A LGEBRAIC MODELS FOR S ASAKIAN MANIFOLDS AND WEIGHTED - HOMOGENEOUS SURFACE SINGULARITIES Alex Suciu Northeastern University Joint work with Stefan Papadima (IMAR) arxiv:1511.08948 Workshop on Singularities and Topology Université de Nice March 9, 2016 A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 1 / 21
O UTLINE A LGEBRAIC MODELS FOR SPACES 1 M ODELS FOR GROUP ACTIONS 2 S ASAKIAN MANIFOLDS 3 W EIGHTED HOMOGENEOUS SINGULARITIES 4 A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 2 / 21
A LGEBRAIC MODELS FOR SPACES CDGA S CDGA S 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. Fix integer q ě 1 (or q “ 8 ). We say that A is q-finite if A is connected, i.e., A 0 “ C . A i is finite-dimensional, for each i ď q (or i ă 8 ). Two CDGA s have the same q -type if there is a zig-zag of connecting morphisms, each one inducing isomorphisms in homology up to degree q and a monomorphism in degree q ` 1. A CDGA p A , d q is q-formal if it has the same q -type as p H ‚ p A q , d “ 0 q . A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 3 / 21
A LGEBRAIC MODELS FOR SPACES R EGULAR SEQUENCES R EGULAR SEQUENCES Let H ‚ be a connected, commutative graded algebra. A sequence t e α u of homogeneous elements in H ą 0 is said to be q-regular if for each α , the class of e α in ÿ H α “ H { e β H β ă α is a non-zero divisor up to degree q . That is, the multiplication map i i ` n α e α ¨ : H ¯ α Ñ H is injective, for all i ď q , where n α “ deg p e α q . α T HEOREM Suppose e 1 , . . . , e r is an even-degree, q-regular sequence in H. Then Ź p t 1 , . . . , t r q with d “ 0 on H and the Hirsch extension A “ H b τ dt α “ τ p t α q “ e α has the same q-type as ´ ¯ ÿ H { α e α H , d “ 0 . In particular, A is q-formal. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 4 / 21
A LGEBRAIC MODELS FOR SPACES T HE S ULLIVAN MODEL T HE S ULLIVAN MODEL To a large extent, the rational homotopy type of a space can be reconstructed from algebraic models associated to it. If the space is a smooth manifold M , the standard model is the de Rham algebra Ω dR p M q . More generally, any space X has an associated Sullivan CDGA , A PL p X q , which serves as the reference algebraic model. In particular, H ˚ p A PL p X qq “ H ˚ p X , C q . A CDGA p A , d q is a q-model for X if A has the same q -type as A PL p X q . We say X is q-formal if A PL p X q has this property. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 5 / 21
A LGEBRAIC MODELS FOR SPACES T HE M ALCEV L IE ALGEBRA T HE M ALCEV L IE ALGEBRA The 1-formality property of a connected CW-complex X with finite 1-skeleton depends only on its fundamental group, π “ π 1 p X q . Let m p π q “ Prim p z C r π sq be the Malcev Lie algebra of π , where p is completion with respect to powers of the augmentation ideal. The 1-formality of the group π is equivalent to m p π q – p L , for some quadratic, finitely generated Lie algebra L , where p is the degree completion. If m p π q – p L , where L is merely assumed to have homogeneous relations, then π is said to be filtered formal (see [SW] for details). A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 6 / 21
A LGEBRAIC MODELS FOR SPACES C OHOMOLOGY JUMP LOCI C OHOMOLOGY JUMP LOCI Let X be a connected, finite-type CW-complex, and let G be a complex linear algebraic group. The characteristic varieties of X with respect to a rational, finite-dimensional representation ϕ : G Ñ GL p V q are the sets ! ) V i ρ P Hom p π, G q | dim H i p X , V ϕ ˝ ρ q ě s s p X , ϕ q “ . In degree i “ 1, these varieties depend only on the group π “ π 1 p X q , and so we may denote them as V 1 s p π, ϕ q . When G “ C ˚ and ϕ “ id C ˚ , we simply write these sets as V i s p X q . A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 7 / 21
M ODELS FOR GROUP ACTIONS C OMPACT L IE GROUP ACTIONS C OMPACT L IE GROUP ACTIONS Let M be a compact, connected, smooth manifold (with B M “ H ). Suppose a compact, connected Lie group K acts smoothly and almost freely on M (i.e., all the isotropy groups are finite). Let K Ñ EK ˆ M Ñ M K be the Borel construction on M . Let τ : H ‚ p K , C q Ñ H ‚` 1 p M K , C q be the transgression in the Serre spectral sequence of this bundle. Let N “ M { K be the orbit space (a smooth orbifold). The projection map pr : M K Ñ N induces an isomorphism pr ˚ : H ‚ p N , C q Ñ H ‚ p M K , C q . By a theorem of H. Hopf, we may dentify H ‚ p K , C q “ Ź P ‚ , where P “ span t t 1 , . . . , t r u where m α “ deg p t α q is odd. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 8 / 21
M ODELS FOR GROUP ACTIONS C OMPACT L IE GROUP ACTIONS T HEOREM There is a map σ : P ‚ Ñ Z ‚` 1 p A PL p N qq such that pr ˚ ˝r σ s “ τ and Ź P . A PL p M q » A PL p N q b σ T HEOREM Suppose that The orbit space N “ M { K is k-formal, for some k ą max t m α u . The characteristic classes e α “ p pr ˚ q ´ 1 p τ p t α qq P H m α ` 1 p N , C q form a q-regular sequence in H ‚ “ H ‚ p N , C q , for some q ď k. Then the CDGA ´ ¯ H ‚ L ÿ e α H ‚ , d “ 0 α is a finite q-model for M. In particular, M is q-formal. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 9 / 21
M ODELS FOR GROUP ACTIONS C OMPACT L IE GROUP ACTIONS T HEOREM Suppose the orbit space N “ M { K is 2 -formal. Then: The group π “ π 1 p M q is filtered-formal. In fact, m p π q is the degree 1 completion of L { r , where L “ Lie p H 1 p π, C qq and r is a homogeneous ideal generated in degrees 2 and 3 . For every complex linear algebraic group G, the germ at 1 of the 2 representation variety Hom p π, G q is defined by quadrics and cubics only. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 10 / 21
M ODELS FOR GROUP ACTIONS C OMPACT L IE GROUP ACTIONS The projection map p : M Ñ M { K induces an epimorphism p 7 : π 1 p M q ։ π orb 1 p M { K q between orbifold fundamental groups. T HEOREM Suppose that the transgression P ‚ Ñ H ‚` 1 p M { K , C q is injective in degree 1 . Then: If the orbit space N “ M { K has a 2 -finite 2 -model, then p 7 induces 1 analytic isomorphisms V 1 s p π orb 1 p N qq p 1 q – V 1 s p π 1 p M qq p 1 q . If N is 2 -formal, then p 7 induces an analytic isomorphism 2 Hom p π orb 1 p N q , SL 2 p C qq p 1 q – Hom p π 1 p M q , SL 2 p C qq p 1 q . A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 11 / 21
S ASAKIAN MANIFOLDS S ASAKIAN MANIFOLDS AND q - FORMALITY S ASAKIAN MANIFOLDS AND q - FORMALITY Sasakian geometry is an odd-dimensional analogue of Kähler geometry. Every compact Sasakian manifold M admits an almost-free circle action with orbit space N “ M { S 1 a Kähler orbifold. The Euler class of the action coincides with the Kähler class of the base, h P H 2 p N , Q q . The class h satisfies the Hard Lefschetz property, i.e., ¨ h k : H n ´ k p N , C q Ñ H n ` k p N , C q is an isomorphism, for each 1 ď k ď n . Thus, t h u is an p n ´ 1 q -regular sequence in H ‚ p N , C q E XAMPLE Let N be a compact Kähler manifold such that the Kähler class is integral, i.e., h P H 2 p N , Z q , and let M be the total space of the principal S 1 -bundle classified by h . Then M is a (regular) Sasakian manifold. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 12 / 21
S ASAKIAN MANIFOLDS S ASAKIAN MANIFOLDS AND q - FORMALITY As shown by Deligne, Griffiths, Morgan, and Sullivan, compact Kähler manifolds are formal. As shown by A. Tievsky, every compact Sasakian manifold M has a finite model of the form p H ‚ p N , C q b Ź p t q , d q , where d vanishes on H ‚ p N , C q and sends t to h . T HEOREM Let M be a compact Sasakian manifold of dimension 2 n ` 1 . Then M is p n ´ 1 q -formal. This result is optimal: for each n ě 1, the p 2 n ` 1 q -dimensional Heisenberg compact nilmanifold (with orbit space T 2 n ) is a Sasakian manifold, yet it is not n -formal. This theorem strengthens a statement of H. Kasuya, who claimed that, for n ě 2, a Sasakian manifold M 2 n ` 1 is 1-formal. The proof of that claim, though, has a gap. A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 13 / 21
S ASAKIAN MANIFOLDS S ASAKIAN GROUPS S ASAKIAN GROUPS A group π is said to be a Sasakian group if it can be realized as the fundamental group of a compact, Sasakian manifold. Open problem: Which finitely presented groups are Sasakian? A first, well-known obstruction is that b 1 p π q must be even. T HEOREM Let π “ π 1 p M 2 n ` 1 q be the fundamental group of a compact Sasakian manifold of dimension 2 n ` 1 . Then: The group π is filtered-formal, and in fact 1 -formal if n ą 1 . 1 All irreducible components of the characteristic varieties V 1 s p π q 2 passing through 1 are algebraic subtori of Hom p π, C ˚ q . If G is a complex linear algebraic group, then the germ at 1 of 3 Hom p π, G q is defined by quadrics and cubics only, and in fact by quadrics only if n ą 1 . A LEX S UCIU (N ORTHEASTERN ) A LGEBRAIC MODELS N ICE , M ARCH 9, 2016 14 / 21
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