G EOMETRY AND TOPOLOGY OF COHOMOLOGY JUMP LOCI L ECTURE 2: R ESONANCE 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 A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 1 / 24
O UTLINE R ESONANCE VARIETIES OF CDGA S 1 Commutative differential graded algebras Resonance varieties Tangent cone inclusion R ESONANCE VARIETIES OF SPACES 2 Algebraic models for spaces Germs of jump loci Tangent cones and exponential maps The tangent cone theorem Detecting non-formality I NFINITESIMAL FINITENESS OBSTRUCTIONS 3 Spaces with finite models Associated graded Lie algebras Holonomy Lie algebras Malcev Lie algebras Finiteness obstructions for groups A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 2 / 24
R ESONANCE VARIETIES OF CDGA S C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS Let A “ p A ‚ , d q be a commutative, differential graded algebra over a field k of characteristic 0. That is: A “ À i ě 0 A i , where A i are k -vector spaces. The multiplication ¨ : A i b A j Ñ A i ` j is graded-commutative, i.e., ab “ p´ 1 q | a || b | ba for all homogeneous a and b . The differential d: A i Ñ A i ` 1 satisfies the graded Leibnitz rule, i.e., d p ab q “ d p a q b ` p´ 1 q | a | a d p b q . A CDGA A is of finite-type (or q-finite ) if it is connected (i.e., A 0 “ k ¨ 1); dim k A i is finite for i ď q . Let H i p A q “ ker p d: A i Ñ A i ` 1 q{ im p d: A i ´ 1 Ñ A i q . Then H ‚ p A q inherits an algebra structure from A . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 3 / 24
R ESONANCE VARIETIES OF CDGA S C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS A cdga morphism ϕ : A Ñ B is both an algebra map and a cochain map. Hence, it induces a morphism ϕ ˚ : H ‚ p A q Ñ H ‚ p B q . A map ϕ : A Ñ B is a quasi-isomorphism if ϕ ˚ is an isomorphism. Likewise, ϕ is a q -quasi-isomorphism (for some q ě 1) if ϕ ˚ is an isomorphism in degrees ď q and is injective in degree q ` 1. Two cdgas, A and B , are (q-)equivalent ( » q ) if there is a zig-zag of ( q -)quasi-isomorphisms connecting A to B . A cdga A is formal (or just q-formal ) if it is ( q -)equivalent to p H ‚ p A q , d “ 0 q . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 4 / 24
R ESONANCE VARIETIES OF CDGA S R ESONANCE VARIETIES R ESONANCE VARIETIES Since A is connected and d p 1 q “ 0, we have Z 1 p A q “ H 1 p A q . For each a P Z 1 p A q , we construct a cochain complex, δ 0 δ 1 δ 2 � A 1 � A 2 � ¨ ¨ ¨ , p A ‚ , δ a q : A 0 a a a with differentials δ i a p u q “ a ¨ u ` d u , for all u P A i . The resonance varieties of A are the sets R i k p A q “ t a P H 1 p A q | dim H i p A ‚ , δ a q ě k u . If A is q -finite, then R i k p A q are algebraic varieties for all i ď q . If A is a CGA (so that d “ 0), these varieties are homogeneous subvarieties of H 1 p A q “ A 1 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 5 / 24
R ESONANCE VARIETIES OF CDGA S R ESONANCE VARIETIES Fix a k -basis t e 1 , . . . , e r u for H 1 p A q , and let t x 1 , . . . , x r u be the dual basis for H 1 p A q “ p H 1 p A qq ˚ . Identify Sym p H 1 p A qq with S “ k r x 1 , . . . , x r s , the coordinate ring of the affine space H 1 p A q . Define a cochain complex of free S -modules, L p A q : “ p A ‚ b k S , δ q , � A i b S δ i � A i ` 1 b S δ i ` 1 � A i ` 2 b S � ¨ ¨ ¨ , ¨ ¨ ¨ where δ i p u b f q “ ř n j “ 1 e j u b fx j ` d u b f . The specialization of p A b k S , δ q at a P A 1 coincides with p A , δ a q . Hence, R i k p A q is the zero-set of the ideal generated by all minors of size b i p A q ´ k ` 1 of the block-matrix δ i ` 1 ‘ δ i . In particular, R 1 k p A q “ V p I r ´ k p δ 1 qq , the zero-set of the ideal of codimension k minors of δ 1 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 6 / 24
R ESONANCE VARIETIES OF CDGA S R ESONANCE VARIETIES E XAMPLE (E XTERIOR ALGEBRA ) Let E “ Ź V , where V “ k n , and S “ Sym p V q . Then L p E q is the Koszul complex on V . E.g., for n “ 3: ˜ x 2 ¸ ˆ x 1 ˙ x 3 0 δ 2 “ ´ x 1 0 x 3 δ 1 “ x 2 � S 3 δ 3 “p x 3 ´ x 2 x 1 q � S . x 3 0 ´ x 1 ´ x 2 � S 3 S Hence, # ` n ˘ t 0 u if k ď , R i i k p E q “ H otherwise . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 7 / 24
R ESONANCE VARIETIES OF CDGA S R ESONANCE VARIETIES E XAMPLE (N ON - ZERO RESONANCE ) Let A “ Ź p e 1 , e 2 , e 3 q{x e 1 e 2 y , and set S “ k r x 1 , x 2 , x 3 s . Then ˆ x 1 ˙ ˆ ˙ x 3 0 ´ x 1 δ 1 “ x 2 δ 2 “ 0 x 3 ´ x 2 x 3 � S 2 . � S 3 L p A q : S $ & t x 3 “ 0 u if k “ 1 , R 1 k p A q “ t 0 u if k “ 2 or 3 , % H if k ą 3 . E XAMPLE (N ON - LINEAR RESONANCE ) Let A “ Ź p e 1 , . . . , e 4 q{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 ˝ ‚ δ 1 “ δ 2 “ x 3 0 x 3 ´ x 2 0 x 4 ´ x 2 x 1 x 4 ´ x 3 � S 3 . � S 4 L p A q : S R 1 1 p A q “ t x 1 x 2 ` x 3 x 4 “ 0 u A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 8 / 24
R ESONANCE VARIETIES OF CDGA S R ESONANCE VARIETIES E XAMPLE (N ON - HOMOGENEOUS RESONANCE ) Let A “ Ź p a , b q with d a “ 0, d b “ b ¨ a . H 1 p A q “ C , generated by a . Set S “ C r x s . Then: δ 1 “ p 0 x q � S 2 δ 2 “p x ´ 1 0 q � S . L p A q : S Hence, R 1 p A q “ t 0 , 1 u , a non-homogeneous subvariety of C . Let A 1 be the sub- CDGA generated by a . The inclusion map, A 1 ã Ñ A , induces an isomorphism in cohomology. But R 1 p A 1 q “ t 0 u , and so the resonance varieties of A and A 1 differ, although A and A 1 are quasi-isomorphic. P ROPOSITION If A » q A 1 , then R i k p A q p 0 q – R i k p A 1 q p 0 q , for all i ď q and k ě 0 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 9 / 24
R ESONANCE VARIETIES OF CDGA S T ANGENT CONE INCLUSION T ANGENT CONE INCLUSION T HEOREM (B UDUR –R UBIO , D ENHAM –S. 2018) If A is a connected k - CDGA A with locally finite cohomology, then TC 0 p R i k p A qq Ď R i k p H ‚ p A qq . In general, we cannot replace TC 0 p R i k p A qq by R i k p A q . E XAMPLE Let A “ Ź p a , b q with d a “ 0 and d b “ b ¨ a . Then H ‚ p A q “ Ź p a q , and so R 1 1 p A q “ t 0 u . Hence R 1 1 p A q “ t 0 , 1 u is not contained in R 1 1 p A q , though TC 0 p R 1 p A qq “ t 0 u is. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 10 / 24
R ESONANCE VARIETIES OF CDGA S T ANGENT CONE INCLUSION In general, the inclusion TC 0 p R i k p A qq Ď R i k p H ‚ p A qq is strict. E XAMPLE Let A “ Ź p a , b , c q with d a “ d b “ 0 and d c “ a ^ b . Writing S “ k r x , y s , we have: ¨ ˛ y ´ x 1 ˆ x ˙ ˝ ‚ δ 2 “ ´ x 0 0 δ 1 “ y 0 0 ´ y 0 � S 3 . � S 3 L p A q : S Hence R 1 1 p A q “ t 0 u . But H ‚ p A q “ Ź p a , b q{p ab q , and so R 1 1 p H ‚ p A qq “ k 2 . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 11 / 24
R ESONANCE VARIETIES OF SPACES A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES Given any space X , there is an associated Sullivan Q -cdga, A PL p X q , such that H ‚ p A PL p X qq “ H ‚ p X , Q q . We say X is q-finite if X has the homotopy type of a connected CW-complex with finite q -skeleton, for some q ě 1. An algebraic (q-)model (over k ) for X is a k -cgda p A , d q which is ( q -) equivalent to A PL p X q b Q k . If M is a smooth manifold, then Ω dR p M q is a model for M (over R ). Examples of spaces having finite-type models include: Formal spaces (such as compact Kähler manifolds, hyperplane arrangement complements, toric spaces, etc). Smooth quasi-projective varieties, compact solvmanifolds, Sasakian manifolds, etc. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 12 / 24
R ESONANCE VARIETIES OF SPACES G ERMS OF JUMP LOCI G ERMS OF JUMP LOCI T HEOREM (D IMCA –P APADIMA 2014) Let X be a q-finite space, and suppose X admits a q-finite, q-model A. Then the map exp: H 1 p X , C q Ñ H 1 p X , C ˚ q induces a local analytic isomorphism H 1 p A q p 0 q Ñ Char p X q p 1 q , which identifies the germ at 0 of R i k p A q with the germ at 1 of V i k p X q , for all i ď q and k ě 0 . C OROLLARY If X is a q-formal space, then V i k p X q p 1 q – R i k p X q p 0 q , for i ď q and k ě 0 . A precursor to corollary can be found in work of Green, Lazarsfeld, and Ein on cohomology jump loci of compact Kähler manifolds. The case when q “ 1 was first established in [DPS 2019]. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 13 / 24
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