Bridgeland stability conditions on some Picard rank one varieties

Transcription

1 Bridgeland stability conditions on some Picard rank one varieties Stavanger workshop Chunyi Li University of Edinburgh September 8, / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

2 1 The CCC Plane 2 2 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

3 Review: Bridgeland Stability Condition Stability conditions: σ = (Z, P) defined on a C-linear triangulated category T. The kernel of the central charge Z avoids all stable characters Support condition: the cone neighborhood of kerz avoids all stable characters T : D b coh (X ) X : Picard rank one (smooth projective connected) varieties over C All examples are of dimension two or three, but statements may also hold for higher dimensional varieties. 3 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

4 Review: Geometric stability conditions Let X be a smooth projective connected surface. Geometric stability condition: all skyscraper sheaves are stable with the same phase. Non-degenerate: the image of the central charge is not contained in a real line. When X is of Picard rank 1 with ample generator H, kerz in K R (X ) is a line: a point in P(K R (X )). Bound for both slope stable and σ-stable: Bogolomov inequality H = (Hch 1 ) 2 2ch 2 (H 2 ch 0 ) 0 Point (avoids stable characters) on P(K R (X )) Kernel of the Central charge 4 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

5 The CCC-plane We use Chern characters {H 2 ch 0, Hch 1, ch 2 } for the bases of the projective plane P(K R (X )), and call it the CCC-plane. The line at infinity: ch 0 = 0. RZ ch2 P H 2 ch0 = 0 > 0 Stab Geo E 0 stable characters P E IZ P Hch1 H 2 ch0 Let (1, s, q) be a point on the CCC-plane, then σ s,q = (Z s,q, P s,q ) is constructed as follows: P s,q ((0, 1]) := Coh s (X ). Z s,q = (ch 2 q ch 0 ) + i(ch 1 s ch 0 ). When a neighborhood of (1, s, q) is not below any stable character on the CCC-plane, σ s,q is a geometric stability condition. 5 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

6 The CCC-plane/ the upper half plane Relation with the upper half plane model: σ s,t σ s, s t2 2 t ch2 H 2 ch0 = 0 s Hch1 H 2 ch0 Upper half plane model: σ s,t = (Z s,t, P s,t) Z s,t = ch s 2 + t2 2 ch 0 + itch s 1 6 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

7 Examples: space of the geometric stability conditions In the Picard rank one case, Stab Geo / GL(2, R) CCC-plane. K3 ( 2H) ch2 H 2 ch0 (2H) ( H) = 0 χ = 0 P 2 Hch1 H 2 ch0 e + ( 3) ch2 ch0 e + (2) = 0 = 1 2 = 1 e + (3) e + ( 5 2 ) e+ ( 5 2 ) e + ( 2) e + ( 1) e + (1) e + (0) ch1 ch0 7 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

8 Potential walls Given two Chern characters w and v, suppose their projections on the CCC-plane are well-defined, the potential wall of them is the straight line across them. The nested wall property is clear. ch 2 H 2 ch 0 = 0 Z(w)//Z(v) kerz, w and v are colinear v Z P v Hch 1 H 2 ch 0 w 8 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

9 Examples: potential walls We consider the first wall of Hilb 2 (X ) for two different X. ch2 ch2 ch0 P 2 = 0 H 2 ch0 X5 = 0 ( H) ch1 ch0 ( H) w (1, 0, 2 5 ) Hch1 H 2 ch0 (1, 0, 2) w 9 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

10 Compare the slope Let P = (1, s, q) be a point on the CCC-plane, E and F be two objects in Coh s. The inequality σ P (E) > σ P (F ) holds if and only if the ray l + PE is above l + PF. ch2 H 2 ch0 = 0 Z P w l + PF F l + PE Hch1 H 2 ch0 E 10 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

11 CCC-plane: From surfaces to higher dimensional varieties When X is a higher dimensional variety with Picard rank one, the same story works for the first tilt slope function. H: Ample divisor. The first tilt heart Coh sh (X ) := T sh (X ), F sh (X )[1] Reduced central charge t > 0: Z s,t (E) = 3tH 2 ch sh 1 (E) + i ( 3H ch sh 3 2 (E) 2 t2 H 2 ch sh 0 (E) ). ν s,t : tilt slope function IZs,t RZ s,t n the CCC-plane, the kernel of Z s,t is (1, s, t2 +s 2 2 ). 11 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

12 Stability condition on three-fold (Review of the progress) The existence of Bridgeland stability conditions on 3-fold. Bayer, Bertram, Macrì, Toda: the double tilting Conjectural Bogolomov-Gieseker type inequality involves the third Chern character P 3 : Macrì; Quadratic hypersurface: Schmidt Abelian 3-fold (Picard rank one): Maciocia, Piyaratne Finite quotient of Abelian 3-fold: Bayer, Macrì, Stellari. 12 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

13 Stability condition on three-fold: reduce to small t Stability conditions on abelian threefolds and some Calabi-Yau threefolds. Bayer, Macrì, Stellari. Picard rank one case: Conjecture 5.3 E stable stable H ch 2 H 3 ch 0 E stable L E+ E = 0 H 2 ch 1 H 3 ch 0 13 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

14 Conjecture 5.3: Definition of β(e) Let E be an object in D b (X ) such that ch 0 (E) and ch 1 (E) are not both 0. β(e) = if ch 0 (E) = 0, H ch 2 (E) H 2 ch 1 (E) if ch 0 (E) 0, H 2 ch 1 (E) H (E) H 3 ch 0 (E) tangent line H ch2 H 3 ch0 = 0 β(f ) w β(e) tangent line E H 2 ch1 H 3 ch0 We call an object E: β-stable if there exists an open neighbor hood U R 2 of (0, β(e)) such that for any (t, s) U with t > 0, either E or E[1] is a ν s,t tilt-stable object of Coh β (E). F 14 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

15 The Conjecture Conjecture (Bayer, Macrì, Stellari) Let E D b (X ) be a β-stable object, then ch β(e)h 3 (E) 0. When X is of Picard rank one, the conjecture implies the existence of stability conditions on D b (X ). 15 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

16 Picard rank one Fano 3-fold Theorem The conjecture holds for Picard rank one Fano 3-folds. Classification of Picard rank one Fano 3-folds: Index: 1,2,3,4 Degree: up to / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

17 Idea of the proof: compare the slope X : Picard rank one Fano 3-fold. H: the ample divisor generator. d: the index, K X = dh. May assume 0 β(e) < 1. H ch 2 H 3 ch 0 H = 0 (dh) β(e) E l + β(2h) l + βe l + β H 2 ch 1 H 3 ch 0. Hom((dH), E) = 0 Hom(E, [1]) = 0 Hom((dH), E[2]) = 0 χ((sh), E) 0, for 1 s d. 17 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

18 Fano 3-folds: HRR formula Hirzebruch-Riemann-Roch: χ(e) = ch 3 (E) + td 1 (X )ch 2 (E) + td 2 (X )ch 1 (E) + td 3 (X )ch 0 (E). X Fano: td i (X ) is positive. Expand χ((sh), E) in terms of ch β i (E), the coefficient for chβ 1 (E) is positive. 18 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

19 Example X : smooth cubic 3-fold in P 4.Hirzebruch-Riemann-Roch: χ(e) = ch 3 (E) + Hch 2 (E) H2 ch 1 (E) H3 ch 0 (E). 0 χ((h), E), χ((2h), E). ch β 3 (E) + βchβ 2 (E) + ( β ) H 2 ch β1 (E) + ( β β 6 ) H 3 ch β 0 (E) ch β 3 (E) + ( β 2 β + 2 ) 3 H 2 ch β 1 (E) + ( β 3 6 β β 3 3) 1 H 3 ch β 0 (E) 19 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

20 Index one case The naive idea fails. A stricter bound for stable characters: ( 3H) ( 2H) ( H) H ch 2 H 3 ch 0 H = 0 H = 3 2d w(h) (2H) H 2 ch 1 H 3 ch 0 χ(e, E) 1. Above the curve H = 3 2d implies the =. Above the tangent lines implies Ext 2 (E, E) = / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

21 Remark Higher Picard rank: partial result. Blpt P 3 : ω and B are parallel to K X, or 2H 0 H 1. Calabi-Yau 3-fold:?? Suppose E and F are both β H -stable such that H ch2 H 3 ch0 = 0 then β H (E) < β H (F ), β(e) Hom(F, E) = 0. β(f ) E w F H 2 ch1 H 3 ch0 21 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces

22 Thank you! 22 / 22 Chunyi Li Workshop: Derived Categories and Moduli Spaces