A 1 Bridgeland A [15] f : X Y = Spec C[x, y, z]/(xy + z n+1 ) A n, Z = f 1 (0) D = D Z (X) X Z C Rf E = 0 E D [15] 1.1 ([15]). D Stab D Stab C Stab C

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1 A 1 Bridgeland A [15] f : X Y = Spec C[x, y, z]/(xy + z n+1 ) A n, Z = f 1 (0)D = D Z (X) X Z C Rf E = 0 E D [15] 1.1 ([15]). D Stab D Stab C Stab C Thomas [22] Bridgeland K3 2 Harder- Narasimhan 2.1

2 2.1. C E E E F E deg F rank F deg E rank E E E x C O x Z(E) = deg E + 1 rank E C (2.1). E or F E arg Z(F ) < (or ) arg Z(E) (2.2) 2.2 Harder-Narasimhan E C Z(E) (2.1) E Harder-Narasimhan (HN) 2.2.E 0 = E 0 E 1 E n 1 E n = E E i /E i 1 arg Z(E i /E i 1 ) > arg Z(E i+1 /E i )

3 E E Jordan-Hölder (JH) E i /E i+1 arg Z(E i /E i+1 ) i E i /E i 1 HNE = E C E {τ <n E } (τ <n+1 E )/(τ <n E ) = H n (E )[ n] (canonical filtraion) [ ] H n (E) HN 0 = E 0 E 1 E 2 E n 1 E n = E A 1 A 2 A n A j i > j Hom D b (C)(A i, A j ) = 0 3 T K(T )Grothendieck Douglas [1, 6, 7, 8, 9] BPS D-brane Bridgeland [4] T [4] T σ = (Z, P) Z : K(T ) C φ R P (φ) T (i) 0 E P(φ) Z(E) = m(e) exp(iπφ) m(e) R >0, Z(E) 0 (ii) φ R, P(φ + 1) = P(φ)[1], (iii) φ 1 > φ 2 A j P(φ j ) (j = 1, 2), Hom T (A 1, A 2 ) = 0,

4 (iv) 0 E T φ 1 > φ 2 > > φ n 0 = E 0 E 1 E 2 E n 1 E n = E A 1 A 2 A n (3.1) A j P(φ j ). Z central charge, (3.1) E Harder-Narasimhan (HN) P (φ)(semi-stable) φphase P (φ) P (φ) (stable) 3.2. C T C Z : K(T ) C (2.1)0 < φ 1P(φ) arg Z(E) = πφ,φ R φ n (0, 1]n Z P(φ) = P(φ n)[n] 2.2 T σ = (Z, P) T I R P(φ) (φ I) extension closed subcategory P(I) P(I) P(φ) (φ I) A B C A[1] A, C P(I)B P(I) T 3.3.φ R P((φ, φ + 1]) t σ t P((0, 1]) σ (t) tz 3.4. T T t A Z : K(T ) C 0 E AIm Z(E) > 0Z(E) R <0,

5 Z A HN property (2.2) A2.2 HN filtraion 3.5. A S 1,..., S n Z : K(A) = i Z[S i ] C Im Z(S i ) > 0 Z(S i ) R <0 (i = 1,..., n) T = D b (A) (T = D b (A) A T t JH 3.6. σ = (Z, P) η > 0 t RP(t η, t + η) ( ) JH T Stab T Z : Stab T Hom(K(T ), C) (Z, P) Z (3.2) 3.7 ([4, Theorem 1.2]). Stab T Σ Hom(K(T ), C) V (Σ)Z Σ K K

6 4 A f : X Spec C[x, y, z]/(xy + z n+1 ) A n Z = f 1 (0) = C 1 C n C i = P 1 ( 2)- i j > 1C i C j = D Z X, C D Rf E = (1) β, ω X ω Z β,ω (E) = χ(e) + (β + 1ω) c 1 (E) (χ(e) = i ( 1)i dim H i (E).) D t (A = coh Z (X)) Z β,ω 3.4, D σ β,ω HNSimpson ω- ββ, ω x Z O x coh Z (X)σ β,ω V = {σ = (Z, P) O x σ-, O x P(1) Z(O x ) = 1 ( x Z)} Stab D 4.1. V = {σ β,ω β, ωω}v U = {σ = (Z, P) O x σ- ( x Z)} Stab D σ U O x phase x Z 4.2. U 4.3. Stab 0 D UStab D

7 4.2 McKay 3 McKay[17], [5] X C 2 G( = Z/(n + 1)Z) C 2 G- coh G (C 2 ) C[x, y] G- McKay D b (coh X) = D b (coh G (C 2 )) (4.1) coh G 0 (C 2 ) coh G (C 2 ){0} G ρ 0,..., ρ n S i := ρ i C O 0 (i = 0,..., n) coh G 0 (C 2 )D (4.1) ω Z [1], O C1 ( 1),..., O Cn ( 1) S 0, S 1,..., S n ρ 0 (4.1) Rf G- D S 1,..., S n C 4.3 (2) D t D b (coh G 0 (C 2 )) t ( McKay ) Z : K(D) C Im Z(S i ) > 0Z(S i ) R <0 3.5 V Stab D V Stab 0 D 4.4. ω A(X/Y ) j ω C j = 1 β = coh Z (X) Z 0,ω σ V Z 0,ω (O x ) = 1, Z 0,ω (O Cj ( 1)) = 1 ( j), Z 0,ω (ω Z ) = 1 + n 1

8 O Cj ( 1)ω Z σ- α (0, 1/2) tan(πα) > n P((α, α + 1]) ω Z [1], O C1 ( 1),..., O Cn ( 1) McKay S 0,..., S n. P((α, α + 1]) = coh G 0 (C2 ). σ V phasev 4.4 T Auteq T 1 Auteq T Stab T ˆX X Z Auteq D (Aut ˆX Pic X) Z Aut ˆXPic X = Pic ˆX 1 Z 4.5. E D Hom D (E, E[i]) = { k if i = 0, 2, 0 otherwise ( 2) O Ci, D 4.2 S i 4.7. E DT E RHom(E, α) C E α T E (α) Seidel-Thomas [21] ET E Br(D) = T S0,..., T Sn Auteq D 1 Out(T )

9 4.8 ([16] + [15]Appendix). Auteq D Auteq D = (Aut ˆX (Br(D) (Z/(n + 1)Z))) Z Z/(n + 1)Z S i S i+1 Pic X Br(D) (Z/(n + 1)Z) Seidel-Thomas [21] Br(D) {T Si } B n (1) σ 0,..., σ 1 σ i σ i+1 σ i = σ i+1 σ i σ i+1, i = 0,..., n, σ i σ j = σ j σ i, i j > 2. σ n+1 = σ 0 σ i T Si ρ : B (1) n Br(D) Auteq D Stab D Br(D) Stab Dρ 4.5 Bridgeland Bridgeland [3] K3 K3 [2] D Grothendieck K(D) E, F K(D) χ(e, F ) = i ( 1) i dim Hom i D (E, F ), (4.2) Riemann-Roch χ(e, F ) = c 1 (E) c 1 (F ). χk(d) McKay C 2 (χ(s i, S j )) Cartan Hom(K(D), C) Cartan ĥ = h C A n h = {(a 1,..., a n+1 ) C n+1 a a n+1 = 0}

10 ĥ ĥreg ĥ reg = {(a 1,..., a n+1, b) h C a i a j + bd 0 for i j and d Z}. ĥ reg Weyl Ŵ ĥreg /Ŵ Z [10] B (1) n 4.9 ([2]). Stab 0 D(3.2)Stab 0 (D) ĥ reg /Ŵ Br(D) Z Stab 0 D Z 2[2] B (1) n Z Br(D) Z ρ idstab 0 (D) ρ Bridgeland K3 Auteq D Stab 0 D Stab D 2. Stab 0 D [3].D C [15] 5 Stab D Stab D (i) Stab D Σ 3.7 V (Σ) Hom(K(D), C)σ = (Z, P) Σ Z(O x ) R <0 E Z(E) R (ii) σ Σσ-ω P(1)σ- E 1, E 2 P((0, 1]),P((0, 1]) 0 E 1 E 2 ω 0 Ext 1 (E 1, E 2 ) = 0 2 ICM 2006

11 (iii) [16] Φ Br(D) Φ(ω) = O x [d] ( x Z, d Z), [16]A (iv) τ Stab(D) x C i Z O x τ- x τ HN xz = C 1 C i 1, Z = C i+1 C n τ D Z (X)D Z (X) (v) nφ Br(D) Φσ O x Φσ U Stab 0 (D) (vi) Br(D) Stab 0 D σ Stab 0 (D) 6 Stab D Br Dρ ρ A n (i) A n [12, 13][11] W = {(x, y, z) C 2 C xyz = z n+1 1} W z C 2 1 n + 1 C 1 n W (ii) ζ = exp[2π 1/(n + 1)] ζ i ζ i+1 c i {L i } n i=0 Fuk W A n D = D b Fuk W

12 (iii) Seidel [20, Proposition 9.1] D T Si D b Fuk W Dehn τ i C n Fuk W D b Fuk W [20, Proposition 9.1] ζ i+2 ζ i+1 ζ i ζ i 1 Figure 1: The Dehn twist τ i (iv) Khovanov-Seidel [19, Theorem 1.3] Floer C 2 2 (v) b B n (1) ρ(b) = id b(s j ) = S j b B n (1) Hom(b (S i ), b(c j )) = Hom(b (S i ), S j ). b (c i ) b(c j ) b (c i ) c j b(c i ) c i (isotopic) (vi) B (1) n ([18])b = id 6.2 k ρ = ρ k 2 ρ = ρ 2 (i) A n Spec Z Z (ii) Z Z ([14])

13 (iii) O Ci (d) (iv) ρ k (b) = id,ρ 2 (b)(o Ci (d)) = O Ci (d) i d ρ 2 (b) = id 2 b = id 7 Stab C Stab D O x Stab C [16] C C S 1,..., S n C 3.5 Br(C) = T S1,..., T Sn Auteq D (i) Stab C Σ 3.7 V (Σ) Hom(K(C), C)σ = (Z, P) Σ E Z(E) R Stab D (ii) σ P((0, 1]) n E 1,..., E n A n (iii) [16]C Br(C) S i ( i, 1 i n) (iv) Khovanov, Seidel, Thomas[19, 21] C A n V = {(x, y, z) C 3 xy = z(z 1) (z n)} () C = D b Fuk V Fuk V z [i 1, i](i = 1,..., n) L i S i L i

14 (v) D b Fuk V (iii) (ii) E i D b Fuk V z (vi) {0,..., n} C Br(C) {E 1,..., E n } {S 1,..., S n } (vii) σ Br(C)C t Stab D Stab C References [1] Paul S. Aspinwall and Michael R. Douglas. D-brane stability and monodromy. J. High Energy Phys., (5):no. 31, 35, [2] Tom Bridgeland. Stability conditions and Kleinian singularities. math.ag/ [3] Tom Bridgeland. Stability conditions on K3 surfaces. math.ag/ [4] Tom Bridgeland. Stability conditions on triangulated categories. math.ag/ [5] Tom Bridgeland, Alastair King, and Miles Reid. The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc., 14(3): (electronic), [6] Michael R. Douglas. D-branes, categories and N = 1 supersymmetry. J. Math. Phys., 42(7): , Strings, branes, and M-theory. [7] Michael R. Douglas. D-branes on Calabi-Yau manifolds. In European Congress of Mathematics, Vol. II (Barcelona, 2000), volume 202 of Progr. Math., pages Birkhäuser, Basel, [8] Michael R. Douglas. Dirichlet branes, homological mirror symmetry, and stability. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pages , Beijing, Higher Ed. Press.

15 [9] Michael R. Douglas, Bartomeu Fiol, and Christian Römelsberger. Stability and BPS branes. J. High Energy Phys., (9):006, 15 pp. (electronic), [10] Nguy ên Viêt Dũng. The fundamental groups of the spaces of regular orbits of the affine Weyl groups. Topology, 22(4): , [11] Jiro Hashiba and Michihiro Naka. Landau-Ginzburg description of D- branes on ALE spaces. Nuclear Phys. B, 599(1-2): , [12] Kentaro Hori, Amer Iqbal, and Cumrun Vafa. D-branes and mirror symmetry. hep-th/ [13] Kentaro Hori and Cumrun Vafa. Mirror symmetry. hep-th/ , [14] Michi-aki Inaba. Toward a definition of moduli of complexes of coherent sheaves on a projective scheme. J. Math. Kyoto Univ., 42(2): , [15] Akira Ishii, Kazushi Ueda, and Hokuto Uehara. Stability conditions on A n -singularities. math.ag/ [16] Akira Ishii and Hokuto Uehara. Autoequivalences of derived categories on the minimal resolutions of A n -singularities on surfaces. J. Differential Geom., 71(3): , [17] M. Kapranov and E. Vasserot. Kleinian singularities, derived categories and Hall algebras. Math. Ann., 316(3): , math.ag/ [18] Richard P. Kent, IV and David Peifer. A geometric and algebraic description of annular braid groups. Internat. J. Algebra Comput., 12(1-2):85 97, International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). [19] Mikhail Khovanov and Paul Seidel. Quivers, Floer cohomology, and braid group actions. J. Amer. Math. Soc., 15(1): (electronic), [20] Paul Seidel. Homological mirror symmetry for the quartic surface. math.ag/

16 [21] Paul Seidel and Richard Thomas. Braid group actions on derived categories of coherent sheaves. Duke Math. J., 108(1):37 108, [22] Richard P. Thomas. Stability conditions and the braid group. math.ag/

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β