A QUICK INTRODUCTION TO COHOMOLOGY JUMP LOCI Alex Suciu Northeastern University Topology Seminar University of California, Berkeley October 11, 2017 A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 1 / 22
C OHOMOLOGY JUMP LOCI S UPPORT VARIETIES S UPPORT VARIETIES Let k be an algebraically closed field. Let S be a commutative, finitely generated k -algebra. Let m Spec ( S ) = Hom k -alg ( S , k ) be the maximal spectrum of S . d i � E i ´ 1 � E i � ¨ ¨ ¨ � E 0 � 0 be an S -chain complex. Let E : ¨ ¨ ¨ The support varieties of E are the subsets of m Spec ( S ) given by ľ s � � W i s ( E ) = supp H i ( E ) . They depend only on the chain-homotopy equivalence class of E . For each i ě 0, m Spec ( S ) = W i 0 ( E ) Ě W i 1 ( E ) Ě W i 2 ( E ) Ě ¨ ¨ ¨ . If all E i are finitely generated S -modules, then the sets W i s ( E ) are Zariski closed subsets of m Spec ( S ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 2 / 22
C OHOMOLOGY JUMP LOCI H OMOLOGY JUMP LOCI H OMOLOGY JUMP LOCI The homology jump loci of the S -chain complex E are defined as V i s ( E ) = t m P m Spec ( S ) | dim k H i ( E b S S / m ) ě s u . They depend only on the chain-homotopy equivalence class of E . For each i ě 0, m Spec ( S ) = V i 0 ( E ) Ě V i 1 ( E ) Ě V i 2 ( E ) Ě ¨ ¨ ¨ . (Papadima–S. 2014) Suppose E is a chain complex of free , finitely generated S -modules. Then: Each V i d ( E ) is a Zariski closed subset of m Spec ( S ) . For each q , ď ď V i W i 1 ( E ) = 1 ( E ) . i ď q i ď q A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 3 / 22
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES R ESONANCE VARIETIES Let A = À i ě 0 A i be a commutative graded k -algebra, with A 0 = k . Let a P A 1 , and assume a 2 = 0 (this condition is redundant if char ( k ) ‰ 2, by graded-commutativity of the multiplication in A ). Consider the cochain complex of k -vector spaces, a a a � A 1 � A 2 � ¨ ¨ ¨ , ( A , δ a ) : A 0 with differentials given by b ÞÑ a ¨ b , for b P A i . The resonance varieties of A are the sets s ( A ) = t a P A 1 | a 2 = 0 and dim k H i ( A , a ) ě s u . R i If A is locally finite (i.e., dim k A i ă 8 , for all i ě 1), then the sets R i s ( A ) are Zariski closed cones inside the affine space A 1 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 4 / 22
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES Fix a k -basis t e 1 , . . . , e n u for A 1 , and let t x 1 , . . . , x n u be the dual basis for A 1 = ( A 1 ) _ . Identify Sym ( A 1 ) with S = k [ x 1 , . . . , x n ] , the coordinate ring of the affine space A 1 . Define a cochain complex of free S -modules, K ( A ) : = ( A ‚ b S , δ ) , δ i δ i + 1 � A i + 2 b k S � A i b k S � A i + 1 b k S � ¨ ¨ ¨ , ¨ ¨ ¨ where δ i ( u b s ) = ř n j = 1 e j u b sx j . The specialization of ( A b S , δ ) at a P A 1 coincides with ( A , δ a ) . s ( A ) = supp ( Ź s H i ( K ( A ))) are The cohomology support loci R i (closed) subvarieties of A 1 . Both R i s ( A ) and R i s ( A ) can be arbitrarily complicated (homogeneous) affine varieties. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 5 / 22
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES E XAMPLE (E XTERIOR ALGEBRA ) Let E = Ź V , where V = k n , and S = Sym ( V ) . Then K ( E ) is the Koszul complex on V . E.g., for n = 3: � x 2 � x 1 x 3 0 � � ´ x 1 0 x 3 x 2 � S 3 ( x 3 ´ x 2 x 1 ) � S . x 3 0 ´ x 1 ´ x 2 � S 3 S This chain complex provides a free resolution ε : K ( E ) Ñ k of the trivial S -module k . Hence, # if s ď ( n t 0 u i ) , R i s ( E ) = H otherwise . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 6 / 22
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES E XAMPLE (N ON - ZERO RESONANCE ) Let A = Ź ( e 1 , e 2 , e 3 ) / x e 1 e 2 y , and set S = k [ x 1 , x 2 , x 3 ] . Then � x 1 � � � x 3 0 ´ x 1 x 2 0 x 3 ´ x 2 x 3 � S 2 . � S 3 K ( A ) : S $ & t x 3 = 0 u if s = 1 , R 1 s ( A ) = t 0 u if s = 2 or 3 , % H if s ą 3 . E XAMPLE (N ON - LINEAR RESONANCE ) Let A = Ź ( e 1 , . . . , e 4 ) / x e 1 e 3 , e 2 e 4 , e 1 e 2 + e 3 e 4 y . Then x 1 � x 4 0 0 ´ x 1 � x 2 x 3 0 x 3 ´ x 2 0 x 4 ´ x 2 x 1 x 4 ´ x 3 � S 3 . � S 4 K ( A ) : S R 1 1 ( A ) = t x 1 x 2 + x 3 x 4 = 0 u A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 7 / 22
T HE TANGENT CONE THEOREM C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite-type CW-complex. Fundamental group π = π 1 ( X , x 0 ) : a finitely generated, discrete group, with π ab – H 1 ( X , Z ) . Fix a field k with k = k (usually k = C ), and let S = k [ π ab ] . Identify m Spec ( S ) with the character group Char ( X ) = Hom ( π , k ˚ ) , also denoted p π = y π ab . The characteristic varieties of X are the homology jump loci of free S -chain complex E = C ˚ ( X ab , k ) : V i s ( X , k ) = t ρ P Char ( X ) | dim k H i ( X , k ρ ) ě s u . Each set V i s ( X , k ) is a subvariety of Char ( X ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 8 / 22
T HE TANGENT CONE THEOREM C HARACTERISTIC VARIETIES E XAMPLE (C IRCLE ) Let X = S 1 . We have ( S 1 ) ab = R . Identify π 1 ( S 1 , ˚ ) = Z = x t y and ZZ = Z [ t ˘ 1 ] . Then: t ´ 1 � Z [ t ˘ 1 ] C ˚ (( S 1 ) ab ) : 0 � Z [ t ˘ 1 ] � 0 For each ρ P Hom ( Z , k ˚ ) = k ˚ , get a chain complex ρ ´ 1 � k C ˚ ( Ă � k � 0 S 1 ) b ZZ k ρ : 0 which is exact, except for ρ = 1, when H 0 ( S 1 , k ) = H 1 ( S 1 , k ) = k . Hence: V 0 1 ( S 1 ) = V 1 1 ( S 1 ) = t 1 u and V i s ( S 1 ) = H , otherwise. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 9 / 22
T HE TANGENT CONE THEOREM C HARACTERISTIC VARIETIES E XAMPLE (T ORUS ) Identify π 1 ( T n ) = Z n , and Hom ( Z n , k ˚ ) = ( k ˚ ) n . Then: " if s ď ( n t 1 u i ) , V i s ( T n ) = H otherwise . E XAMPLE (W EDGE OF CIRCLES ) Identify π 1 ( Ž n S 1 ) = F n , and Hom ( F n , k ˚ ) = ( k ˚ ) n . Then: $ ( k ˚ ) n & if s ă n , ł n S 1 � = V 1 � t 1 u if s = n , s % H if s ą n . E XAMPLE (O RIENTABLE SURFACE OF GENUS g ą 1 ) $ ( k ˚ ) 2 g & if s ă 2 g ´ 1 , V 1 s ( Σ g ) = if s = 2 g ´ 1 , 2 g , t 1 u % H if s ą 2 g . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 10 / 22
T HE TANGENT CONE THEOREM C HARACTERISTIC VARIETIES Homotopy invariance: If X » Y , then V i s ( Y , k ) – V i s ( X , k ) . Product formula: 1 ( X 1 ˆ X 2 , k ) = Ť p + q = i V p 1 ( X 1 , k ) ˆ V q V i 1 ( X 2 , k ) . Degree 1 interpretation: The sets V 1 s ( X , k ) depend only on π = π 1 ( X ) —in fact, only on π / π 2 . Write them as V 1 s ( π , k ) . ϕ : p Functoriality: If ϕ : π ։ G is an epimorphism, then ˆ Ñ p G ã π restricts to an embedding V 1 Ñ V 1 s ( G , k ) ã s ( π , k ) , for each s . Universality: Given any subvariety W Ă ( k ˚ ) n defined over Z , there is a finitely presented group π such that π ab = Z n and V 1 1 ( π , k ) = W . Alexander invariant interpretation: Let X ab Ñ X be the maximal abelian cover. View H ˚ ( X ab , k ) as a module over S = k [ π ab ] . Then: � à ď �� V j � X ab , k 1 ( X , k ) = supp H j . j ď i j ď i A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 11 / 22
T HE TANGENT CONE THEOREM T HE TANGENT CONE THEOREM T HE TANGENT CONE THEOREM The resonance varieties of X (with coefficients in k ) are the loci R i d ( X , k ) associated to the cohomology algebra A = H ˚ ( X , k ) . Each set R i s ( X ) : = R i s ( X , C ) is a homogeneous subvariety of H 1 ( X , C ) – C n , where n = b 1 ( X ) . Recall that V i s ( X ) : = V i s ( X , C ) is a subvariety of H 1 ( X , C ˚ ) – ( C ˚ ) n ˆ Tors ( H 1 ( X , Z )) . (Libgober 2002) TC 1 ( V i s ( X )) Ď R i s ( X ) . Given a subvariety W Ă H 1 ( X , C ˚ ) , let τ 1 ( W ) = t z P H 1 ( X , C ) | exp ( λ z ) P W , @ λ P C u . (Dimca–Papadima–S. 2009) τ 1 ( W ) is a finite union of rationally defined linear subspaces, and τ 1 ( W ) Ď TC 1 ( W ) . Thus, τ 1 ( V i s ( X )) Ď TC 1 ( V i s ( X )) Ď R i s ( X ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 12 / 22
T HE TANGENT CONE THEOREM F ORMALITY F ORMALITY X is formal if there is a zig-zag of cdga quasi-isomorphisms from ( A PL ( X , Q ) , d ) to ( H ˚ ( X , Q ) , 0 ) . X is k-formal (for some k ě 1) if each of these morphisms induces an iso in degrees up to k , and a monomorphism in degree k + 1. X is 1-formal if and only if π = π 1 ( X ) is 1-formal, i.e., its Malcev Lie algebra, m ( π ) = Prim ( y Q π ) , is quadratic. For instance, compact Kähler manifolds and complements of hyperplane arrangements are formal. (Dimca–Papadima–S. 2009) Let X be a 1-formal space. Then, for each s ą 0, τ 1 ( V 1 s ( X )) = TC 1 ( V 1 s ( X )) = R 1 s ( X ) . Consequently, R 1 s ( X ) is a finite union of rationally defined linear subspaces in H 1 ( X , C ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI UCB T OPOLOGY S EMINAR 13 / 22
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