GEOMETRY AND TOPOLOGY OF COHOMOLOGY JUMP LOCI LECTURE 1: CHARACTERISTIC VARIETIES Alex Suciu Northeastern University MIMS Summer School: New Trends in Topology and Geometry Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia July 9–12, 2018 ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 1 / 25 OUTLINE 1 CAST OF CHARACTERS The character group The equivariant chain complex Characteristic varieties Degree1 characteristic varieties 2 EXAMPLES AND COMPUTATIONS Warm-up examples Toric complexes and RAAGs Quasi-projective manifolds 3 APPLICATIONS Homology of finite abelian covers Dwyer–Fried sets Duality and propagation ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 2 / 25 CAST OF CHARACTERS THE CHARACTER GROUP THE CHARACTER GROUP Throughout, X will be a connected CW-complex with finite q-skeleton, for some q ¥ 1. We may assume X has a single 0-cell, call it e0. 0 Let G = p1(X, e ) be the fundamental group of X: a finitely 1 1 generated group, with generators x1 = [e1], ... , xm = [em]. The character group, G = Hom(G, Cˆ) Ă (Cˆ)m is a (commutative) algebraicp group, with multiplication r ¨ r1(g) = r(g)r1(g), and identity G Ñ Cˆ, g ÞÑ 1. 1 Let Gab = G/G – H1(X, Z) be the abelianization of G. The » projection ab: G Ñ Gab induces an isomorphism Gab ÝÑ G. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMERp SCHOOLp2018 3 / 25 CAST OF CHARACTERS THE CHARACTER GROUP The identity component, G0, is isomorphic to a complex algebraic torus of dimension n = rank Gab. p The other connected components are all isomorphic to G0 = (Cˆ)n, and are indexed by the finite abelian group Tors(Gab). p Char(X ) = G is the moduli space of rank1 local systems on X: ˆ p r : G Ñ C Cr the complex vector space C, viewed as a right module over the group ring ZG via a ¨ g = r(g)a, for g P G and a P C. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 4 / 25 CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX THE EQUIVARIANT CHAIN COMPLEX Let p : X Ñ X be the universal cover. The cell structure on X lifts to a cell structure on X. 0 ´1 0 0 Fixing ar lift e˜ P p (e ) identifies G = p1(X, e ) with the group of deck transformationsr of X. Thus, we may view the cellular chain complex of X as a chain complex of left ZG-modules,r r B˜i+1 B˜i ¨ ¨ ¨ / Ci+1(X, Z) / Ci (X, Z) / Ci´1(X, Z) / ¨ ¨ ¨ . ˜ 1 0 B1(e˜ i ) = (xi ´ 1)er˜ . r r ˜ 2 m f 1 B2(e˜ ) = i=1 Br/Bxi ¨ e˜ i , where r P F = xx , , x y is the word traced by the attaching map of e2; mř 1 ... m Br/Bxi P ZFm are the Fox derivatives of r; f : ZFm Ñ ZG is the linear extension of the projection Fm G. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 5 / 25 CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX H˚(X, Cr) is the homology of the chain complex of C-vector spaces Cr bZG C‚(X, Z): rB˜i+1(r) B˜i (r) ¨ ¨ ¨ / Ci+1(X, C) / Ci (X, C) / Ci´1(X, C) / ¨ ¨ ¨ , where the evaluation of B˜i at r is obtained by applying the ring homomorphism ZG Ñ C, g ÞÑ r(g) to each entry of B˜i . Alternatively, consider the universal abelian cover, X ab, and its ab equivariant chain complex, C‚(X , Z) = ZGab bZG C‚(X, Z), ab with differentials Bi = id b Bi . r Then H˚(X, Cr) is computed from the resulting C-chain complex, ab ˜r with differentials Bi (r) = Bi (r). The identity1 P Char(X ) yields the trivial local system, C1 = C, and H˚(X, C) is the usual homology of X with C-coefficients. Denote by bi (X ) = dimC Hi (X, C) the ith Betti number of X. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 6 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES CHARACTERISTIC VARIETIES DEFINITION The characteristic varieties of X are the sets i Vk (X ) = tr P Char(X ) | dimC Hi (X, Cr) ¥ ku. i i i For each i, get stratification Char(X ) = V0 Ě V1 Ě V2 Ě ¨ ¨ ¨ i 1 P Vk (X ) ðñ bi (X ) ¥ k. 0 0 V1 (X ) = t1u and Vk (X ) = H, for k ¡ 1. i ˆ Define analogously Vk (X, |) Ă Hom(G, | ), for arbitrary field |. i i ˆ Then Vk (X, |) = Vk (X, K) X Hom(G, | ), for any | Ď K. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 7 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES LEMMA i For each 0 ¤ i ¤ q andk ¥ 0, the set Vk (X ) is a Zariski closed subset of the algebraic group G = Char(X ). PROOF (FOR i ă q). p Let R = C[Gab] be the coordinate ring of G = Gab. By definition, a i character r belongs to Vk (X ) if and only if p p ab ab rank Bi+1(r) + rank Bi (r) ¤ ci ´ k, where ci = ci (X ) is the number of i-cells of X. Hence, i ab ab Vk (X ) = tr P G | rank Bi+1(r) ¤ r ´ 1 or rank Bi (r) ¤ s ´ 1u r+s=c ´k+1; r,s¥0 i £ p ab ab = V Ir (Bi ) ¨ Is(Bi+1) , r+s=c ´k+1; r,s¥0 i ¸ where Ir (j) = ideal of r ˆ r minors of j. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 8 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES The characteristic varieties are homotopy-type invariants of a space: LEMMA SupposeX » X 1. There is then an isomorphism G 1 – G, which i 1 i restricts to isomorphisms Vk (X ) – Vk (X ), for alli ¤ q andk ¥ 0. p p PROOF. Let f : X Ñ X 1 be a (cellular) homotopy equivalence. 0 1 10 The induced homomorphism f7 : p1(X, e ) Ñ p1(X , e ), yields an 1 isomorphism of algebraic groups, fˆ7 : G Ñ G. Lifting f to a cellular homotopy equivalence, f˜ X X 1, defines x p : Ñ 1 1 isomorphisms Hi (X, Cr˝f7 ) Ñ Hi (X , Cr), for each r P G . r r ˆ i 1 i Hence, f7 restricts to isomorphisms Vk (X ) Ñ Vk (X ). p ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 9 / 25 CAST OF CHARACTERS DEGREE 1 CHARACTERISTIC VARIETIES DEGREE 1 CHARACTERISTIC VARIETIES 1 2 Vk (X ) depends only on G = p1(X ) (in fact, only on G/G ), so we 1 may write these sets as Vk (G). Suppose G = xx1, ... , xm | r1, ... , rpy is finitely presented 1 ab Away from1 P G, we have that Vk (G) = V (Ek (B1 )), the zero-set of the ideal of codimension k minors of the Alexander matrix p ab ab p m B1 = Bri /Bxj : ZGab Ñ ZGab. If j : G Q is an epimorphism, then, for each k ¥ 1, the induced monomorphism between character groups, j˚ : Q ãÑ G, restricts 1 1 to an embedding Vk (Q) ãÑ Vk (G). p p Given any subvariety W Ă (Cˆ)n defined over Z, there is a finitely n 1 presented group G such that Gab = Z and V1 (G) = W . ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 10 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES WARM-UP EXAMPLES EXAMPLE (THE CIRCLE) We have S1 = R. 1 ˘1 Identify p1(S , ˚) = Z = xty and ZZ = Z[t ]. Then: Ă 1 ˘1 t´1 ˘1 C‚(S ) : 0 / Z[t ] / Z[t ] / 0 For r P Hom(Z, CĂˆ) = Cˆ, we get r´1 1 Cr bZZ C‚(S ) : 0 / C / C / 0 1 1 which is exact, except for rĂ= 1, when H0(S , C) = H1(S , C) = C. Hence: 0 1 1 1 V1 (S ) = V1 (S ) = t1u i 1 Vk (S ) = H, otherwise. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 11 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES EXAMPLE (THE n-TORUS) n n n ˆ ˆ n Identify p1(T ) = Z , and Hom(Z , C ) = (C ) . Using the Koszul n resolution C‚(T ) as above, we get n Ă i n t1u if k ¤ ( i ), Vk (T ) = #H otherwise. EXAMPLE (NILMANIFOLDS) More generally, let M be a nilmanifold. An inductive argument on the nilpotency class of p1(M), based on the Hochschild-Serre spectral sequence, yields i t1u if k ¤ bi (M), Vk (M) = #H otherwise ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 12 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES EXAMPLE (WEDGE OF CIRCLES) n 1 ˆ ˆ n Identify p1( S ) = Fn, and Hom(Fn, C ) = (C ) . Then: Ž ˆ n n (C ) if k ă n, 1 1 Vk S = $t1u if k = n, ' ª &H if k ¡ n. %' EXAMPLE (ORIENTABLE SURFACE OF GENUS g ¡ 1) Write p1(Sg ) = xx1, ... , xg, y1, ... , yg | [x1, y1] ¨ ¨ ¨ [xg, yg ] = 1y, and ˆ ˆ 2g identify Hom(p1(Sg ), C ) = (C ) . Then: (Cˆ)2g if i = 1, k ă 2g ´ 1, i Vk (Sg ) = $t1u if i = 1, k = 2g ´ 1, 2g; or i = 2, k = 1, &'H otherwise. %' ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 13 / 25 EXAMPLES AND COMPUTATIONS TORIC COMPLEXES TORIC COMPLEXES AND RAAGS Given L simplicial complex on n vertices, define the toric complex n TL as the subcomplex of T obtained by deleting the cells corresponding to the missing simplices of L: s s n TL = T , where T = tx P T | xi = ˚ if i R su.