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A LGEBRAIC MODELS AND THE
TANGENT C ONE THEOREM
Alex Suciu
Representation Theory and Related Topics Seminar
Northeastern University
April 10, 2015
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A LGEBRAIC MODELS
A PRIL 10, 2015
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R ESONANCE VARIETIES
R ESONANCE VARIETIES OF A CDGA
R ESONANCE VARIETIES OF A CDGA
Let A “ pA‚ , dq be a commutative, differential graded C-algebra.
Multiplication ¨ : Ai b Aj Ñ Ai`j is graded-commutative.
Differential d : Ai Ñ Ai`1 satisfies the graded Leibnitz rule.
Assume
A is connected, i.e., A0 “ C.
A is of finite-type, i.e., dim Ai ă 8 for all i ě 0.
For each a P Z 1 pAq – H 1 pAq, we get a cochain complex,
pA‚ , δa q : A0
δa0
/ A1
δa1
/ A2
δa2
/ ¨¨¨ ,
with differentials δai puq “ a ¨ u ` d u, for all u P Ai .
Resonance varieties:
Ri pAq “ ta P H 1 pAq | H i pA‚ , δa q ‰ 0u.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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R ESONANCE VARIETIES
R ESONANCE VARIETIES OF A CDGA
Fix C-basis te1 , . . . , en u for H 1 pAq, and let tx1 , . . . , xn u be dual
basis for H1 pAq “ H 1 pAq_ .
Identify SympH1 pAqq with S “ Crx1 , . . . , xn s, the coordinate ring of
the affine space H 1 pAq.
Define a cochain complex of free S-modules,
pA‚ b S, δq : ¨ ¨ ¨
where
δ i pu b sq “
/ Ai b S
δi
řn
b sxj ` d u b s.
j“1 ej u
i`1
/ Ai`1 b S δ / Ai`2 b S
/ ¨¨¨ ,
The specialization of A b S at a P H 1 pAq coincides with pA, δa q.
r i pAq “ supppH i pA‚ b S, δqq are
The cohomology support loci R
subvarieties of H1 pAq.
A LEX S UCIU
A LGEBRAIC MODELS
A PRIL 10, 2015
3 / 20
R ESONANCE VARIETIES
R ESONANCE VARIETIES OF A CDGA
Let pA‚ b S, Bq be the dual chain complex.
r i pAq “ supppHi pA‚ b S, Bqq are
The homology support loci R
subvarieties of H1 pAq.
Using a result of [Papadima–S. 2014], we obtain:
T HEOREM
For each q ě 0,Ťthe duality isomorphism
H 1 pAq – H1 pAq restricts to an
Ť
i
r
isomorphism
iďq R pAq –
iďq Ri pAq.
We also have Ri pAq – Ri pAq.
r i pAq fl R
r i pAq.
In general, though, R
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 pAq “ C, generated by a. Set S “ Crxs. Then:
´
B2 “
A‚ b S : S
0
x´1
¯
/ S2
B1 “p x 0 q
/S.
r 1 pAq “ t1u. Using the
Hence, H1 pA‚ b Sq “ S{px ´ 1q, and so R
1
above theorem, we conclude that R pAq “ t0, 1u.
R1 pAq is a non-homogeneous subvariety of C.
r 1 pAq “ t0u ‰ R
r 1 pAq.
H 1 pA‚ b Sq “ S{pxq, and so R
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 ˚ pX , Cq with d “ 0, and get the usual
resonance varieties, Ri pX q :“ Ri pAq.
Or, we may take pA, dq to be a finite-type cdga, weakly equivalent
to Sullivan’s model APL pX q, if such a cdga exists.
If X is formal, then (H ˚ pX , Cq, 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.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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R ESONANCE VARIETIES
R ESONANCE VARIETIES OF A SPACE
T HEOREM (M ACINIC , PAPADIMA , P OPESCU , S. – 2013)
Suppose there is a finite-type CDGA pA, dq such that APL pX q » A.
Then, for each i ě 0, the tangent cone at 0 to the resonance variety
Ri pAq is contained in Ri pX q.
In general, we cannot replace TC0 pRi pAqq by Ri pAq.
E XAMPLE
Let X “ S 1 , and take A “
Ź
pa, bq with d a “ 0, d b “ b ¨ a.
R1 pAq
Then
“ t0, 1u is not contained in R1 pX q “ t0u, though
1
TC0 pR pAqq “ t0u is.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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R ESONANCE VARIETIES
R ESONANCE VARIETIES OF A SPACE
A rationally defined CDGA pA, dq has positive weights if each Ai
can be decomposed into weighted pieces Aiα , with positive weights
in degree 1, and in a manner compatible with the CDGA structure:
1
2
3
À
Ai “ αPZ Aiα .
A1α “ 0, for all α ď 0.
i`1
If a P Aiα and b P Ajβ , then ab P Ai`j
α`β and d a P Aα .
A space X is said to have positive weights if APL pX q does.
If X is formal, then X has positive weights, but not conversely.
T HEOREM (D IMCA –PAPADIMA 2014, MPPS)
Suppose there is a rationally defined, finite-type CDGA pA, dq with
positive weights, and a q-equivalence between APL pX q and A
preserving Q-structures. Then, for each i ď q,
1
2
Ri pAq is a finite union of rationally defined linear subspaces of
H 1 pAq.
Ri pAq Ď Ri pX q.
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,
R1 pX q “ H 1 pX , Cq “ C2 .
Ź
Model A “ pa, b, cq generated in degree 1, with d a “ d b “ 0
and d c “ a ¨ b.
This is a finite-dimensional model, with positive weights:
wtpaq “ wtpbq “ 1, wtpcq “ 2.
Writing S “ Crx, y s, we get
¨
A‚ b S : ¨ ¨ ¨
/ S3
˛
y 0 0
˝ ´x 0 0 ‚
1 ´x ´y
/ S3
px y 0q
/S.
Hence H1 pA‚ b Sq “ S{px, y q, and so R1 pAq “ t0u.
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 “ π1 pX , x0 q: a finitely generated group.
CharpX q “ Hompπ, C˚ q: an abelian, algebraic group.
CharpX q0 – pC˚ qn , where n “ b1 pX q.
Characteristic varieties of X :
V i pX q “ tρ P CharpX q | H i pX , Cρ q ‰ 0u.
T HEOREM (L IBGOBER 2002, D IMCA –PAPADIMA –S. 2009)
τ1 pV i pX qq Ď TC1 pV i pX qq Ď Ri pX q
Here, if W Ă pC˚ qn is an algebraic subset, then
τ1 pW q :“ tz P Cn | exppλzq P W , for all λ P Cu.
This is a finite union of rationally defined linear subspaces of Cn .
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 –PAPADIMA 2014)
Suppose APL pX q is q-equivalent to a finite-type model pA, dq. Then
V i pX qp1q – Ri pAqp0q , for all i ď q.
C OROLLARY
If X is a q-formal space, then V i pX qp1q – Ri pX qp0q , 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 APL pX q is q-equivalent to a finite-type CDGA A. Then, @i ď q,
1
2
TC1 pV i pX qq “ TC0 pRi pAqq.
If, moreover, A has positive weights, and the q-equivalence
APL pX q » A preserves Q-structures, then TC1 pV i pX qq “ Ri pAq.
T HEOREM (DPS-2009, DP-2014)
Suppose X is a q-formal space. Then, for all i ď q,
τ1 pV i pX qq “ TC1 pV i pX qq “ Ri pX q.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY IN LOCAL SYSTEMS
T HE TANGENT CONE THEOREM
C OROLLARY
If X is q-formal, then, for all i ď q,
1
2
All irreducible components of Ri pX q are rationally defined
subspaces of H 1 pX , Cq.
All irreducible components of V i pX q which pass through the origin
are algebraic subtori of CharpX q0 , of the form exppLq, where L
runs through the linear subspaces comprising Ri pX q.
The Tangent Cone theorem can be used to detect non-formality.
E XAMPLE
Let π “ xx1 , x2 | rx1 , rx1 , x2 ssy.
Then V 1 pπq “ tt1 “ 1u, and so
τ1 pV 1 pπqq “ TC1 pV 1 pπqq “ tx1 “ 0u.
On the other hand, R1 pπq “ C2 , and so π is not 1-formal.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY IN LOCAL SYSTEMS
T HE TANGENT CONE THEOREM
E XAMPLE (DPS 2009)
Let π “ xx1 , . . . , x4 | rx1 , x2 s, rx1 , x4 srx2´2 , x3 s, rx1´1 , x3 srx2 , x4 sy. Then
R1 pπq “ tz P C4 | z12 ´ 2z22 “ 0u: a quadric which splits into two linear
subspaces over R, but is irreducible over Q. Thus, π is not 1-formal.
E XAMPLE (S.–YANG –Z HAO 2015)
Let π be a finitely presented group with πab “ Z3 and
(
V 1 pπq “ pt1 , t2 , t3 q P pC˚ q3 | pt2 ´ 1q “ pt1 ` 1qpt3 ´ 1q ,
This is a complex, 2-dimensional torus passing through the origin, but
this torus does not embed as an algebraic subgroup in pC˚ q3 . Indeed,
τ1 pV 1 pπqq “ tx2 “ x3 “ 0u Y tx1 ´ x3 “ x2 ´ 2x3 “ 0u.
Hence, π is not 1-formal.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY JUMP LOCI IN ALGEBRAIC GEOMETRY
G YSIN MODELS
G YSIN MODELS
Let X be a (connected) smooth quasi-projective variety.
Let X be a “good" compactification, i.e., X “ X zD, for some
normal-crossings divisor D “ tD1 , . . . , Dm u.
Algebraic model: A “ ApX , Dq (Morgan’s Gysin
À model): a
rationally defined, bigraded CDGA, with Ai “ p`q“i Ap,q and
¯
à p´ č
Ap,q “
H
Dk , C p´qq
|S|“q
1
1
k PS
1
1
Multiplication Ap,q ¨ Ap ,q Ď Ap`p ,q`q from cup-product in X .
Differential d : Ap,q Ñ Ap`2,q´1 from intersections of divisors.
Model has positive weights: wtpAp,q q “ p ` 2q.
Improved version by Dupont [2013]: divisor D is allowed to have
“arangement-like" singularities.
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A LGEBRAIC MODELS
A PRIL 10, 2015
15 / 20
C OHOMOLOGY JUMP LOCI IN ALGEBRAIC GEOMETRY
G YSIN MODELS
Suppose X “ Σ is a connected, smooth algebraic curve.
Then Σ admits a canonical compactification, Σ, and thus, a
canonical Gysin model, ApΣq.
E XAMPLE
Let Σ “ E ˚ be a once-punctured elliptic curve. Then Σ “ E, and
ľ
ApΣq “
pa, b, eq{pae, beq
where a, b are in bidegree p1, 0q and e in bidegree p0, 1q, while
d a “ d b “ 0 and d e “ ab.
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A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY JUMP LOCI IN ALGEBRAIC GEOMETRY
T HE TANGENT CONE THEOREM
T HE TANGENT CONE THEOREM
T HEOREM (B UDUR , WANG 2013)
Let X be a smooth quasi-projective variety. Then each characteristic
variety V i pX q is a finite union of torsion-translated subtori of CharpX q.
T HEOREM
Let ApX q be a Gysin model for X . Then, for each i ě 0,
τ1 pV i pX qq “ TC1 pV i pX qq “ Ri pApX qq Ď Ri pX q.
Moreover, if X is q-formal, the last inclusion is an equality, for all i ď q.
E XAMPLE
Let X be the C˚ -bundle over E “ S 1 ˆ S 1 with e “ 1. Then
V 1 pX q “ t1u, and so τ1 pV 1 pX qq “ TC1 pV 1 pX qq “ t0u. On the other
hand, R1 pX q “ C2 , and so X is not 1-formal.
A LEX S UCIU
A LGEBRAIC MODELS
A PRIL 10, 2015
17 / 20
C OHOMOLOGY JUMP LOCI IN ALGEBRAIC GEOMETRY
T HE TANGENT CONE THEOREM
A holomorphic map f : X Ñ Σ is admissible if f is surjective, has
connected generic fiber, and the target Σ is a connected, smooth
complex curve with χpX q ă 0.
T HEOREM (A RAPURA 1997)
The map f ÞÑ f ˚ pCharpΣqq yields a bijection between the set EX of
equivalence classes of admissible maps X Ñ Σ and the set of
positive-dimensional, irreducible components of V 1 pX q containing 1.
T HEOREM (DP 2014, MPPS 2013)
ď
R1 pApX qq “
f ˚ pH 1 pApΣqqq.
f PEX
T HEOREM (DPS 2009)
Ť
Suppose X is 1-formal. Then R1 pX q “ f PEX f ˚ pH 1 pΣ, Cqq. Moreover,
all the linear subspaces in this decomposition have dimension ě 2,
and any two distinct ones intersect only at 0.
A LEX S UCIU
A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY JUMP LOCI
A PPLICATIONS
A PPLICATIONS OF COHOMOLOGY JUMP LOCI
Homological and geometric finiteness of regular abelian covers
Bieri–Neumann–Strebel–Renz invariants
Dwyer–Fried invariants
Obstructions to (quasi-) projectivity
Right-angled Artin groups and Bestvina–Brady groups
3-manifold groups, Kähler groups, and quasi-projective groups
Resonance varieties and representations of Lie algebras
Homological finiteness in the Johnson filtration of automorphism
groups
Homology of finite, regular abelian covers
Homology of the Milnor fiber of an arrangement
Rational homology of smooth, real toric varieties
Lower central series and Chen Lie algebras
The Chen ranks conjecture
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A LGEBRAIC MODELS
A PRIL 10, 2015
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C OHOMOLOGY JUMP LOCI
A PPLICATIONS
R EFERENCES
A.I. Suciu, Around the tangent cone theorem, arxiv:1502.02279.
A. Dimca, S. Papadima, A.I. Suciu, Algebraic models, Alexander-type
invariants, and Green–Lazarsfeld sets, Bull. Math. Soc. Sci. Math.
Roumanie (to appear), arxiv:1407.8027.
A. Măcinic, S. Papadima, R. Popescu, A.I. Suciu, Flat connections and
resonance varieties: from rank one to higher ranks, arxiv:1312.1439.
S. Papadima, A.I. Suciu, Jump loci in the equivariant spectral sequence,
Math. Res. Lett. 21 (2014), no. 4, 863–883.
A.I. Suciu, Y. Yang, G. Zhao, Homological finiteness of abelian covers,
Ann. Sc. Norm. Super. Pisa 14 (2015), no. 1, 101–153.
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