STABILITY CONDITIONS ON 1 AND WALL-CROSSING

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1 STIITY CONDITIONS ON ND W-CROSSING CHRMINE SI STRCT. We sketch a proof that the stability manifold of the bounded derived category of coherent sheaves on, D b Coh( ), is isomorphic to 2 as a complex manifold. We also discuss the notion of wall-crossing for three hearts for the t-structure on D b Coh( ), namely Coh( ), Rep( ) and Rep( ).. PREIMINRIES In [ri07], ridgeland introduced the notion of a stability condition (Z, ) on a triangulated category. This is an abstraction of the properties of µ-stability for sheaves on projective varieties. Here the central charge Z : K() is a group homomorphism from the Grothendieck group of to the complex numbers and is a slicing of, that is, there are full additive subcategories (φ) for each φ, satisfying the following compability conditions: (a) if E (φ), then Z(E) = m(e)e iπφ for some m(e) >0, (b) for all φ, (φ + ) = (φ)[], (c) if φ > φ 2 and j (φ j ), then Hom (, 2 ) = 0, (d) (Harder-Narasimhan property) for each nonzero object E, there are a finite sequence of real numbers and a collection of triangles 0 = E 0 E φ > φ 2 > > φ n E 2 E n E n = E with j (φ j ) for all j. 2 n ridgeland showed that the set of stability conditions Stab() on a fixed category has a natural topology, and that Stab() is in fact a manifold under this topology. Macrì [Mac04] showed that for a smooth projective curve C over of positive genus, Stab(D b Coh(C)) = G + (2, ), the universal cover of the group of rank two matrices with positive determinant. The main theorem of this talk is the following: Theorem (Okada [Oka04]). The stability manifold for the bounded derived category of coherent sheaves on, Stab(D b Coh( )), is isomorphic to 2 as a complex manifold. The strategy is to show that the quotient of Stab(D b Coh( )) for a certain action of is isomorphic to. The -action is defined as follows: Definition 2. et (Z, ) be a stability condition and z = x + i y. Then z (Z, ) is defined by z Z = e z Z and (z )(φ) = (φ y/π). Remark 3. The real and imaginary parts of z act as rescaling and rotation respectively. Rotation affects the heart of a triangulated category but preserves semistable objects (defined below), while wall-crossing, which we will discuss later, preserves the heart but affects semistable objects. Definition 4. stability function on an abelian category is a group homomorphism Z : K( ) such that for all 0 E, the complex number Z(E) lies in H := {re iπφ : r > 0 and 0 < φ } = <0, Notes prepared for Mathematics 280x: ridgeland Stability Conditions class at Harvard University, October 24, 203. Some figures borrowed from [Oka04].

2 2 CHRMINE SI where = {x + i y x, y, y > 0} is the complex upper half-plane. Given a stability function Z : K( ), the phase of an object 0 E is defined to be φ(e) = arg Z(E) (0, ]. π n object 0 E is said to be semistable if every subobject 0 E satisfies φ() φ(e). Example 5. Recall that the heart of a triangulated category is always an abelian category. stability function on the heart Coh( ) of the triangulated category D b Coh( ) is given by a choice of Z(k x ) <0 and Z( ), where k x is a skyscraper sheaf. We illustrate the standard stability function Z std : K(Coh( )), which is given by Z(k x ) = and Z( ) = i, below. E deg E + i rk E, (n) (2) () ( ) k x ( )[] FIGURE. Standard stability function Z std (E) = deg E + i rk E on Coh( ). Every object in Coh( ) has image in H (shaded gray). y a lemma of ridgeland, since Z satisfies the Harder-Narasimhan property, it extends to a stability condition (Z, ) on D b Coh( ). Rotation by z = iφ( ( ))π gives a stability condition z (Z, ) = (Z, ) such that the heart ((0, ]) is equivalent to Rep( ). We see this as follows. theorem of eilinson says that there is an equivalence of triangulated categories Recall from class that D b Coh( ) D b Rep( ). Rep( α 2) β is equivalent to the category of left modules on the path algebra k( ) of the quiver (i.e. the free (non-unital) associative algebra generated by the vertices and arrows of the quiver subject to the expected rules of composition), and the quiver representation corresponding to k( ) as a k( )-module is k e f α k e 2 k α k β. f β Note that elements of the algebra act on elements of the module by postcomposition. Writing this out in tabular format, we have k( ) as module k( ) as algebra e e 2 f α f β e e e 2 0 e 2 f α f β f α f α f β f β TE. ction of k( ) on itself

3 STIITY CONDITIONS ON ND W-CROSSING 3 eilinson s equivalence is given by et us do a few calculations. We have D b Coh( ) D b Rep( ) D b (k e f α k e 2 k α k β ) E Ext ( (), E). Hom( (), ) = Hom(, ) Hom( (), ) = k id, Ext ( (), ) = Hom(, ( ()) ( 2)) = Hom(, ( ) ( 2)) = 0 by Serre duality, since the dualizing sheaf of is ( 2), and we know that Ext i ( (), ) = 0 for i 0, because the abelian category of coherent sheaves on a smooth projective variety X of dimension n has homological dimension at most n (i.e. Ext i (, ) = 0 for i > n and any two coherent sheaves and on X ). We also have Hom( (), ( )[]) = Ext ( (), ( )) = Hom( ( ), ( ()) ( 2)) = k id ( ) by Serre duality and using Hom( (m)[a], (n)[b]) = Ext b a ( (m), (n)), Ext ( (), ( )[]) = Hom( ( )[], ( ( )) ( 2)) = Ext ( ( ), ( 2) ( 3)) = 0. gain, using the result on the homological dimension of Coh( ), we conclude that Ext i ( (), ( )[]) = 0 for i 0. There is a two-dimensional space of maps (). For a suitable choice of coordinates x 0, x on, let us abuse notation and call the corresponding generators of this space of maps x 0 and x also. Similarly, there is an action of End( ()) = End( ) End( ()) Hom(, ()) Hom( (), ) = k k () k x 0 k x on Hom( (), E) by precomposition: End( ()) as module End( ()) as algebra id () id x 0 x id () id () id 0 id x 0 x x 0 x x x TE 2. ction of End( ()) on itself f β Comparing Table 2 with Table, we conclude that corresponds to the subquiver 0 k and () corresponds to k 0. Consider the result of rotating the heart in Figure by z = iφ( ( ))π. We obtain the heart in Figure 2a, which we showed above corresponds to the heart in Figure 2b. Thus, using the equivalence above, we see that (0, ] is generated by and ( )[]. () k x 0 k ( )[] ( ) k 0 () Z std on Coh( ) rotated by z = iφ( ( )[])π () Stability function on Rep( ) FIGURE 2. Equivalence of hearts

4 4 CHRMINE SI We can express this using the commutative diagram below: K(D b Coh( )) K(D b Rep( )) E k a k b From the diagrams 2 2 ( deg E, rk E) (a, b) 0 k ( )[] k 0 k x k k (0, ) (0, ) (, ) (, 0) we see that the matrix of the linear transformation is 0 =. (, 0) (, ) 2. W CROSSING We now consider what happens to the semistable objects when we fix ((0, ]) and vary the stability function Z on it. From the above example, we note that Z is determined by a choice of values z 0 = Z( ), z = Z( ( )[]) H. There are three cases, whose loci partition Stab Rep( ) (D b Coh( )), the locus of stability conditions on D b Coh( ) with heart (0, ] Rep( ). φ(z 0 ) < φ(z ). This is the case we considered above, where Z is equivalent to the standard stability function Z std (E) = deg(e) + i rk(e) up to reparametrization of by an element in G + (2, ) (and accordingly adjusting the phases of objects). φ(z 0 ) = φ(z ). Then any non-zero object in ((0, ]) is semistable since all non-zero objects have the same slope. ( )[] FIGURE 3. Case when the generators z 0 = Z( ) and z = Z( ( )[]) of im Z have the same slope. φ(z 0 ) > φ(z ). Then any quiver k a k b not of the form k a 0 has a subobject 0 k of larger phase unless the original quiver had the form 0 k b, so the multiples of and ( )[] are the only semistable objects in ((0, ]). k a k b ( )[] 0 k k 0 () On (0, ] () On Rep( ) FIGURE 4. The multiples of and ( )[] are the only semistable objects when z 0 = Z( ) has larger phase than z = Z( ( )[]).

5 STIITY CONDITIONS ON ND W-CROSSING 5 Thus we see that the semistable objects change when as we pass through the locus φ(z 0 ) = φ(z ). In particular, the skyscraper sheaf k x is an example of an object for which the property of being semistable changes as we pass through the locus φ(z 0 ) = φ(z ). This leads us to the following definition, of which the locus {(Z, ) Stab(D b Coh( )) φ(z 0 ) = φ(z )} is an example. Definition 6. wall is a codimension one submanifold of a stability manifold such that as one varies a stability condition, a semistable object can only become non-semistable if one crosses a wall. side. Suppose that we are in the last case φ(z 0 ) > φ(z ) and we deform Z such that z leaves the upper half plane by crossing the positive real line. In this case, the semistable objects do not change; however, we obtain a new heart = ((0, ]) generated by the stable objects and ( )[2]. ( )[2] FIGURE 5. Case when the generators z 0 = Z( ) and z = Z( ( )[]) of im Z have the same slope. Using Hom( (m)[a], (n)[b]) = Ext b a ( (m), (n)) and the result on the homological dimension of Coh( ) a last time, we see that there are no morphisms or extensions between these two objects, so is isomorphic to the category of pairs of vector spaces and thus it is isomorphic to Rep( ). This is an example of a heart of D b Coh( ) such that D b ( ) D b Coh( ). The following is a generalization of our observations above to any heart. emma 7. Up to the action of ut(d b Coh( )), for any stability condition on Stab(D b Coh( )) (i.e. not necessarily one with heart (0, ]), there exists some p > 0 such that ( )[p] and are semistable and φ( ( )[p]), φ( ) (r, r + ] (i.e. φ( ) φ( ( )[]) > ) for some r. If φ(z( ( )[])) < φ(z( ), the multiples of the shifts of ( ) and are the only semistable objects. If φ(z( ( )[])) > φ(z( ), then all line bundles and torsion sheaves are semistable. In fact all cases in the above lemma exist: Proposition 8. For any α, β such that α β > and any m α, m β >0, there exists a unique stability condition (Z, ) such that φ( ( )[]) = β and φ( ) = α, and z 0 = m α e iπα, z = m β e iπβ. Moreover, we have the following cases: if α > β, then for any r, there exist p, q such that ((r, r]) = ( )[p + ], [q] and p q (α β, α β + ); if α = β, then for any r, there exists j such that ( )[j + ], (j) = ((r, r]); if α < β, then for any r, either there exist i, j such that ((r, r]) = (i )[j + ], (i)[j] and φ( (i 2)[j + ]) > r φ( (i )[j + ]), or there exists j such that ((r, r]) = Coh [ j] and r = φ(k x [ j]). Corollary 9. The hearts on which there exists a stability function satisfying the Harder-Narasimhan property are Coh [ j] and (i )[p + j], (i)[j] for all i, j and p > PROOF OF THE MIN THEOREM We now return to the -action on Stab(D b Coh( )). s mentioned in the previous talk, G + (2, ) acts on Stab() for any triangulated category as follows. Regard G + (2, ) as the set of pairs (T, f ) where f : is an increasing map with f (φ + ) = f (φ) + and T : 2 2 is an orientation-preserving linear isomorphism such that the induced maps on S = /2 = ( 2 \ {0})/ >0 are the same. Given a stability condition (Z, ) Stab() and a pair (T, f ) G + (2, ), define a new stability condition (Z, ) by setting Z = T Z and (φ) = (f (φ)). Note that the semistable objects do not change, but the phases have been relabeled.

6 We have z i (Żi, Ṗi) = (Z i, P i ) X, since (z i φ i )(O) = φ(o(i )[]) φ(o(i)) < implies (z i φ i )(O) (0, ]. Graphically we can explain actions above as follows. 6 CHRMINE SI Proposition 0. The -action is holomorphic, free, coincides with the action of a subgroup of contains the shifts. The quotient Stab( )/ is a complex manifold. G + (2, ) and Proof. ridgeland s main theorem says that for each connected component Σ Stab(), there are a linear subspace V (Σ) Hom (K(, ), with a well-defined linear topology, and a local homeomorphism : Σ V (Σ) which maps a stability condition (Z, ) to its central charge Z. Holomorphicity thus follows from the holomorphicity of the -action on a vector space via multiplication by e z. The action is free since z Z = Z implies that e x =, that is, x = 0, and z = implies that y = 0. Note that the -action by z has the same effect as the action of (e x, y) G + (2, ), where is rotation by the angle πy. The shift [] can be realized at the action of iπ. We omit the proof of the second sentence. STIITY MNIFOD OF P 9 elow, we state The slope some facts and the without length proof. of O( )[] (Proofs can are befixed foundby in (b). [Oka04, So each Section (Z, 4].) P) Denote X in by () the copy of that acts theon stability D b Coh( manifold ) by tensoring can be uniquely with linerepresented bundles. et byx Z(O) be the onsubset the n-th of Stab(D sheet of b Coh( the )) consisting of all stability Riemann conditions surface (Z, of) og satisfying z, where thenfollowing is greatest properties: integer such that φ(o)/2 n. (a), ( )[] Here m(o), are semistable, φ(o) > 0 by (a) and (c). (b) φ(z 0 ) > 0, (c) φ(z ) = and m(z ) = (m is defined as on page ). Then () emma X = Stab(D 4.3. b Coh( fundamental )) (so X domain containsof astab(d(p fundamental ))/(Z)C domain) is isomorphic and X isomorphic to K def = to the upper half-plane {x by + the iy map C sending y > 0, cos (Z, y ) toe x log(m(z } as 0 in )) + the iπ(z shaded 0 ). domain in the figure below. fundamental When passing domaintofstab(d(p b Coh( ))/(Z)C ))/() one identifies is isomorphic points toon K := the{x boundary + i y that y > have 0, cos y e x }, as shaded in Figure the same 6 below. imaginary Whenpart. passing to Stab(D b Coh( ))/(), one identifies points on the boundary with the same imaginary part. K y = π e x cos y = e x cos y = + y = π 2 Figure FIGURE fundamental domain of of Stab(D(P b Coh( ))/(Z)C ))/() y the Riemann mapping theorem and the reflection principle, this fundamental domain is conformally equivalent to, as in Figure 7 on the following page. (K Proof. Notice that the actions by line bundles u is the left half of the shaded region and K and by C commute. et (Z, P) X. l is the right half of the shaded region. We shall abuse notation and label the image under each map by the same letters.) First, we will see that if φ(o) >, there are no repetitions; i.e., C(Z) (Z, P) We can now prove the main result of this talk. X = {(Z, P)}. y Theorem.2, all indecomposable semistable objects are shifts of O( ) and O. The action of O(i) (x + iπψ) (Z)C changes (Z, P) into Proof. The action (Z, P of () on X = Stab(D ) such that P (φ(o) b Coh( + ψ) = ))/ gives an exact O(i) and P sequence (φ(o( )[]) + ψ) = O( + i)[] m (O(i))/m(O) = 0 e x π and m (O( + i)[])/m(o( )) = e x. Hence, O and (X) π (X/) α π 0 () π 0 (X). O( )[] are not semistable unless i = 0, and even if i = 0 we have φ (O( )[]) We shall show or m that (O( )[]) X is connected. unless Since φ = X = x = 0. is connected, so is X. Now, Stab(D b Coh( )) = () X, so it suffices to check In thethat remaining () fixescase some 0 < connected φ(o) φ(o( )[]) component of = Stab(D ; repetitions b Coh( C(Z) )). This (Z, isp) true Xbecause () fixes {(Z, ) Stab(D are indexed b Coh(by )) Z ((0, ı ]) (Z i =, PCoh( i ) = (z )}, i, O(i)) which lies (Z, in P). X. Here, Hencez X i = is connected m(o(i )[]) and + the map α is a surjective map iπ( φ(o(i, hence )[])). an isomorphism. et us denote It (Żi, follows Ṗi) that = O(i) π (X) (Z, isp); 0, so i.e., X is the universal covering of, that is, X = Stab(D b Coh( ))/ =. Since H (, ) = 0, there is a unique extension Stab(D b Coh( )) = 2. This completes the proof. Z i (O( )[]) = Z(O(i )[]), φ i (O( )[]) = φ(o(i )[]), Z i (O) = Z(O(i)), φ i (O) = φ(o(i)).

7 from from to the to unit the unit disk. disk. We can We extend can extend any any isomorphism of bounded of bounded domains domains to t a homeomorphism a on their on their closures closures by [7, by Theorem [7, Theorem -]. -]. y ay linear a linear fractional transformation, we can we rearrange can rearrange three three points points on the on boundary the boundary in arbitrary in arbitrary way way as a long long as we askeep we keep their their order. order. Hence Hence is conformally is equivalent equivalent to the to unit the unit diskdis where where and and correspond correspond to, to, and, and the upper-half the upper-half circle circle and and the lower-half the circle circle correspond correspond and and. See. Figure See Figure emma 4.3. fundamental domain of Stab(D(P ))/(Z)C is isomorphic to K def = {x + iy C y > 0, cos y e x } as in the shaded domain in the figure below. When passing to Stab(D(P ))/(Z)C one identifies points on the boundary that have the same imaginary part. STIITY CONDITIONS ON ND W-CROSSING 7 K y = π y = π 2 z z i z+i - - K l K l Riemann mapping theorem applied to - - e x cos y = e x cos y = SO OKD ( ) zi+ Next, by the composition of z 2i Figure Figure 9. K9. ink the in the Figure Figure 0. compose zi, z z+ z 2 4 z, and z z, Next, by the composition 0. K asof u zas 2i z 2i zi+, as the unit disk is conformally equivalent tounit theunit disk lower-half disk disk, where as andthe unit thedisk unit theis unit disk conformally diskzi equivalent to z z+ z 2 4 correspond and, and the lower-half circle corresponds to. See Figure correspond. and, andz the lower-half circle Figure 2. fundamental domain of Stab(D(P ))/(Z)C Proof. Notice that the actions by line bundles and by C commute. et (Z, P) X. First, we will see that if φ(o) >, there are no repetitions; i.e., C(Z) (Z, P) X = {(Z, P)}. y Theorem.2, all indecomposable semistable objects are shifts of O( ) and O. The action of O(i) (x + iπψ) (Z)C changes (Z, P) into (Z, P ) such that P (φ(o) + ψ) = O(i) and P (φ(o( )[]) + ψ) = O( + i)[] m (O(i))/m(O) = e x and m (O( + i)[])/m(o( )) = e x. Hence, O and O( )[] are not semistable unless i = 0, and even if i = 0 we have φ (O( )[]) 2 or m (O( )[]) unless φ = x = 0. In the remaining case 0 < φ(o) φ(o( )[]) z+ z - = ; repetitions C(Z) (Z, P) X are indexed by Z ı (Z i, P i ) = (z i, O(i)) (Z, KP). Here, z i = m(o(i )[]) + u iπ( φ(o(i )[])). et us denote (Żi, Ṗi) = O(i) (Z, P); i.e., - K l + 2 SO OKD extend the mapping to K by Schwarz reflection principle - and z z ( zi zi Z i (O( )[]) = Z(O(i )[]), Z i (O) = Z(O(i)), φ i (O( )[]) = φ(o(i )[]), φ i (O) = φ(o(i)). Figure. as Figure 2. K as the Figure. as FIGURE the7. half Conformal disk equivalence between K := unit {x disk + i y y > 0, cos y e x } with the half points disk on the boundary with the same imaginary part identified and. The diagrams in the righthand column show how the conformal equivalence acts on. Now, by the Reflection Principle we can extend the bijective conformal mapping Now, by the Reflection Principle we can ex from K to the unit disk, where the two points on ± with the same imaginary from part K to the unit disk, where the two points go to the points on the boundary on the unitreferences disk with the same real part. go to the points on the boundary on the unit Finally, z i( z+ z ) sends K as the unit disk to the upper-half plane, wherefinally, two z i( z+ z ) sends K as the unit d points E. Macrì, on the Some boundary examples with of spaces theofsame stabilityreal conditions valueon are derived mapped categories, to two preprint, points arxiv:math/0463v3 points on theon the boundary [math.g]. with the same real va boundary S. Okada, with Stability the manifold same absolute of, preprint, value. arxiv:math/04220v3 Then, z z 2 sends [math.g]. K as the upper-half boundary with the same absolute value. The plane to C and identifies on the real axis. plane to C and identifies on the real axis. We have z i (Żi, Ṗi) = (Z i, P i ) X, since (z i φ i )(O) = φ(o(i )[]) φ(o(i)) < implies (z i φ i )(O) (0, ]. Graphically we can explain actions above as follows. [ri07] T. ridgeland, Stability conditions on triangulated categories, nn. of Math. (2) 66 (2007), [Mac04] [Oka04] Proof of Theorem.. y emma 4.4, we have Stab(D(P ))/(Z)C = C. The Proof ac-otion of Z on X = Stab(D(P ))/C gives an exact sequence 0 π (X) π (X/Z) tion of α Z on X = Stab(D(P ))/C gives an exa Theorem.. y emma 4.4, we ha π 0 (Z) π 0 (X). We will show that X is connected. Recall that Stab(D(P )) = (Z)C X and, by emma 4.2, X π 0 (Z) π 0 (X). We will show that X is co = H is connected, hence so is C X. It remains (Z)C X and, by emma 4.2, X = H is con to check that (Z) fixes some connected component of Stab(D(P )), but it to fixes check that (Z) fixes some connected com {(Z, P) Stab(D(P )) P((0, ]) = Coh P }, which lives in X. So X is connected {(Z, P) Stab(D(P )) P((0, ]) = Coh P }, and the map α is a surjective map Z Z, therefore α is injective and π (X) = 0. Hence X is the universal covering of C ; i.e., Stab(D(P ))/C and the map α is a surjective map Z Z, th = C. Moreover, since H (C, O) = 0, Stab(D(P )) Hence X is the universal covering of C ; i.e., = C 2. H (C, O) = 0, Stab(D(P )) = C Walls and hearts of Stab(D(P )) 5. Walls and hearts We define the cell Stab C (T ) for a heart C by Stab C (T ) = {(Z, P) Stab(T We ) define the cell Stab C (T ) for a heart P((0, ]) = C} (S C (T ) for short). In this section, we describe all cells S C (D(P P((0, )) and]) = C} (S C (T ) for short). In this sectio how they fit together. We describe a fundamental domain of Stab(D(P ))/(Z)R how as they fit together. We describe a fundam a real manifold, this shows that the quotient Stab(D(P ))/(Z)R is an open torus. a real manifold, this shows that the quotient For the hearts C j and C p,i,j from Corollary 3.4, let S j = S Cj and S p,i,j = S Cp,i,j For. the hearts C j and C p,i,j from Corollary Then they can be described as Then they can be described as S j = {(Z, P) Stab(D(P )) φ(o x [j]) =, 0 < φ(o[j]) < } = R <0 H; S j = {(Z, P) Stab(D(P )) φ(o x [j]) = S p,i,j = {(Z, P) Stab(D(P )) 0 < φ(o(i )[p + j]), φ(o(i)[j]) } = HS 2. p,i,j = {(Z, P) Stab(D(P )) 0 < φ(o(i