A LGEBRAIC MODELS AND THE T ANGENT C ONE THEOREM Alex Suciu Representation Theory and Related Topics Seminar Northeastern University April 10, 2015 A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 1 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A CDGA R ESONANCE VARIETIES OF A CDGA Let A “ p A ‚ , d q be a commutative, differential graded C -algebra. Multiplication ¨ : A i b A j Ñ A i ` j is graded-commutative. Differential d : A i Ñ A i ` 1 satisfies the graded Leibnitz rule. Assume A is connected, i.e., A 0 “ C . A is of finite-type, i.e., dim A i ă 8 for all i ě 0. For each a P Z 1 p A q – H 1 p A q , we get 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 . Resonance varieties : R i p A q “ t a P H 1 p A q | H i p A ‚ , δ a q ‰ 0 u . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 2 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A CDGA Fix C -basis t e 1 , . . . , e n u for H 1 p A q , and let t x 1 , . . . , x n u be dual basis for H 1 p A q “ H 1 p A q _ . Identify Sym p H 1 p A qq with S “ C r x 1 , . . . , x n s , the coordinate ring of the affine space H 1 p A q . Define a cochain complex of free S -modules, p A ‚ b S , δ q : ¨ ¨ ¨ � A i b S δ i � A i ` 1 b S δ i ` 1 � A i ` 2 b S � ¨ ¨ ¨ , δ i p u b s q “ ř n where j “ 1 e j u b sx j ` d u b s . The specialization of A b S at a P H 1 p A q coincides with p A , δ a q . R i p A q “ supp p H i p A ‚ b S , δ qq are The cohomology support loci r subvarieties of H 1 p A q . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 3 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A CDGA Let p A ‚ b S , Bq be the dual chain complex. The homology support loci r R i p A q “ supp p H i p A ‚ b S , Bqq are subvarieties of H 1 p A q . Using a result of [Papadima–S. 2014], we obtain: T HEOREM For each q ě 0 , the duality isomorphism H 1 p A q – H 1 p A q restricts to an Ť i ď q R i p A q – Ť i ď q r R i p A q . isomorphism We also have R i p A q – R i p A q . In general, though, r R i p A q fl r R i p A q . If d “ 0, then all the resonance varieties of A are homogeneous. In general, though, they are not. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 4 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A CDGA E XAMPLE Let A be the exterior algebra on generators a , b in degree 1, endowed with the differential given by d a “ 0 and d b “ b ¨ a . H 1 p A q “ C , generated by a . Set S “ C r x s . Then: ´ ¯ 0 B 2 “ B 1 “p x 0 q x ´ 1 � S 2 � S . A ‚ b S : S Hence, H 1 p A ‚ b S q “ S {p x ´ 1 q , and so r R 1 p A q “ t 1 u . Using the above theorem, we conclude that R 1 p A q “ t 0 , 1 u . R 1 p A q is a non-homogeneous subvariety of C . H 1 p A ‚ b S q “ S {p x q , and so r R 1 p A q “ t 0 u ‰ r R 1 p A q . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 5 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A SPACE R ESONANCE VARIETIES OF A SPACE Let X be a connected, finite-type CW-complex. We may take A “ H ˚ p X , C q with d “ 0, and get the usual resonance varieties, R i p X q : “ R i p A q . Or, we may take p A , d q to be a finite-type cdga, weakly equivalent to Sullivan’s model A PL p X q , if such a cdga exists. If X is formal , then ( H ˚ p X , C q , d “ 0) is such a finite-type model. Finite-type cdga models exist even for possibly non-formal spaces, such as nilmanifolds and solvmanifolds, Sasakian manifolds, smooth quasi-projective varieties, etc. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 6 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A SPACE T HEOREM (M ACINIC , P APADIMA , P OPESCU , S. – 2013) Suppose there is a finite-type CDGA p A , d q such that A PL p X q » A. Then, for each i ě 0 , the tangent cone at 0 to the resonance variety R i p A q is contained in R i p X q . In general, we cannot replace TC 0 p R i p A qq by R i p A q . E XAMPLE Let X “ S 1 , and take A “ Ź p a , b q with d a “ 0, d b “ b ¨ a . Then R 1 p A q “ t 0 , 1 u is not contained in R 1 p X q “ t 0 u , though TC 0 p R 1 p A qq “ t 0 u is. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 7 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A SPACE A rationally defined CDGA p A , d q has positive weights if each A i can be decomposed into weighted pieces A i α , with positive weights in degree 1, and in a manner compatible with the CDGA structure: A i “ À α P Z A i α . 1 A 1 α “ 0, for all α ď 0. 2 α and b P A j β , then ab P A i ` j If a P A i α ` β and d a P A i ` 1 α . 3 A space X is said to have positive weights if A PL p X q does. If X is formal, then X has positive weights, but not conversely. T HEOREM (D IMCA –P APADIMA 2014, MPPS) Suppose there is a rationally defined, finite-type CDGA p A , d q with positive weights, and a q-equivalence between A PL p X q and A preserving Q -structures. Then, for each i ď q, R i p A q is a finite union of rationally defined linear subspaces of 1 H 1 p A q . R i p A q Ď R i p X q . 2 A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 8 / 20
R ESONANCE VARIETIES R ESONANCE VARIETIES OF A SPACE E XAMPLE Let X be the 3-dimensional Heisenberg nilmanifold. All cup products of degree 1 classes vanish; thus, R 1 p X q “ H 1 p X , C q “ C 2 . Model A “ Ź p a , b , c q generated in degree 1, with d a “ d b “ 0 and d c “ a ¨ b . This is a finite-dimensional model, with positive weights: wt p a q “ wt p b q “ 1, wt p c q “ 2. Writing S “ C r x , y s , we get ¨ ˛ y 0 0 ˝ ‚ ´ x 0 0 1 ´ x ´ y p x y 0 q � S 3 � S 3 � S . A ‚ b S : ¨ ¨ ¨ Hence H 1 p A ‚ b S q “ S {p x , y q , and so R 1 p A q “ t 0 u . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 9 / 20
C OHOMOLOGY IN LOCAL SYSTEMS C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a finite-type, connected CW-complex. π “ π 1 p X , x 0 q : a finitely generated group. Char p X q “ Hom p π, C ˚ q : an abelian, algebraic group. Char p X q 0 – p C ˚ q n , where n “ b 1 p X q . Characteristic varieties of X : V i p X q “ t ρ P Char p X q | H i p X , C ρ q ‰ 0 u . T HEOREM (L IBGOBER 2002, D IMCA –P APADIMA –S. 2009) τ 1 p V i p X qq Ď TC 1 p V i p X qq Ď R i p X q Here, if W Ă p C ˚ q n is an algebraic subset, then τ 1 p W q : “ t z P C n | exp p λ z q P W , for all λ P C u . This is a finite union of rationally defined linear subspaces of C n . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 10 / 20
C OHOMOLOGY IN LOCAL SYSTEMS C HARACTERISTIC VARIETIES T HEOREM (D IMCA –P APADIMA 2014) Suppose A PL p X q is q-equivalent to a finite-type model p A , d q . Then V i p X q p 1 q – R i p A q p 0 q , for all i ď q. C OROLLARY If X is a q-formal space, then V i p X q p 1 q – R i p X q p 0 q , for all i ď q. A precursor to corollary can be found in work of Green–Lazarsfeld on the cohomology jump loci of compact Kähler manifolds. The case when q “ 1 was first established in [DPS-2009]. Further developments in work of Budur–Wang [2013]. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 11 / 20
C OHOMOLOGY IN LOCAL SYSTEMS T HE TANGENT CONE THEOREM T HE TANGENT CONE THEOREM T HEOREM Suppose A PL p X q is q-equivalent to a finite-type CDGA A. Then, @ i ď q, TC 1 p V i p X qq “ TC 0 p R i p A qq . 1 If, moreover, A has positive weights, and the q-equivalence 2 A PL p X q » A preserves Q -structures, then TC 1 p V i p X qq “ R i p A q . T HEOREM (DPS-2009, DP-2014) Suppose X is a q-formal space. Then, for all i ď q, τ 1 p V i p X qq “ TC 1 p V i p X qq “ R i p X q . A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 12 / 20
C OHOMOLOGY IN LOCAL SYSTEMS T HE TANGENT CONE THEOREM C OROLLARY If X is q-formal, then, for all i ď q, All irreducible components of R i p X q are rationally defined 1 subspaces of H 1 p X , C q . All irreducible components of V i p X q which pass through the origin 2 are algebraic subtori of Char p X q 0 , of the form exp p L q , where L runs through the linear subspaces comprising R i p X q . The Tangent Cone theorem can be used to detect non-formality. E XAMPLE Let π “ x x 1 , x 2 | r x 1 , r x 1 , x 2 ssy . Then V 1 p π q “ t t 1 “ 1 u , and so τ 1 p V 1 p π qq “ TC 1 p V 1 p π qq “ t x 1 “ 0 u . On the other hand, R 1 p π q “ C 2 , and so π is not 1-formal. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 13 / 20
C OHOMOLOGY IN LOCAL SYSTEMS T HE TANGENT CONE THEOREM E XAMPLE (DPS 2009) Let π “ x x 1 , . . . , x 4 | r x 1 , x 2 s , r x 1 , x 4 sr x ´ 2 2 , x 3 s , r x ´ 1 1 , x 3 sr x 2 , x 4 sy . Then R 1 p π q “ t z P C 4 | z 2 1 ´ 2 z 2 2 “ 0 u : a quadric which splits into two linear subspaces over R , but is irreducible over Q . Thus, π is not 1-formal. E XAMPLE (S.–Y ANG –Z HAO 2015) Let π be a finitely presented group with π ab “ Z 3 and � ( p t 1 , t 2 , t 3 q P p C ˚ q 3 | p t 2 ´ 1 q “ p t 1 ` 1 qp t 3 ´ 1 q V 1 p π q “ , This is a complex, 2-dimensional torus passing through the origin, but this torus does not embed as an algebraic subgroup in p C ˚ q 3 . Indeed, τ 1 p V 1 p π qq “ t x 2 “ x 3 “ 0 u Y t x 1 ´ x 3 “ x 2 ´ 2 x 3 “ 0 u . Hence, π is not 1-formal. A LEX S UCIU A LGEBRAIC MODELS A PRIL 10, 2015 14 / 20
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