TOPOLOGY HYPERPLANE ARRANGEMENTS. Alex Suciu. Northeastern University.
|
|
- Fay Todd
- 5 years ago
- Views:
Transcription
1 TOPOLOGY OF HYPERPLANE ARRANGEMENTS Alex Suciu Northeastern University A.M.S. Fall Eastern Section Meeting Columbia University, New York, NY November 5,
2 Hyperplane arrangements Arrangement: A collection A = {H 1,...,H n } of hyperplanes, H i =ker α i,inc l. Defining polynomial: Q A = α 1 α n. Intersection lattice: L(A) = { H B H B A}. ordered by reverse inclusion, ranked by codimension Complement: X(A) =C l \ H A H. Example. Boolean arrangement D n in C n. Q Dn = z 1 z n L(D n ): X(D n )=(C ) n π 1 (X(D n )) = Z n subsets of [n] :={1,...,n} complex n-torus Example. Braid arrangement B l in C l. Q Bl = 1 i<j l (z i z j ) L(B l ): X(B l )=F (l, C) π 1 (X(B l )) = P l partition lattice of [l] configuration space of l ordered points in C pure braid group on l strings; in fact: X(B l ) K(P l, 1) 2
3 Types of arrangements Complexified A = A R C (i.e., Q A has real coefficients) Reflection Complexification of reflecting hyperplanes of a Coxeter group. E.g., B l of type A l-1. Simplicial Complexification of real arrangement, all of whose complementary regions are open simplices. Deligne: A simplicial = X(A) K(π, 1). Fiber-type (Falk-Randell) There is a tower of linear fibrations X(A) p l X(A l 1 ) p l 1 X(A 2 ) p 2 X(A 1 )=C with fiber(p i )=C \{d i points}. Thus, X K(G, 1), with G = F dl ρl ρ2 F d1, where ρ i : π 1 (X(A i 1 )) P di Aut(F di ). Terao: A fiber-type L(A) supersolvable Cohen-S.: Explicit ZG-resolution C ( X) Z (using the Fox Jacobians of ρ i ) 3
4 Generic A = {H 1,...,H n } in C l (2 < l < n), s.t. codim H B H = B, for all B Awith B l. E.g.: Boolean Hattori: X(A) S 1 (T n 1 ) (l 1). Thus: π 1 (X) =Z n, π i (X) =0 (if 1 <i<l 1) π l 1 (X) 0. Hypersolvable (Jambu-Papadima) Combinatorial condition, generalization of both supersolvable and generic. Hattori s result generalizes in this context (Papadima-S.) Graphic Each simple graph G with vertices {1,...,m} defines an arrangement in C m : A G = {ker(z i z j ) {i, j} edge in G} G diagram of type A = A G Boolean G = K m = A G = B m G polygon = A G generic every cycle in G has a chord A G fiber-type (Stanley, Fulkerson-Gross) 4
5 Cohomology ring H (X(B l )) = H (P l ): Arnol d H (X(A)): Brieskorn, Orlik-Solomon H (F (l, R k )): F. Cohen H (F (l, S k )): Feichtner-Ziegler, Xicoténcatl H (X(subspace arrangement)): Goresky-MacPherson, Björner-Ziegler, De Concini-Procesi, Yuzvinsky, de Longueville Theorem. (Orlik-Solomon) Let A = {H 1,...,H n } be a hyperplane arrangement. Then L(A) determines the cohomology ring of X = X(A): H (X) = Z n /( e B codim H B A ) H< B where Z n = exterior algebra over Z on generators e i (dual to meridian of H i ) in degree 1, and, for B = {H i1,...,h ir }, e B = e i1 e ir and e B = q ( 1)q 1 e i1 ê iq e ir. H (X) is torsion-free; basis: no broken circuits (nbc). Poincaré polynomial: P (X, t) = µ(y )( t) codim Y where µ : L(A) Z Möbius function. Y L(A) 5
6 Resonance varieties Let A = H (X(A), C) be the Orlik-Solomon algebra of A. Forλ C n, let e λ = n i=1 λ ie i A 1 = C n. Aomoto complex : 0 A 0 e λ A 1 e λ A 2 e λ A l 0 The resonance varieties of A were defined by Falk as: R k d(a) ={λ C n dim C H k (A,e λ ) d} They are actually subsets of n := {λ n i=1 λ i =0}, and depend only on A (up to linear iso of C n ). Theorem. Each component of Rd k (A) is a linear subspace in C n. Conjectured by Falk. Proved by Cohen-S., Libgober (k = 1); Cohen-Orlik, Libgober (all k). The varieties R d (A) =Rd 1 (A) admit a purely combinatorial description started by Falk, completed by Libgober-Yuzvinsky, using Vinberg s classification of affine Kac-Moody algebras. 6
7 A partition P =(p 1 p q )ofa is neighborly if ( p j I I 1) = I p j, I L 2 (A) It defines a linear subspace of C n : L P = n {λ {I L 2 (A) I p j, j} i I λ i =0}. Theorem. (Libgober-Yuzvinsky) All components L i of R 1 (A) arise from neighborly partitions of sub-arrangements A A. dim L i 2. L i L j = {0} for i j. R d (A) ={0} dim L i d+1 L i. E.g., for each I L 2 (A) with I 3, there is a local component L I = {λ n i=1 λ i =0 and λ i =0 for i/ I} corresponding to the partition (I)ofA I = {H i i I}. Note that dim L I = I 1, and thus L I R I 2 (A). 7
8 Example. (Braid arrangement B = B 4 ) L 156 L 246 L 345 L 123 L ( ) The OS-algebra A = H (X(B), C) has generators e 1,...,e 6, and relations e 2 i = e ie j + e j e i =0 and e 1 e 2 e 1 e 3 + e 2 e 3 =0, e 1 e 5 e 1 e 6 + e 5 e 6 =0, e 2 e 4 e 2 e 6 + e 4 e 6 =0, e 3 e 4 e 3 e 5 + e 4 e 5 =0. The resonance variety R 1 (B) C 6 has 5 components (4 local, and 1 non-local), all 2-dimensional: L 123 = {λ λ 1 + λ 2 + λ 3 = λ 4 = λ 5 = λ 6 =0} L 156 = {λ λ 1 + λ 5 + λ 6 = λ 2 = λ 3 = λ 4 =0} L 246 = {λ λ 2 + λ 4 + λ 6 = λ 1 = λ 3 = λ 5 =0} L 345 = {λ λ 3 + λ 4 + λ 5 = λ 1 = λ 2 = λ 6 =0} L ( ) = {λ λ 1 λ 4 = λ 2 λ 5 = λ 3 λ 6 = λ i =0} 8
9 Characteristic varieties X finite CW-complex, G = π 1 (X). Assume G ab = Z n (with basis t 1,...,t n ). Character variety: Hom(G, C ) = (C ) n algebraic torus, with coordinate ring C[t ±1 1,...,t±1 n ] Characteristic varieties: V k d (X) ={t (C ) n dim C H k (X, C t ) d} where C t is the G-module C with action given by the representation t : G C. Vd k (X) depends only on the homotopy type of X (up to a monomial isomorphism of (C ) n ). Theorem. (Arapura) Suppose X is the complement of a normal-crossing divisor in a compact Kähler manifoldwith b 1 =0. Then the components of V k d (X) are subtori of (C ) n, possibly translatedby roots of 1. Uses Deligne s mixed Hodge structures. Generalizes results of Green-Lazarsfeld, Simpson. 9
10 Characteristic varieties of A = {H 1,...,H n }: V k d (A) :=V k d (X(A)) (C ) n Recall the resonance varieties Rd k(a) Cn. Theorem. (Cohen-S., Cohen-Orlik, Libgober) TC 1 (V k d (A)) = R k d(a) As a consequence, the components of Vd k (A) passing through 1 are combinatorially determined (by L(A)). In general, though, there do exist components that do not pass through 1 (i.e., translated subtori). Question. Are such components combinatorially determined? Remark. Rd k (X) may be defined for arbitrary X. Then (Libgober): TC 1 (V k d (X)) R k d(x). But the inclusion is strict in general, e.g., for link complements (Matei), and even for complements of real subspace arrangements (M.-S.). 10
11 Fundamental groups of arrangements A = {H 1,...,H n } hyperplane arrangement, with complement X, and fundamental group G = π 1 (X). A = {l 1,...,l n } generic 2-section. By Lefschetz-type theorem of Hamm and Lê :π 1 (X) = π 1 (X ). So reduce to the case where A is an arrangement of affine lines in C 2. Let v 1,...,v s be the intersection points of the lines. v q = l i1 l ir I q := {i 1,...,i r } Lattice: L 1 (A) =[n], L 2 (A) ={I 1,...,I s }. Group: G(A) =π 1 (X(A)). Question. Is G(A) combinatorially determined? I.e.: L(A 1 ) = L(A 2 ) = G(A 1 ) = G(A 2 )? According to Rybnikov, the answer is no. 11
12 Presentation for G = G(A) (Van Kampen, Artin/Randell, Salvetti, Arvola/Moishezon, Libgober, E. Hironaka, Cordovil-Fachada, Cohen-S.,... ): G = x 1,...,x n α 1 (x i )=x i,...,α s (x i )=x i (1 i n) where α q = A δ q I q P n acting on F n = x 1,...x n via the Artin representation (A I =full twist on I-strands, and δ q B n can be read from a braided wiring diagram ). Example. l 4 v 3 v 1 v 4 v 2 l 1 l 3 l 2 I 1 I 2 I 3 I α 1 = A 23, α 2 = A A 23 13, α 3 = A 124, α 4 = A
13 G = x1,x 2, x 1 x 2 x 4 = x 4 x 1 x 2 = x 2 x 4 x 1, x 3,x 4 [x 1,x 3 ]=[x 2,x 3 ]=[x 4,x 3 ]=1 = F 2 Z 2 13
14 Characteristic varieties of G (over field K): V d (G, K) ={t Hom(G, K ) dim K H 1 (G, K t ) d} For d<n, we have (E. Hironaka): V d (G, K) ={t (K ) n rank K A G (t) <n d} where A G = J ab G is the Alexander matrix of G, obtained by abelianizing the Fox Jacobian J G = ( r i x j ). Resonance varieties of G (over K): R d (G, K) = { λ H 1 (G, K) subspace W H 1 (G, K), dim W = d +1,λ W =0 } Then (Matei-S.): R d (G, K) ={λ K n rank K A (1) G (λ) <n d} where A (1) G (λ) =(A G ti =1 λ i ) linear terms is the linearizedalexander matrix of G. Remark. If G = π 1 (X(A)), then R d (A) =R d (G, C). But: R d (G, C) modp R d (G, F p ), TC 1 (V d (G, F p )) R d (G, F p ). 14
15 Homology of finite covers Theorem. (Libgober, Sakuma) G f.p.group, G ab = Z n, 1 K G γ Γ 1. IfΓ finite abelian, then: b 1 (K) =n + (corank J ρ γ G 1) 1 ρ Hom(Γ,C ) More generally, let b (q) 1 (G) :=dim K H 1 (G, K) where K is a field of characteristic q. Theorem. (Matei-S.) If Γ finite and q Γ, then: b (q) 1 (K) =b(q) 1 (G)+ n ρ (corank J ρ γ G n ρ) ρ 1 (sum over all non-trivial irreps ρ :Γ GL(n ρ, K), K field of char. q containing all roots of 1 of order exp Γ). 15
16 Corollary. (M.-S.) Let K =ker(γ : G Z N ). Then: b (q) 1 (K) =b(q) 1 (G)+ 1 k N φ(k) depth K (γ N/k ) where depth K (t) :=max{d t V d (G, K)}. Application to homology of Milnor fiber of a (central) arrangement A. Milnor fibration: F (A) X(A) Q A C H (F (A)) studied by Randell, Orlik-Randell, Artal Bartolo, Cohen-S., Denham, Cohen-Orlik,... F (A) isthen-fold cyclic cover of X(A), given by γ : G Z n, γ(x i )=1 Thus: b (q) 1 (F )=n 1+ φ(k) depth K (γ n/k ) 1 k n Question. Is H (F (A)) combinatorially determined? Question. Is H (F (A)) torsion-free? In particular, is b (q) 1 (F )=b 1(F ), for all q n? 16
17 Corollary. (M.-S.) K =ker(γ : G Z p ). Then: b (q) 1 (K) =b 1(G)+(p 1) depth K (γ) where K = C if q =0,orK = F q s if q prime, q p. s = ord p (q) =smallest positive integer s.t. p q s 1 F s q = F q (ζ), where ζ is a primitive p-th root of 1 Define: β (q) p,d (G) = {K G [G : K] =p and b (q) 1 (K) =b(q) 1 (G)+(p 1)d } Then: β (q) p,d (G) = Tors p(v d (G, K) \ V d+1 (G, K)) p 1 where for V K n. Tors N (V )={t V t N = 1 and t 1} 17
18 Representations onto finite groups G f.g. group, Γ finite group. δ Γ (G) := Epi(G, Γ)/ Aut(Γ) = {factor groups of G that are isomorphic to Γ} May compute δ Γ (G) when Γ abelian, or a semidirect product of (certain) abelian groups (Matei-S.) Γ abelian Write Γ = p Γ Γ p, where Γ p is a finite abelian p-group. Clearly, δ Γ (G) = p Γ δ Γ p (G). Write Γ p = Z p ν 1 Z p ν k,ν=(ν 1 ν k ). If G ab = Z n, then: δ Γp (G) = p ν (n 1) 2 ν ϕ n (p 1 ) ϕ n k (p 1 ) r 1 ϕ m r (ν)(p 1 ) where: ν = k i=1 ν i, ν = k i=1 (i 1)ν i, m r (ν) = {j ν j = r}, ϕ m (t) = m i=1 (1 ti ). 18
19 Γ=Z s q σ Z p (p q primes, s = ord p (q), σ Aut(Z s q) of order p) δ Z s q σ Z p (G) = p 1 s(q s 1) n d=1 β (q) p,d (G)(qsd 1) Thus, we may compute δ Z s q σ Z p (G) from V d (G, F q s). E.g., for S 3 = Z 3 ( 1) Z 2 and A 4 = Z 2 2 ( ) 01 Z3 : δ S3 (G) = 1 2 δ A4 (G) = 1 3 d 1 d 1 11 Tors 2 (V d (G, F 3 ) \ V d+1 (G, F 3 ))(3 d 1) Tors 3 (V d (G, F 4 ) \ V d+1 (G, F 4 ))(4 d 1) This gives info about a k (G) = {index k subgroups of G}, a k (G) = {index k normal subgroups of G}, e.g.: a 3 = 1 2 (3n 1) + 3δ S3 a 4 = 1 3 (2n+1 1)(2 n 1)+4(δ D8 + δ A4 + δ S4 ) a 3 = 1 2 (3n 1) a 4 = 1 3 (2n+1 1)(2 n 1) a 6 = 1 2 (3n 1)(2 n 1) + δ S3 a 8 = 1 21 (2n+2 1)(2 n+1 1)(2 n 1) + δ D8 + δ Q8 19
20 Congruence covers X finite cell complex, with H 1 (X) = Z n. X N is the regular (Z N ) n -cover of X determined by π 1 (X) ab H 1 (X, Z) modn H 1 (X; Z N ). By Libgober and Sakuma: b 1 (X N )=n + t Tors N (C ) n depth C (t). Theorem. (Sarnak-Adams, Sakuma) The sequence {b 1 (X N )} N N is polynomially periodic, i.e., there is T 1, andpolynomials P 1 (x),...,p T (x), such that b 1 (X N )=P i (N), if N i mod T. Follows from above formula, and Theorem. (Laurent) If V is a subvariety of (C ) n, then Tors N (V )= v i=1 Tors N(S i ), where S i are subtori of (C ) n, possibly translatedby roots of unity. For n 6: b 1 (X N (A)) = P A (N). For n =7: As.t. b 1 (X N (A)) has period T =2. 20
21 Hirzebruch covering surfaces A arrangement of n planes in C 3. A projectivized arrangement of lines in CP 2. X N (A) congruence cover of X(A). X N (A) associated branched cover CP 2. M N (A) minimal desingularization of X N (A). Hirzebruch computed the Chern numbers of M N (A): c 2 1 = ( (3b 2 s 5n+9)N 2 4(b 2 n)n +(b 2 +n+m 2 ) ) N n 3 c 2 = ( (b 2 2n +3)N 2 2(b 2 n)n +(b 2 + s m 2 ) ) N n 3 where m r = {I L 2 (A) I = r}, s = m r, b 2 = r m r(r 1) Sakuma computed the first Betti number of M N (A): b 1 (M N (A)) = depth C (t At ) t Tors N (C ) n where A t = {H i A t(x i ) 1}. Theorem. (Hironaka, Sakuma) The sequence {b 1 (M N (A))} N N is polynomially periodic. For n 7: b 1 (M N (A)) = P A (N). For n =8: As.t. b 1 (M N (A)) has period T =4. 21
22 Lower central series quotients G f.g. group. Lower central series: G = γ 1 G γ 2 G, where γ k+1 G =[γ k G, G] LCS quotients: gr k G = γ k G/γ k+1 G (f.g. abelian) Chen groups: gr k (G/G ). φ k (G) =rank(gr k G) θ k (G) =rank(gr k G/G ) Clearly, φ 1 = θ 1, φ 2 = θ 2, φ 3 = θ 3, φ k θ k. E.g.: φ k (F n )=w k (n) := 1 k µ(d)n k/d (Witt) θ k (F n )= ( n+k 2 k d k ) (k 1), for k 2 (Murasugi, Massey-Traldi) Massey: gr k (G/G )=I k 2 B/I k 1 B B = G /G : Alexander invariant (Z[G/G ]-module), I = ker ɛ: augmentation ideal. 22
23 Hence, if G/G = Z n : k 0 θ k+2 t k =Hilb(gr B) A presentation for gr B = k 0 gr k B (as module over gr Z[G/G ] = Z[λ 1,...,λ n ]) can be obtained from a presentation for B via a Gröbner basis algorithm. Let A be arrangement of n hyperplanes, G = G(A). Falk: φ k = φ k (G) combinatorially determined. Cohen-S.: Presentation for the Alexander invariant B = G /G. This gives algorithm for computing the Chen groups of G(A). E.g., for P n = n 1 i=1 F i: θ k (P n )=(k 1) ( ) n+1 4, for k 3. It follows that θ k (P n ) θ k (Π n ), where Π n = n 1 i=1 F i. On the other hand, φ k (P n )=φ k (Π n ), by Theorem. (LCS formula of Falk andrandell) If A fiber-type, with exponents d 1,...,d l : (1 t k ) φ k = P (X, t) = k 1 l (1 d i t) i=1 i.e., φ k (G) = l i=1 φ k(f di ). 23
24 Kohno: First proved LCS formula for A = B l. Shelton-Yuzvinsky: Consequence of Koszul duality. Papadima-Yuz.: Extend LCS formula to formal rational K(G, 1) spaces. Also, if A arrangement in C 3 : LCS formula holds A fiber-type. Jambu-Papadima: Extend LCS formula to hypersolvable arrangements. Let R 1 (A) = v i=1 L i, with L i linear subspaces of C n. Let h r = {L i dim L i = r}. Conjecture. The Chen groups of G = G(A) are free abelian, of rank θ k (G) = r 2 h r θ k (F r ), for k 4. (This would imply that the Chen groups of an arrangement are combinatorially determined.) Conjecture. If φ 4 = θ 4, then gr k G is free abelian, of rank φ k (G) = r 2 h r φ k (F r ), for k 4. 24
25 Resonance varieties & nilpotent quotients Let X be a finite CW-complex, with H (X) torsionfree, and H (X) generated in degree 1. E.g.: X = X(A), or X = X(link in S 3 with lk i,j = ±1). Let G = π 1 (X). Theorem. (Matei-S.) If X, X as above, then: H 2 (X) = H 2 (X ) G/γ 3 G = G /γ 3 G. For p prime, d 0, define: { ν p,d (G/γ 3 G)= K G/γ 3 G Then: [G/γ 3 G : K] =p and dim Fp (Tors H 1 (K)) F p = d } ν p,d (G/γ 3 G)= (R p,d(g, F p ) \ R p,d+1 (G, F p )) p 1 One may define higher-order resonance varieties, S (k) d (G, K), using higher-order Massey products <x,λ,...,λ>,or higher-order truncations of A G (D. Matei, Ph.D. thesis). One may also define ν p,d (G/γ k+2 G), as above. An analogue of the framed formula remains to be found. 25
26 Higher homotopy groups Let G be a graph, A = A G, X = X(A). Assume G has no 3-cycles. Then π 1 (X) =Z n, where n = edges. Let S be the set of 4-cycles. Proposition. (Papadima-S.) If S =, then π 2 (X) =0. Otherwise, the Zπ 1 -module π 2 (X) is combinatorially determined (by the graph G), and π 2 (X) π1 = Z[S] 0. Let V d (π 2 )=V ( d π 2 ) (C ) n. These varieties may distinguish π 2 s with the same coinvariants. Example. e 6 e 1 e 1 e 7 e 2 e 7 e 2 e 9 e e 9 e 8 8 e 6 e3 e3 e 5 e 5 e 4 The graphs G 1 and G 2 have no 3-cycles. Each graph has exactly two 4-cycles. The complements X i = X(A Gi )have: π 1 = Z 9, (π 2 ) π1 = Z 2, b 2 =36, b 3 =82. V 1 (π 2 (X 1 )) has 2 components, V 1 (π 2 (X 1 )) has 3. Hence: e 4 π 2 (X 1 ) = π 2 (X 2 ) (as Zπ 1 -modules). 26
27 Example (Braid arrangement) C 156 C 246 C 345 Π C 123 Q = xyz(x y)(x z)(y z). n =6, s =7, m 2 =3, m 3 =4. P (X, t) =(1 + t)(1 + 2t)(1 + 3t). G = P 4 = F 3 F 2 F 1. V 1 (G, K) =C 124 C 135 C 236 C 456 Π, where Π=C ( ) = {(s, t, (st) 1,s,t,(st) 1 ) s, t K }. V 2 (G, K) ={1} β (q) p,1 = ν p,1 =5(p + 1). b 1 (X N )=5N b 1 (M N )=5(N 1)(N 2) c 2 1(M N )=5N 3 (N 2) 2, c 2 (M N )=N 3 (2N 2 10N+15). δ S3 = 15, δ A4 = 20, a 2 = 63, a 3 = 364, a 3 = 409. φ 1 =6,φ 2 =4,φ 3 = 10, φ 4 = 21, φ k = w k (2) + w k (3). θ 1 =6,θ 2 =4,θ 3 = 10, θ 4 = 15, θ k =5(k 1). 27
28 Example (Non-Fano plane) ρ Π 1 Π 2 Π 3 Q = xyz(x y)(x z)(y z)(x + y z). n =7,s =9,m 2 =3,m 3 =6. P (X, t) =(1+t)(1+3t) 2. V 1 (G, K) has nine 2-dim components. V 2 (G, K) =Π 1 Π 2 Π 3 = {1,ρ}, where ρ 2 = 1 R 1 (G, F 2 ) has 3-dim component Υ TC 1 (V 1 (G, F 2 )). R 2 (G, F 2 ) has 1-dim component Υ TC 1 (V 2 (G, F 2 )). β (q) p,1 = ν p,1 =9(p + 1), except for: β (q) 2,1 = ν 2,1 = 24, β (q) 2,2 = ν 2,2 =1. 9N 2 3 if N even, b 1 (X N )= 9N 2 2 if N odd. b 1 (M N )=9(N 1)(N 2). δ S3 = 28, δ A4 = 36, a 2 = 127, a 3 =1, 093, a 3 =1, 177. φ 1 =7,φ 2 =6,φ 3 = 17, φ 4 = 42, φ 5 = 123, φ 6 = 341, φ 7 =1, 041. θ 1 =7,θ 2 =6,θ 3 = 17, θ 4 = 27, θ k =9(k 1). Υ Υ 28
29 Example (Deleted B 3 arrangement) ρ + ρ Ω Π 1 Π 2 Π 3 Π 4 t 1 1 t t 2 t 2 t 1 t 1 Q = xyz(x y)(x z)(y z)(x y z)(x y + z). n =8, s =11, m 2 =4, m 3 =6, m 4 =1. V 1 (G, K) has a 1-dim component which does not pass through 1 (unless char K =2): Ω={(t, t 1, t 1,t,t 2, 1,t 2, 1) t K } Π 5 29
30 ν p,1 =11(p + 1), ν p,2 = p3 1 p 1, β(q) p,d = ν p,d, except: β (q) 2,1 =27, β(q) 2,2 =9,β(2) 3,1 =45. Distribution of index 3, normal subgroups in G: K ab Z 8 Z 8 Z 2 2 Z 10 Z 12 K 1, ( ) ( ) K = ker(γ : G Z 3 ), γ =(ω, ω 2,ω 2,ω,ω 2, 1,ω,1) (Z 3 ) 8. Since γ/ V 1 (G, C), but γ Ω V 1 (G, F 2 (ω)): b 1 (X N )= b 1 (K) =8, b (2) 1 (K)=8+(3 1) 1=10. 2N 3 +11N 2 + N 9 2N 3 +11N 2 5 if N even, if N odd, b 1 (M N )= (N 1)(2N 2 +9N 24) + N 2 if N 0 mod 4 (N 1)(2N 2 +9N 24) + 1 (N 2) if N 2 mod 4 2 (N 1)(2N 2 +9N 24) if N odd. δ S3 = 63, δ A4 = 110, a 2 = 255, a 3 =3, 280, a 3 =3, 469. φ 1 =8,φ 2 =9,φ 3 = 28, φ 4 = 78, φ k = w k (3) + w k (4). θ 1 =8,θ 2 =9,θ 3 = 28, θ 4 = 48, θ k =(k + 12)(k 1). 30
31 Example (MacLane arrangement) Q = xyz(y x)(z x)(z+ωy)(z+ω 2 x+ωy)(z x ω 2 y). n =8,s = 12, m 2 =4,m 3 =8. P (X, t) =(1+t)(1+7t +13t 2 ). V d (G, K) has only local components R 1 (G, F 3 ) has one non-local, 2-dim component: Ξ={(λ µ, λ, µ, λ, µ, λ + µ, λ µ, µ λ) λ, µ F 3 } β (q) p,1 =8(p+1) and ν p,1 =8(p+1), except for ν 3,1 = 36. b 1 (X N )=8N 2. b 1 (M N )=8(N 1)(N 2). δ S3 = 24, δ A4 = 32, a 2 = 255, a 3 =3, 280, a 3 =3, 352. φ 1 =8,φ 2 =8,φ 3 = 21, φ 4 = 42, φ 5 = 87, φ 6 = 105. θ 1 =8,θ 2 =8,θ 3 = 21, θ 4 = 24, θ k =8(k 1). Note: There is torsion in the LCS quotients of G. E.g.: gr 5 G = Z 87 Z 4 2 Z 3 Ξ 31
32 Example (Ziegler arrangements). A 1 1 A 1 2 n = 13, s = 31, m 2 = 20, m 3 =9,m 5 =1,m 7 =1. P (X i,t)=(1+t)(1+6t) 2. φ 1 = 13, φ 2 = 30, φ 3 = 140, φ k =2w k (6). θ k = (k 1)(k4 +10k 3 +47k 2 +86k+696) 24. R d (G 1, K) and R d (G 2, K) are (abstractly) isomorphic. Hence, ν p,d (G 1 )=ν p,d (G 2 ). a 2 =8, 191, a 3 = 797, 161, a 3 = 820,
33 V 1 (G 1, K) has τ 1 = 3 translated subtori V 1 (G 2, K) has τ 2 = 2 translated subtori β (q) 2 = (69, 4, 15, 0, 63) β (q) p = (27(p +1), 0, p4 1 p 1, 0, p6 1 p 1 ) except if p =3,q =2,d =1 Thus: δ Γ (G 1 )=δ Γ (G 2 ), if Γ = Z s q σ Z p = A 4. But: Hence: β (2) 3,1 (G 1) =111, β (2) 3,1 (G 2) =110. δ A4 (G 1 ) =124, 435, δ A4 (G 2 ) =124, 434. b 1 (X N (A i )) = 5N 6 +3N 4 +27N 2 + τ i (N 2) 26 if N even 5N 6 +3N 4 +27N 2 22 if N odd f(n)+τ i (N 2) if N 0 mod 4 b 1 (M N (A i )) = f(n)+ τ i 2 (N 2) if N 2 mod 4 f(n) if N odd, where f(n) =(N 1)(5N 5 2N 4 + N 3 4N 2 +23N 58) 33
TOPOLOGY OF LINE ARRANGEMENTS. Alex Suciu. Northeastern University. Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014
TOPOLOGY OF LINE ARRANGEMENTS Alex Suciu Northeastern University Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014 ALEX SUCIU (NORTHEASTERN) TOPOLOGY OF LINE ARRANGEMENTS CORTONA,
More informationHomotopy types of the complements of hyperplane arrangements, local system homology and iterated integrals
Homotopy types of the complements of hyperplane arrangements, local system homology and iterated integrals Toshitake Kohno The University of Tokyo August 2009 Plan Part 1 : Homotopy types of the complements
More informationLower central series, free resolutions, and homotopy Lie algebras of arrangements Alex Suciu
Lower central series, free resolutions, and homotopy Lie algebras of arrangements Alex Suciu Northeastern University www.math.neu.edu/~suciu NSF-CBMS Regional Research Conference on Arrangements and Mathematical
More informationAlgebra and topology of right-angled Artin groups
Algebra and topology of right-angled Artin groups Alex Suciu Northeastern University Boston, Massachusetts (visiting the University of Warwick) Algebra Seminar University of Leeds October 19, 2009 Alex
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More informationPROPAGATION OF RESONANCE. Alex Suciu. Northeastern University. Joint work with Graham Denham and Sergey Yuzvinsky
COMBINATORIAL COVERS, ABELIAN DUALITY, AND PROPAGATION OF RESONANCE Alex Suciu Northeastern University Joint work with Graham Denham and Sergey Yuzvinsky Algebra, Topology and Combinatorics Seminar University
More informationPartial products of circles
Partial products of circles Alex Suciu Northeastern University Boston, Massachusetts Algebra and Geometry Seminar Vrije University Amsterdam October 13, 2009 Alex Suciu (Northeastern University) Partial
More informationA short introduction to arrangements of hyperplanes
A short introduction to arrangements of hyperplanes survey Sergey Yuzvinsky University of Oregon Pisa, May 2010 Setup and notation By a hyperplane arrangement we understand the set A of several hyperplanes
More informationResonance varieties and Dwyer Fried invariants
Resonance varieties and Dwyer Fried invariants Alexander I. Suciu Abstract. The Dwyer Fried invariants of a finite cell complex X are the subsets Ω i r (X) of the Grassmannian of r-planes in H 1 (X, Q)
More informationH A A}. ) < k, then there are constants c t such that c t α t = 0. j=1 H i j
M ath. Res. Lett. 16 (2009), no. 1, 171 182 c International Press 2009 THE ORLIK-TERAO ALGEBRA AND 2-FORMALITY Hal Schenck and Ştefan O. Tohǎneanu Abstract. The Orlik-Solomon algebra is the cohomology
More informationMilnor Fibers of Line Arrangements
Milnor Fibers of Line Arrangements Alexandru Dimca Université de Nice, France Lib60ber Topology of Algebraic Varieties Jaca, Aragón June 25, 2009 Outline Outline 1 Anatoly and me, the true story... 2 Definitions,
More informationarxiv:math/ v2 [math.ag] 27 Nov 2000
Contemporary Mathematics arxiv:math/0010105v2 [math.ag] 27 Nov 2000 Fundamental groups of line arrangements: Enumerative aspects Alexander I. Suciu Abstract. This is a survey of some recent developments
More informationTORSION IN MILNOR FIBER HOMOLOGY
TORSION IN MILNOR FIBER HOMOLOGY DANIEL C. COHEN 1, GRAHAM DENHAM 2, AND ALEXANDER I. SUCIU 3 Abstract. In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial f such
More informationGeneric section of a hyperplane arrangement and twisted Hurewicz maps
arxiv:math/0605643v2 [math.gt] 26 Oct 2007 Generic section of a hyperplane arrangement and twisted Hurewicz maps Masahiko Yoshinaga Department of Mathematice, Graduate School of Science, Kobe University,
More informationALGEBRAIC MODELS, DUALITY, AND RESONANCE. Alex Suciu. Topology Seminar. MIT March 5, Northeastern University
ALGEBRAIC MODELS, DUALITY, AND RESONANCE Alex Suciu Northeastern University Topology Seminar MIT March 5, 2018 ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 1 / 24 DUALITY
More informationOn the topology of matrix configuration spaces
On the topology of matrix configuration spaces Daniel C. Cohen Department of Mathematics Louisiana State University Daniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 1 Configuration spaces
More informationMathematisches Forschungsinstitut Oberwolfach. Miniworkshop: Cohomology Jumping Loci
Mathematisches Forschungsinstitut Oberwolfach Report No. 11/2002 Miniworkshop: Cohomology Jumping Loci March 3rd March 9th, 2002 1. Overview This Mini-Workshop was organized by A. Suciu (Boston) and S.
More informationHodge theory for combinatorial geometries
Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that
More informationThe Orlik-Solomon Algebra and the Supersolvable Class of Arrangements
International Journal of Algebra, Vol. 8, 2014, no. 6, 281-292 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4216 The Orlik-Solomon Algebra and the Supersolvable Class of Arrangements
More informationFUNDAMENTAL GROUPS. Alex Suciu. Northeastern University. Joint work with Thomas Koberda (U. Virginia) arxiv:
RESIDUAL FINITENESS PROPERTIES OF FUNDAMENTAL GROUPS Alex Suciu Northeastern University Joint work with Thomas Koberda (U. Virginia) arxiv:1604.02010 Number Theory and Algebraic Geometry Seminar Katholieke
More informationISSN (on-line) (printed) 511 Algebraic & Geometric Topology Volume 3 (2003) 511{535 Published: 15 June 2003 ATG Torsion in Milnor
ISSN 1472-2739 (on-line) 1472-2747 (printed) 511 Algebraic & Geometric Topology Volume 3 (2003) 511{535 Published: 15 June 2003 ATG Torsion in Milnor ber homology Daniel C. Cohen Graham Denham Alexander
More informationMULTINETS, PARALLEL CONNECTIONS, AND MILNOR FIBRATIONS OF ARRANGEMENTS
MULTINETS, PARALLEL CONNECTIONS, AND MILNOR FIBRATIONS OF ARRANGEMENTS GRAHAM DENHAM 1 AND ALEXANDER I. SUCIU 2 Abstract. The characteristic varieties of a space are the jump loci for homology of rank
More informationFUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY. Alex Suciu AND THREE-DIMENSIONAL TOPOLOGY. Colloquium University of Fribourg June 7, 2016
FUNDAMENTAL GROUPS IN ALGEBRAIC GEOMETRY AND THREE-DIMENSIONAL TOPOLOGY Alex Suciu Northeastern University Colloquium University of Fribourg June 7, 2016 ALEX SUCIU (NORTHEASTERN) FUNDAMENTAL GROUPS IN
More informationLOWER CENTRAL SERIES AND FREE RESOLUTIONS OF HYPERPLANE ARRANGEMENTS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 9, Pages 409 4 S 000-9947(0001-0 Article electronically published on May 8, 00 LOWER CENTRAL SERIES AND FREE RESOLUTIONS OF HYPERPLANE
More informationThe Milnor fiber associated to an arrangement of hyperplanes
University of Iowa Iowa Research Online Theses and Dissertations Summer 2011 The Milnor fiber associated to an arrangement of hyperplanes Kristopher John Williams University of Iowa Copyright 2011 Kristopher
More informationALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS
ALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS LAURENTIU MAXIM Abstract. These are notes I wrote for a series of lectures I gave at Tokyo University of Sciences and The University of Tokyo, Tokyo, Japan,
More informationOn the Alexander invariants of hypersurface complements
1 On the Alexander invariants of hypersurface complements Laurentiu Maxim Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois, 60607, USA E-mail: lmaxim@math.uic.edu
More informationMathematical Research Letters 7, (2000) ARRANGEMENTS AND LOCAL SYSTEMS. Daniel C. Cohen and Peter Orlik
Mathematical Research Letters 7, 299 316 (2000) ARRANGEMENTS AND LOCAL SYSTEMS Daniel C. Cohen and Peter Orlik Abstract. We use stratified Morse theory to construct a complex to compute the cohomology
More informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationAnnihilators of Orlik Solomon Relations
Advances in Applied Mathematics 28, 231 249 (2002) doi:10.1006/aama.2001.0779, available online at http://www.idealibrary.com on Annihilators of Orlik Solomon Relations Graham Denham 1 and Sergey Yuzvinsky
More informationarxiv:math/ v1 [math.ag] 19 Jul 1999
ARRANGEMENTS AND LOCAL SYSTEMS DANIEL C. COHEN AND PETER ORLIK arxiv:math/9907117v1 [math.ag] 19 Jul 1999 Abstract. We use stratified Morse theory to construct a complex to compute the cohomology of the
More informationABELIAN ARRANGEMENTS
ABELIAN ARRANGEMENTS by CHRISTIN BIBBY A DISSERTATION Presented to the Department of Mathematics and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree
More informationThe rational cohomology of real quasi-toric manifolds
The rational cohomology of real quasi-toric manifolds Alex Suciu Northeastern University Joint work with Alvise Trevisan (VU Amsterdam) Toric Methods in Homotopy Theory Queen s University Belfast July
More informationFreeness of hyperplane arrangement bundles and local homology of arrangement complements
University of Iowa Iowa Research Online Theses and Dissertations Summer 2010 Freeness of hyperplane arrangement bundles and local homology of arrangement complements Amanda C. Hager University of Iowa
More informationFORMALITY OF THE COMPLEMENTS OF SUBSPACE ARRANGEMENTS WITH GEOMETRIC LATTICES
FORMALITY OF THE COMPLEMENTS OF SUBSPACE ARRANGEMENTS WITH GEOMETRIC LATTICES EVA MARIA FEICHTNER AND SERGEY YUZVINSKY Abstract. We show that, for an arrangement of subspaces in a complex vector space
More informationPOLYHEDRAL PRODUCTS, TORIC MANIFOLDS, AND. Alex Suciu TWISTED COHOMOLOGY. Princeton Rider workshop Homotopy theory and toric spaces.
POLYHEDRAL PRODUCTS, TORIC MANIFOLDS, AND TWISTED COHOMOLOGY Alex Suciu Northeastern University Princeton Rider workshop Homotopy theory and toric spaces February 23, 2012 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL
More informationarxiv:math/ v2 [math.at] 2 Oct 2004
arxiv:math/0409412v2 [math.at] 2 Oct 2004 INTERSECTION HOMOLOGY AND ALEXANDER MODULES OF HYPERSURFACE COMPLEMENTS LAURENTIU MAXIM Abstract. Let V be a degree d, reduced, projective hypersurface in CP n+1,
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationarxiv: v1 [math.ag] 13 Mar 2019
THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show
More informationStratified Morse Theory: Past and Present
Stratified Morse Theory: Past and Present David B. Massey In honor of Robert MacPherson on his 60th birthday 1 Introduction In 1974, Mark Goresky and Robert MacPherson began their development of intersection
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationExercises on characteristic classes
Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives
More informationFAKE PROJECTIVE SPACES AND FAKE TORI
FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.
More informationContents Preliminaries Non-abelian curve groups A special curve Characteristic Varieties
Orbifolds and fundamental groups of plane curves Enrique ARTAL BARTOLO Departmento de Matemáticas Facultad de Ciencias Instituto Universitario de Matemáticas y sus Aplicaciones Universidad de Zaragoza
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationarxiv: v2 [math.at] 17 Sep 2009
ALGEBRAIC MONODROMY AND OBSTRUCTIONS TO FORMALITY arxiv:0901.0105v2 [math.at] 17 Sep 2009 STEFAN PAPADIMA 1 AND ALEXANDER I. SUCIU 2 Abstract. Given a fibration over the circle, we relate the eigenspace
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationSpecial cubic fourfolds
Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows
More informationDegeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements
Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements Pauline Bailet Hokkaido University Computational Geometric Topology in Arrangement Theory July 6-10, 2015 1 / 15 Reference
More informationJUMP LOCI. Alex Suciu. Northeastern University. Intensive research period on Algebraic Topology, Geometric and Combinatorial Group Theory
ALGEBRAIC MODELS AND COHOMOLOGY JUMP LOCI Alex Suciu Northeastern University Intensive research period on Algebraic Topology, Geometric and Combinatorial Group Theory Centro di Ricerca Matematica Ennio
More informationApplications of the Serre Spectral Sequence
Applications of the Serre Spectral Seuence Floris van Doorn November, 25 Serre Spectral Seuence Definition A Spectral Seuence is a seuence (E r p,, d r ) consisting of An R-module E r p, for p, and r Differentials
More informationL 2 BETTI NUMBERS OF HYPERSURFACE COMPLEMENTS
L 2 BETTI NUMBERS OF HYPERSURFACE COMPLEMENTS LAURENTIU MAXIM Abstract. In [DJL07] it was shown that if A is an affine hyperplane arrangement in C n, then at most one of the L 2 Betti numbers i (C n \
More informationHYPERPLANE ARRANGEMENTS AND LINEAR STRANDS IN RESOLUTIONS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 2, Pages 609 618 S 0002-9947(02)03128-8 Article electronically published on September 6, 2002 HYPERPLANE ARRANGEMENTS AND LINEAR STRANDS
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationarxiv:math/ v1 [math.ac] 24 Nov 2006
arxiv:math/061174v1 [math.ac] 4 Nov 006 TE CARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT TAKURO ABE, IROAKI TERAO, AND MAX WAKEFIELD Abstract. Given a multiarrangement of hyperplanes we define a series
More informationON THE ISOMORPHISM CONJECTURE FOR GROUPS ACTING ON TREES
ON THE ISOMORPHISM CONJECTURE FOR GROUPS ACTING ON TREES S.K. ROUSHON Abstract. We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions
More informationarxiv: v2 [math.gr] 8 Jul 2012
GEOMETRIC AND HOMOLOGICAL FINITENESS IN FREE ABELIAN COVERS ALEXANDER I. SUCIU arxiv:1112.0948v2 [math.gr] 8 Jul 2012 Abstract. We describe some of the connections between the Bieri Neumann Strebel Renz
More informationarxiv: v1 [math.ag] 28 Sep 2016
LEFSCHETZ CLASSES ON PROJECTIVE VARIETIES JUNE HUH AND BOTONG WANG arxiv:1609.08808v1 [math.ag] 28 Sep 2016 ABSTRACT. The Lefschetz algebra L X of a smooth complex projective variety X is the subalgebra
More informationCHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents
CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally
More informationEKT of Some Wonderful Compactifications
EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some
More informationSOME EXERCISES. By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra.
SOME EXERCISES By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra. 1. The algebraic thick subcategory theorem In Lecture 2,
More informationSTEENROD OPERATIONS IN ALGEBRAIC GEOMETRY
STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group
More informationDivisor class groups of affine complete intersections
Divisor class groups of affine complete intersections Lambrecht, 2015 Introduction Let X be a normal complex algebraic variety. Then we can look at the group of divisor classes, more precisely Weil divisor
More informationOrlik-Solomon Algebras and Tutte Polynomials
Journal of Algebraic Combinatorics 10 (1999), 189 199 c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Orlik-Solomon Algebras and Tutte Polynomials CARRIE J. ESCHENBRENNER Independent
More informationHyperplane arrangements and K-theory 1
Hyperplane arrangements and K-theory 1 Nicholas Proudfoot 2 Department of Mathematics, University of California, Berkeley, CA 94720 Abstract. We study the Z 2-equivariant K-theory of MA, where MA is the
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationStructure of elliptic curves and addition laws
Structure of elliptic curves and addition laws David R. Kohel Institut de Mathématiques de Luminy Barcelona 9 September 2010 Elliptic curve models We are interested in explicit projective models of elliptic
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
More informationLecture VI: Projective varieties
Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still
More informationMost rank two finite groups act freely on a homotopy product of two spheres
Most rank two finite groups act freely on a homotopy product of two spheres Michael A. Jackson University of Rochester mjackson@math.rochester.edu Conference on Pure and Applied Topology Isle of Skye,
More informationBraid groups, their applications and connections
Braid groups, their applications and connections Fred Cohen University of Rochester KITP Knotted Fields July 1, 2012 Introduction: Artin s braid groups are at the confluence of several basic mathematical
More informationSpherical varieties and arc spaces
Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected
More informationTHE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
More informationAn introduction to calculus of functors
An introduction to calculus of functors Ismar Volić Wellesley College International University of Sarajevo May 28, 2012 Plan of talk Main point: One can use calculus of functors to answer questions about
More informationTopological Combinatorics * * * * * *
Topological Combinatorics * * * * * * Anders Björner Dept. of Mathematics Kungl. Tekniska Högskolan, Stockholm * * * * * * MacPherson 60 - Fest Princeton, Oct. 8, 2004 Influence of R. MacPherson on topological
More informationCOMPLETELY REDUCIBLE HYPERSURFACES IN A PENCIL
COMPLETELY REDUCIBLE HYPERSURFACES IN A PENCIL J. V. PEREIRA AND S. YUZVINSKY Abstract. We study completely reducible fibers of pencils of hypersurfaces on P n and associated codimension one foliations
More informationCOURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA
COURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA JAROD ALPER WEEK 1, JAN 4, 6: DIMENSION Lecture 1: Introduction to dimension. Define Krull dimension of a ring A. Discuss
More informationCoxeter Groups and Artin Groups
Chapter 1 Coxeter Groups and Artin Groups 1.1 Artin Groups Let M be a Coxeter matrix with index set S. defined by M is given by the presentation: A M := s S sts }{{ } = tst }{{ } m s,t factors m s,t The
More informationMultiplicity of singularities is not a bi-lipschitz invariant
Multiplicity of singularities is not a bi-lipschitz invariant Misha Verbitsky Joint work with L. Birbrair, A. Fernandes, J. E. Sampaio Geometry and Dynamics Seminar Tel-Aviv University, 12.12.2018 1 Zariski
More informationLECTURE 6: THE ARTIN-MUMFORD EXAMPLE
LECTURE 6: THE ARTIN-MUMFORD EXAMPLE In this chapter we discuss the example of Artin and Mumford [AM72] of a complex unirational 3-fold which is not rational in fact, it is not even stably rational). As
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationSplitting criterion for reflexive sheaves
Splitting criterion for reflexive sheaves TAKURO ABE MASAHIKO YOSHINAGA April 6, 2005 Abstract The purpose of this paper is to study the structure of reflexive sheaves over projective spaces through hyperplane
More informationHyperplane arrangements, local system homology and iterated integrals
Hyperplane arrangements, local system homology and iterated integrals Abstract. Toshitake Kohno We review some aspects of the homology of a local system on the complement of a hyperplane arrangement. We
More information1 Moduli spaces of polarized Hodge structures.
1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an
More informationRational Curves On K3 Surfaces
Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:
More informationTopology of Toric Varieties, Part II
Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences
More informationp-divisible Groups and the Chromatic Filtration
p-divisible Groups and the Chromatic Filtration January 20, 2010 1 Chromatic Homotopy Theory Some problems in homotopy theory involve studying the interaction between generalized cohomology theories. This
More informationarxiv:math/ v3 [math.ag] 11 Apr 1998
CHARACTERISTIC VARIETIES OF ARRANGEMENTS arxiv:math/9801048v3 [math.ag] 11 Apr 1998 DANIEL C. COHEN 1 AND ALEXANDER I. SUCIU 2 Abstract. The k th Fitting ideal of the Alexander invariant B of an arrangement
More informationNONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) aar. Heidelberg, 17th December, 2008
1 NONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Heidelberg, 17th December, 2008 Noncommutative localization Localizations of noncommutative
More informationOutline. Some Reflection Group Numerology. Root Systems and Reflection Groups. Example: Symmetries of a triangle. Paul Renteln
Outline 1 California State University San Bernardino and Caltech 2 Queen Mary University of London June 13, 2014 3 Root Systems and Reflection Groups Example: Symmetries of a triangle V an n dimensional
More information3. Signatures Problem 27. Show that if K` and K differ by a crossing change, then σpk`q
1. Introduction Problem 1. Prove that H 1 ps 3 zk; Zq Z and H 2 ps 3 zk; Zq 0 without using the Alexander duality. Problem 2. Compute the knot group of the trefoil. Show that it is not trivial. Problem
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationCONFIGURATION SPACES AND BRAID GROUPS
CONFIGURATION SPACES AND BRAID GROUPS FRED COHEN AND JONATHAN PAKIANATHAN Abstract. The main thrust of these notes is 3-fold: (1) An analysis of certain K(π, 1) s that arise from the connections between
More information1.1 Definition of group cohomology
1 Group Cohomology This chapter gives the topological and algebraic definitions of group cohomology. We also define equivariant cohomology. Although we give the basic definitions, a beginner may have to
More informationOral exam practice problems: Algebraic Geometry
Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n
More informationEvgeniy V. Martyushev RESEARCH STATEMENT
Evgeniy V. Martyushev RESEARCH STATEMENT My research interests lie in the fields of topology of manifolds, algebraic topology, representation theory, and geometry. Specifically, my work explores various
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationTranscendental L 2 -Betti numbers Atiyah s question
Transcendental L 2 -Betti numbers Atiyah s question Thomas Schick Göttingen OA Chennai 2010 Thomas Schick (Göttingen) Transcendental L 2 -Betti numbers Atiyah s question OA Chennai 2010 1 / 24 Analytic
More informationMath 6510 Homework 10
2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained
More informationA geometric solution of the Kervaire Invariant One problem
A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :
More information