal H of a strict epimorphism A\twoheadrightarrow B along B'\to B is a subobject of A\oplus B', hence lies in \mathcal F, and its projection to B' is an ambient epimorphism, hence strict. These ambient universal properties remain universal in the full subcategory \mathcal F. Thus kernels and cokernels exist, pushouts preserve strict monomorphisms, and pullbacks preserve strict epimorphisms. This proves quasi-abelianity and the stated description of strict exact sequences.□

Proof. Let I have endpoints a\le b and width w=b-a<1, and put

where \ell is the full-lattice norm lower bound used in Lemma 4.4. A nonzero semistable F of phase in I has nonzero numerical class, since its charge is nonzero. Therefore (4.13) gives

Every nonzero object of \mathcal P(I) is a finite extension of nonzero semistable objects with phases in I. Additivity thus gives the same lower bound q_I(Z(E))\ge d_I for every such object.

By Lemma 4.8, a strict exact sequence in \mathcal P(I) is exact in its ambient heart, so the projection q_I Z is additive on it. Every nonzero successive quotient in a strict filtration of a fixed E\in\mathcal P(I) is again an object of \mathcal P(I) and consumes at least d_I of this projection. The number of such quotients is consequently at most q_I(Z(E))/d_I. This bounds both increasing and decreasing strict subobject chains, and proves finite length in the quasi-abelian exact structure. The argument uses discreteness of the full numerical lattice; it requires no discreteness of its image under Z.□

Proof of Lemma 2.1. Write T_j=T_{\OO_X(j)} and L=(-\otimes\OO_X(1)). We interpret the twist as the Fourier–Mukai transform whose kernel is the cone of the evaluation map

We take cones in the standard enhancement by complexes of perfect kernels, so that the evaluation squares below induce maps of cones. Their transforms give natural transformations of functors. Tensoring the evaluation kernel gives LT_jL^{-1}\simeq T_{j+1}, and hence

We will identify the kernel of T_0T_1T_2T_3T_4 with \OO_\Delta(-5)[2], where the twist is in the second factor.

Keep the first copy of X as a source parameter. Set

Here p and q are the first projections. The kernel convention is that F\in\Db(\Coh(X\times Y)) sends E to R\pi_{Y*}(\pi_X^*E\otimes^{\mathbf L}F). For such a kernel, let

The arrow is the adjunction evaluation. Projection formula and proper base change show that \mathsf L_j^Y(F) represents postcomposition of the transform of F with the evaluation cone for \OO_Y(j). Indeed, on an input E the first kernel in this cone gives

In particular the source coefficient here is a_j^Y(F) itself.

These coefficient functors let us compare the evaluation cones on X with those on \mathbf P^4. We will show that the five projective-space evaluations remove all coefficients and leave the zero kernel, while the comparison preserves the cone of the divisor counit. Computing that cone will give the required shift.

Start with

given by the counit. For j=4,3,\ldots,0, form successively

We construct compatible maps u_j\colon Lf^*K_j\to G_j and show that their cones are all isomorphic to \operatorname{Cone}(u_5).

For any map u\colon Lf^*K\to G, the unit followed by f_*u gives K\to f_*G, hence comparison maps

For (K_5,G_5,u_5) these are isomorphisms for every l: the composite f_*\OO_\Delta\to f_*Lf^*f_*\OO_\Delta\to f_*\OO_\Delta is the identity by the adjunction identity. Suppose that, before the step indexed by j, the maps \theta_l are isomorphisms for 0\le l\le j. The evaluation square

commutes by adjunction. Taking its cones defines u_j; because its left vertical arrow is an isomorphism, the cone of u_j is isomorphic to the cone of u_{j+1}.

For l<j, apply the coefficient functor a_l to the two evaluation triangles. The comparison on their first terms is

It is an isomorphism: \theta_j is an isomorphism, and the divisor sequence

identifies the two cohomology complexes for 1\le m\le4. The comparison on the middle terms is also an isomorphism by induction, so the remaining maps \theta_l stay isomorphisms after the mutation. This proves the induction and the constancy of the cone.

We next show that K_0=0. The step indexed by j kills a_j^{\mathbf P^4}, since R\Gamma(\mathbf P^4,\OO)=\C. It preserves the already vanishing coefficients with indices k>j, since R\Gamma(\mathbf P^4,\OO(j-k))=0 for 1\le k-j\le4. Thus a_j^{\mathbf P^4}(K_0)=0 for 0\le j\le4. All these kernels are perfect, and proper base change implies that every derived source fiber F of K_0 satisfies

The exact Koszul complex of the five homogeneous coordinates propagates these five consecutive vanishings to R\Gamma(\mathbf P^4,F(n))=0 for every integer n. For sufficiently large n, Serre vanishing and global generation, applied to the finitely many cohomology sheaves of F, force all those sheaves to vanish. Therefore every derived source fiber is zero, and the derived Nakayama lemma gives K_0=0. It follows that

Finally, f is a Cartier divisor embedding with normal bundle \OO_X(5) in the target variable. Its two-term divisor resolution computes

with no other cohomology sheaves. The counit u_5 induces the identity in degree zero. Consequently its cone has the single cohomology sheaf \OO_\Delta(-5) in degree -2, and

Since G_0 is the composition kernel for T_0T_1T_2T_3T_4, this is an isomorphism of kernels and therefore a natural functor isomorphism. Equation (A.1) now gives \Phi^5\simeq[2]. The asserted inverse follows by composing this relation with [-2].□