Alex Suciu Northeastern University Online Algebraic Geometry - - PowerPoint PPT Presentation

POINCARÉ DUALITY AND COHOMOLOGY

JUMP LOCI

Alex Suciu

Northeastern University

Online Algebraic Geometry Seminar Humboldt University Berlin July 15, 2020

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 1 / 33

RESONANCE VARIETIES RESONANCE VARIETIES

RESONANCE VARIETIES

Let A = (A•, dA) be a connected, locally finite, graded-commutative graded differential graded algebra (cdga)

• ver a field k, and let M = (M•, dM) be an A-cdgm.

Since A0 = k, we have Z 1(A) ∼ = H1(A). Set Q(A) = {a ∈ Z 1(A) | a2 = 0 ∈ A2}. For each a ∈ Q(A), we then have a cochain complex, (M•, δa): M0

δ0

a

M1

δ1

a

M2

δ2

a

· · · ,

with differentials δi

a(m) = a · m + dM(m), for all m ∈ Mi.

The resonance varieties of M (in degree i ≥ 0 and depth k ≥ 0): Ri

k(M) = {a ∈ Q(A) | dimk Hi(M•, δa) ≥ k}.

TC0(Ri

k(M)) ⊆ Ri k(H•(M)), but not = in general.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 2 / 33

RESONANCE VARIETIES RESONANCE VARIETIES OF GRADED ALGEBRAS

RESONANCE VARIETIES OF GRADED ALGEBRAS

Now let A be a graded, graded-commutative, connected, locally finite k-algebra (char k = 2). For each a ∈ A1 we have a2 = −a2, and so a2 = 0. We then

• btain a cochain complex of k-vector spaces,

(A, δa): A0

δ0

a

A1

δ1

a

A2

δ2

a

· · · ,

with differentials δi

a(u) = a · u, for all u ∈ Ai.

The resonance varieties of A are the affine varieties Ri

k(A) = {a ∈ A1 | dimk Hi(A, δa) ≥ k}.

An element a ∈ A1 belongs to Ri

k(A) if and only if there exist

u1, . . . , uk ∈ Ai such that au1 = · · · = auk = 0 in Ai+1, and the set {au, u1, . . . , uk} is linearly independent in Ai, for all u ∈ Ai−1.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 3 / 33

RESONANCE VARIETIES RESONANCE VARIETIES OF GRADED ALGEBRAS

Set bj = bj(A). For each i ≥ 0, we have a descending filtration, A1 = Ri

0(A) ⊇ Ri 1(A) ⊇ · · · ⊇ Ri bi(A) = {0} ⊃ Ri bi+1(A) = ∅.

A linear subspace U ⊂ A1 is isotropic if the restriction of A1 ∧ A1

·

− → A2 to U ∧ U is the zero map (i.e., ab = 0, ∀a, b ∈ U). If U ⊆ A1 is an isotropic subspace of dimension k, then U ⊆ R1

k−1(A).

R1

1(A) is the union of all isotropic planes in A1.

If k ⊂ K is a field extension, then the k-points on Ri

k(A ⊗k K)

coincide with Ri

k(A).

Let ϕ: A → B be a morphism of graded, connected algebras. If the map ϕ1 : A1 → B1 is injective, then ϕ1(R1

k(A)) ⊆ R1 k(B), ∀k.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 4 / 33

RESONANCE VARIETIES THE BGG CORRESPONDENCE

THE BGG CORRESPONDENCE

Fix a k-basis {e1, . . . , en} for A1, and let {x1, . . . , xn} be the dual basis for A1 = (A1)∗. Identify Sym(A1) with S = k[x1, . . . , xn], the coordinate ring of the affine space A1. The BGG correspondence yields a cochain complex of finitely generated, free S-modules, L(A) := (A• ⊗ S, δ), · · ·

Ai ⊗ S

δi

A Ai+1 ⊗ S

δi+1

A

Ai+2 ⊗ S · · · ,

where δi

A(u ⊗ s) = ∑n j=1 eju ⊗ sxj.

The specialization of (A ⊗ S, δ) at a ∈ A1 coincides with (A, δa), that is, δi

A

• xj=aj = δi

a.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 5 / 33

RESONANCE VARIETIES THE BGG CORRESPONDENCE

By definition, an element a ∈ A1 belongs to Ri

k(A) if and only if

rank δi−1

a

+ rank δi

a ≤ bi(A) − k.

Let Ir(ψ) denote the ideal of r × r minors of a p × q matrix ψ with entries in S, where I0(ψ) = S and Ir(ψ) = 0 if r > min(p, q). Then: Ri

k(A) = V

• Ibi(A)−k+1
• δi−1

A

⊕ δi

A

• =
• s+t=bi(A)−k+1
• V
• Is(δi−1

A

) ∪ V

• It(δi

A)

• .

In particular, R1

k(A) = V(In−k(δ1 A)) (0 ≤ k < n) and R1 n(A) = {0}.

The (degree i, depth k) resonance scheme Ri

k(A) is defined by

the ideal Ibi(A)−k+1

• δi−1

A

⊕ δi

A

• ; its underlying set is Ri

k(A).

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 6 / 33

POINCARÉ DUALITY ALGEBRAS POINCARÉ DUALITY ALGEBRAS

POINCARÉ DUALITY ALGEBRAS

Let A be a connected, locally finite k-cga. A is a Poincaré duality k-algebra of dimension m if there is a k-linear map ε: Am → k (called an orientation) such that all the bilinear forms Ai ⊗k Am−i → k, a ⊗ b → ε(ab) are non-singular. That is, A is a graded, graded-commutative Gorenstein Artin algebra of socle degree m. If A is a PDm algebra, then:

bi(A) = bm−i(A), and Ai = 0 for i > m. ε is an isomorphism. The maps PD: Ai → (Am−i)∗, PD(a)(b) = ε(ab) are isomorphisms.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 7 / 33

POINCARÉ DUALITY ALGEBRAS POINCARÉ DUALITY ALGEBRAS

Each a ∈ Ai has a Poincaré dual, a∨ ∈ Am−i, such that ε(aa∨) = 1. The orientation class is ωA := 1∨. We have ε(ωA) = 1, and thus aa∨ = ωA. The class of k-PD algebras is closed under taking tensor products and connected sums:

If A is PDm and B is PDn, then A ⊗k B is PDm+n. If A and B are PDm, then A#B is PDm, where (ω)

ω→ωA ω → ωB

A

• B

A#B

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 8 / 33

POINCARÉ DUALITY ALGEBRAS THE ASSOCIATED ALTERNATING FORM

THE ASSOCIATED ALTERNATING FORM

Associated to a k-PDm algebra there is an alternating m-form, µA : mA1 → k, µA(a1 ∧ · · · ∧ am) = ε(a1 · · · am). Assume now that m = 3, and set n = b1(A). Fix a basis {e1, . . . , en} for A1, and let {e∨

1 , . . . , e∨ n } be the dual basis for A2.

The multiplication in A, then, is given on basis elements by eiej =

r

∑

k=1

µijk e∨

k ,

eie∨

j = δijω,

where µijk = µ(ei ∧ ej ∧ ek). Let Ai = (Ai)∗. We may view µ dually as a trivector, µ = ∑ µijk ei ∧ ej ∧ ek ∈ 3A1, which encodes the algebra structure of A. For instance, µA#B = µA + µB.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 9 / 33

POINCARÉ DUALITY ALGEBRAS CLASSIFICATION OF ALTERNATING FORMS

CLASSIFICATION OF ALTERNATING FORMS

(Following J. Schouten, G. Gurevich, D. Djokovi´ c, A. Cohen–A. Helminck, . . . )

Let V be a k-vector space of dimension n. The group GL(V) acts

• n m(V ∗) by (g · µ)(a1 ∧ · · · ∧ am) = µ
• g−1a1 ∧ · · · ∧ g−1am
• .

The orbits of this action are the equivalence classes of alternating m-forms on V. (We write µ ∼ µ′ if µ′ = g · µ.) Over k, the closures of these orbits are affine algebraic varieties. There are finitely many orbits over k only if n2 ≥ ( n

m), that is,

m ≤ 2 or m = 3 and n ≤ 8. For k = C, each complex orbit has only finitely many real forms. When m = 3, and n = 8, there are 23 complex orbits, which split into either 1, 2, or 3 real orbits, for a total of 35 real orbits.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 10 / 33

POINCARÉ DUALITY ALGEBRAS CLASSIFICATION OF ALTERNATING FORMS

Let A and B be two PDm algebras. We say that a morphism of graded algebras ϕ: A → B has non-zero degree if the linear map ϕm : Am → Bm is non-zero. (Equivalently, ϕ is injective.) A and B are isomorphic as PDm algebras if and only if they are isomorphic as graded algebras, in which case µA ∼ µB. PROPOSITION For two PD3 algebras A and B, the following are equivalent.

1

A ∼ = B, as PD3 algebras.

2

A ∼ = B, as graded algebras.

3

µA ∼ µB. We thus have a bijection between isomorphism classes of 3-dimensional Poincaré duality algebras and equivalence classes

• f alternating 3-forms, given by A µA.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 11 / 33

POINCARÉ DUALITY ALGEBRAS CLASSIFICATION OF ALTERNATING FORMS

POINCARÉ DUALITY IN ORIENTABLE MANIFOLDS

If M is a compact, connected, orientable, m-dimensional manifold, then the cohomology ring A = H.(M, k) is a PDm algebra over k. Sullivan (1975): for every finite-dimensional Q-vector space V and every alternating 3-form µ ∈ 3V ∗, there is a closed 3-manifold M with H1(M, Q) = V and cup-product form µM = µ. Such a 3-manifold can be constructed via “Borromean surgery." E.g., 0-surgery on the Borromean rings in S3 yields M = T 3, with µM = e1e2e3. If M is the link of an isolated surface singularity (e.g., if M = Σ(p, q, r) is a Brieskorn manifold), then µM = 0.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 12 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS

RESONANCE VARIETIES OF PD-ALGEBRAS

Let A be a PDm algebra. For 0 ≤ i ≤ m and a ∈ A1, the square (Am−i)∗

(δm−i−1

−a

)∗

(Am−i−1)∗

Ai

δi

a

• PD ∼

=

• Ai+1

PD ∼ =

• commutes up to a sign.

Consequently,

• Hi(A, δa)

∗ ∼ = Hm−i(A, δ−a). Hence, for all i and k, Ri

k(A) = Rm−i k

(A). In particular, Rm

1 (A) = R0 k(A) = {0}.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 13 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS RESONANCE VARIETIES OF PD3 ALGEBRAS

RESONANCE VARIETIES OF PD3 ALGEBRAS

COROLLARY Let A be a PD3 algebra with b1(A) = n. Then Ri

k(A) = ∅, except for:

1

Ri

0(A) = A1 for all i ≥ 0.

2

R3

1(A) = R0 1(A) = {0} and R2 n(A) = R1 n(A) = {0}.

3

R2

k(A) = R1 k(A) for 0 < k < n.

THEOREM Every PD3 algebra A decomposes as A ∼ = B#C, where B are C are PD3 algebras such that µB is irreducible and has the same rank as µA, and µC = 0. Furthermore, A1 ∼ = B1 ⊕ C1 restricts to isomorphisms R1

k(A) ∼

= R1

k−r+1(B) × C1 ∪ R1 k−r(B) × {0}

(∀k ≥ 0), where r = corank µA. In particular, R1

k(A) = A1 for all k < corank µA.

(The rank of µ: 3 V → k is the minimum dimension of a linear subspace W ⊂ V such that µ factors through 3 W.)

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 14 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS NULLITY AND RESONANCE

A linear subspace U ⊂ V is 2-singular with respect to a 3-form µ: 3V → k if µ(a ∧ b ∧ c) = 0 for all a, b ∈ U and c ∈ V. If dim U = 2, we simply say U is a singular plane. The nullity of µ, denoted null(µ), is the maximum dimension of a 2-singular subspace U ⊂ V. Clearly, V contains a singular plane if and only if null(µ) ≥ 2. Let A be a PD3 algebra. A linear subspace U ⊂ A1 is 2-singular (with respect to µA) if and only if U is isotropic. Using a result of A. Sikora [2005], we obtain: THEOREM Let A be a PD3 algebra over an algebraically closed field k with char(k) = 2, and let ν = null(µA). If b1(A) ≥ 4, then dim R1

ν−1(A) ≥ ν ≥ 2.

In particular, dim R1

1(A) ≥ ν.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 15 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS REAL FORMS AND RESONANCE

REAL FORMS AND RESONANCE

Sikora made the following conjecture: If µ: 3V → k is a 3-form with dim V ≥ 4 and if char(k) = 2, then null(µ) ≥ 2. Conjecture holds if n := dim V is even or equal to 5, or if k = k. Work of J. Draisma and R. Shaw [2010, 2014] implies that the conjecture does not hold for k = R and n = 7. We obtain: THEOREM Let A be a PD3 algebra over R. Then R1

1(A) = {0}, except when

n = 1, µA = 0. n = 3, µA = e1e2e3. n = 7, µA = −e1e3e5 + e1e4e6 + e2e3e6 + e2e4e5 + e1e2e7 + e3e4e7 + e5e6e7. Sketch: If R1

1(A) = {0}, then the formula (x × y) · z = µA(x, y, z)

defines a cross-product on A1 = Rn, and thus a division algebra structure on Rn+1, forcing n = 1, 3 or 7 by Bott–Milnor/Kervaire [1958].

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 16 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS REAL FORMS AND RESONANCE

EXAMPLE Let A be the real PD3 algebra corresponding to octonionic multiplication (the case n = 7 above). Let A′ be the real PD3 algebra with µA′ = e1e2e3 + e4e5e6 + e1e4e7 + e2e5e7 + e3e6e7. Then µA ∼ µA′ over C, and so A ⊗R C ∼ = A′ ⊗R C. On the other hand, A ∼ = A′ over R, since µA ∼ µA′ over R, but also because R1

1(A) = {0}, yet R1 1(A′) = {0}.

Both R1

1(A ⊗R C) and R1 1(A′ ⊗R C) are projectively smooth

conics, and thus are projectively equivalent over C, but R1

1(A ⊗R C) = {x ∈ C7 | x2 1 + · · · + x2 7 = 0}

has only one real point (x = 0), whereas R1

1(A′ ⊗R C) = {x ∈ C7 | x1x4 + x2x5 + x3x6 = x2 7}

contains the real (isotropic) subspace {x4 = x5 = x6 = x7 = 0}.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 17 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS PFAFFIANS AND RESONANCE

PFAFFIANS AND RESONANCE

For a k-PD3 algebra A, the complex L(A) = (A ⊗k S, δA) looks like A0 ⊗k S

δ0

A A1 ⊗k S

δ1

A A2 ⊗k S

δ2

A A3 ⊗k S ,

where δ0

A =

• x1 · · · xn
• and δ2

A = (δ0 A)⊤, while δ1 A is the skew-

symmetric matrix whose are entries linear forms in S given by δ1

A(ei) = ∑ n j=1 ∑ n k=1 µjike∨ k ⊗ xj .

Recall that R1

k(A) = V(In−k(δ1 A)). Using work of Buchsbaum and

Eisenbud [1977] on Pfaffians of skew-symmetric matrices, we get: THEOREM R1

2k(A) = R1 2k+1(A) = V(Pfn−2k(δ1 A)),

if n is even, R1

2k−1(A) = R1 2k(A) = V(Pfn−2k+1(δ1 A)),

if n is odd. Hence, A1 = R1

0 = R1 1 ⊇ R1 2 = R1 3 ⊇ R1 4 = · · · if b1(A) is even,

and A1 = R1

0 ⊇ R1 1 = R1 2 ⊇ R1 3 = R1 4 ⊇ · · · if b1(A) is odd.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 18 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS BOTTOM-DEPTH RESONANCE

BOTTOM-DEPTH RESONANCE

THEOREM Let A be a PD3 algebra. If µA has maximal rank n ≥ 3, then R1

n−2(A) = R1 n−1(A) = R1 n(A) = {0}.

Otherwise, write A = B # C, where µB is irreducible and µC = 0. If n = dim A1 is at least 3, then R1

n−2(A) = R1 n−1(A) = C1.

LEMMA (TURAEV 2002) Suppose n ≥ 3. There is then a polynomial Det(µA) ∈ Sym(A1) such that, if δ1

A(i; j) is the sub-matrix obtained from δ1 A by deleting the i-th

row and j-th column, then det δ1

A(i; j) = (−1)i+jxixj Det(µA).

Moreover, if n is even, then Det(µA) = 0, while if n is odd, then Det(µA) = Pf(µA)2, where pf(δ1

A(i; i)) = (−1)i+1xi Pf(µA).

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 19 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS TOP-DEPTH RESONANCE

TOP-DEPTH RESONANCE

Suppose dimk V = 2g + 1 > 1. We say µ: 3V → k is generic (in the sense of Berceanu–Papadima [1994]) if there is a v ∈ V such that the 2-form γv ∈ V ∗ ∧ V ∗ given by γv(a ∧ b) = µA(a ∧ b ∧ v) for a, b ∈ V has rank 2g, that is, γg

v = 0 in 2gV ∗.

THEOREM Let A be a PD3 algebra with b1(A) = n. Then R1

1(A) =

         ∅ if n = 0; {0} if n = 1 or n = 3 and µ has rank 3; V(Pf(µA)) if n is odd, n > 3, and µA is BP-generic; A1

• therwise.

Example: Let M = Σg × S1, where g ≥ 2. Then µM = ∑g

i=1 aibic is

BP-generic, and Pf(µM) = xg−1

2g+1. Hence, R1 1(M) = {x2g+1 = 0}. In fact,

R1

1 = · · · = R1 2g−2 and R1 2g−1 = R1 2g = R1 2g+1 = {0}.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 20 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS TOP-DEPTH RESONANCE

As a corollary, we recover a closely related result, proved by Draisma and Shaw [2010] by very different methods. COROLLARY Let V be a k-vector space of odd dimension n ≥ 5 and let µ ∈ 3 V ∗. Then the union of all singular planes is either all of V or a hypersurface defined by a homogeneous polynomial in k[V] of degree (n − 3)/2. For µ ∈ 3 V ∗, there is another genericity condition, due to P . De Poi,

• D. Faenzi, E. Mezzetti, and K. Ranestad [2017]: rank(γv) > 2, for all

non-zero v ∈ V. We may interpret some of their results, as follows. THEOREM (DFMR) Let A be a PD3 algebra over C, and suppose µA is generic. Then: If n is odd, then R1

1(A) is a hypersurface of degree (n − 3)/2

which is smooth if n ≤ 7, and singular in codimension 5 if n ≥ 9. If n is even, then R1

2(A) has codim 3 and degree 1 4(n−2 3 ) + 1; it is

smooth if n ≤ 10, and singular in codimension 7 if n ≥ 12.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 21 / 33

RESONANCE VARIETIES OF PD-ALGEBRAS TOP-DEPTH RESONANCE

RESONANCE VARIETIES OF 3-FORMS OF LOW RANK

C µ R1 R2 R3 I ∅ ∅ ∅ II 123 III 125 + 345 {x5 = 0} {x5 = 0} C R µ R1 R2 = R3 R4 IV 135 + 234 + 126 k6 {x1 = x2 = x3 = 0} V a 123 + 456 k6 {x1 = x2 = x3 = 0} ∪ {x4 = x5 = x6 = 0} b −135 + 146 + 236 + 245 k6 V(x2

1 + x2 2 , x2 3 + x2 4 , x2 5 + x2 6 , x4x5 − x3x6, x3x5 + x4x6,

x2x5 − x1x6, x1x5 + x2x6, x2x3 − x1x4, x1x3 + x2x4)

C R µ R1 = R2 R3 = R4 R5 VI 123 + 145 + 167 {x1 = 0} {x1 = 0} VII 125 + 136 + 147 + 234 {x1 = 0} {x1 = x2 = x3 = x4 = 0} VIII a 134 + 256 + 127 {x1 = 0} ∪ {x2 = 0} {x1 = x2 = x3 = x4 = 0} ∪ {x1 = x2 = x5 = x6 = 0} b −135 + 146 + 236 + 245 + 127 {x2

1 + x2 2 = 0}

V(x1, x2, x2

3 + x2 4 , x2 5 +

x2

6 , x3x5 + x4x6, x4x5 − x3x6)

IX a 125 + 346 + 137 + 247 {x1x4 + x2x5 = 0} V(x2

7 − x3x6, x1, x2, x4, x5)

b −135 + 146 + 236 + 245 + 127 + 347 {x1x3 + x2x4 = 0} V(x2

7 − x5x6, x1, x2, x3, x4)

X a 123 + 456 + 147 + 257 + 367 {x1x4 + x2x5 + x3x6 = x2

7 }

b −135 + 146 + 236 + 245 + 127 + 347 + 567 {x2

1 + x2 2 + x2 3 + x2 4 + x2 5 + x2 6 + x2 7 = 0}

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 22 / 33

CHARACTERISTIC VARIETIES CHARACTERISTIC VARIETIES OF SPACES

CHARACTERISTIC VARIETIES OF SPACES

Let X be a connected, finite-type CW-complex. Then G = π1(X, x0) is a finitely presented group, with Gab ∼ = H1(X, Z). The ring R = C[Gab] is the coordinate ring of the character group, Char(X) = Hom(G, C∗) ∼ = (C∗)n × Tors(Gab), where n = b1(X). The characteristic varieties of X are the homology jump loci Vi

k(X) = {ρ ∈ Char(X) | dim Hi(X, Cρ) ≥ k}.

These varieties are homotopy-type invariants of X, with V1

k (X)

depending only on G = π1(X). Set V1(G) := V1

1(K(G, 1)); then V1(G = V1(G/G′′).

Let f ∈ Z[t±1

1 , . . . , t±1 n ], f(1) = 0. There is then a finitely

presented group G with Gab = Zn such that V1(G) = V(f).

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 23 / 33

CHARACTERISTIC VARIETIES TANGENT CONES

TANGENT CONES

Let exp: H1(X, C) → H1(X, C∗) be the coefficient homomorphism induced by C → C∗, z → ez. Let W = V(I), a Zariski closed subset of Char(G) = H1(X, C∗). The tangent cone at 1 to W is TC1(W) = V(in(I)). The exponential tangent cone at 1 to W: τ1(W) = {z ∈ H1(X, C) | exp(λz) ∈ W, ∀λ ∈ C}. Both tangent cones are homogeneous subvarieties of H1(X, C); are non-empty iff 1 ∈ W; depend only on the analytic germ of W at 1; commute with finite unions and arbitrary intersections. τ1(W) ⊆ TC1(W), with = if all irred components of W are subtori, but = in general. τ1(W) is a finite union of rationally defined subspaces.

ALEX SUCIU (NORTHEASTERN) POINCARÉ DUALITY AND JUMP LOCI HU BERLIN, JULY 15, 2020 24 / 33

CHARACTERISTIC VARIETIES ALGEBRAIC MODELS FOR SPACES

ALGEBRAIC MODELS FOR SPACES

A cdga map ϕ: A → B is a quasi-isomorphism if ϕ∗ : H.(A) → H.(B) is an isomorphism. ϕ is a q-quasi-isomorphism (for some q ≥ 1) if ϕ∗ is an isomorphism in degrees ≤ q and is injective in degree q + 1. Two cdgas, A and B, are (q-) equivalent if there is a zig-zag of (q-) quasi-isomorphisms connecting A to B. A is formal (or just q-formal) if it is (q-) equiv. to (H•(A), d = 0).

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CHARACTERISTIC VARIETIES ALGEBRAIC MODELS FOR SPACES

Given any (path-connected) space X, there is an associated Sullivan Q-cdga, APL(X), such that H•(APL(X)) = H•(X, Q). An algebraic (q-)model (over k) for X is a k-cgda (A, d) which is (q-) equivalent to APL(X) ⊗Q k. If X is q-formal, then X admits a q-finite q-model, but the converse is not true. If M is a smooth manifold, then ΩdR(M) 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, polyhedral products ZK (S1, ∗), etc. Smooth quasi-projective varieties, compact solvmanifolds, Sasakian manifolds, etc.

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CHARACTERISTIC VARIETIES THE TANGENT CONE THEOREM

THE TANGENT CONE THEOREM

Let X be a connected CW-complex with finite q-skeleton. Suppose X admits a q-finite q-model A. THEOREM For all i ≤ q and all k ≥ 0: (DPS 2009, Dimca–Papadima 2014) Vi

k(X)(1) ∼

= Ri

k(A)(0).

In particular, if X is q-formal, then Vi

k(X)(1) ∼

= Ri

k(X)(0).

(Budur–Wang 2017) All the irreducible components of Vi

k(X)

passing through the origin of Char(X) are algebraic subtori. Consequently, τ1(Vi

k(X)) = TC1(Vi k(X)) = Ri k(A).

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CHARACTERISTIC VARIETIES THE TANGENT CONE THEOREM

THEOREM (QUILLEN, SULLIVAN) A finitely generated group G is 1-formal if and only if its Malcev Lie algebra, m(G) := Prim( Q[G]), is the LCS completion of a quadratic Lie algebra. THEOREM (PAPADIMA–S. 2019) A finitely generated group G admits a 1-finite 1-model if and only if m(G) is the LCS completion of a finitely presented Lie algebra. COROLLARY Suppose G is a finitely generated group whose Malcev Lie algebra is the LCS completion of a finitely presented Lie algebra. Then τ1(V1

k (G)) = TC1(V1 k (G)), for all k ≥ 0.

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CHARACTERISTIC VARIETIES OF 3-MANIFOLDS ALEXANDER POLYNOMIALS

ALEXANDER POLYNOMIALS OF 3-MANIFOLDS

Let H = H1(X, Z)/Tors. Let X H → X be the maximal torsion-free abelian cover of X, with cellular chain complex C•(X H, ∂H). The Alexander polynomial ∆X ∈ Z[H] is the gcd of the codimension 1 minors of the Alexander matrix ∂H

1 .

PROPOSITION Let λ be a Laurent polynomial in n ≤ 3 variables such that ¯ λ . = λ and λ(1) = 0. Then λ can be realized as the Alexander polynomial ∆M of a closed, orientable 3-manifold M with b1(M) = n. Set W1

1(M) = V1 1(M) ∩ Char0(M). Using work of McMullen [2002] and

Turaev [2002], we find: PROPOSITION Let M be a closed, orientable, 3-dimensional manifold. Then W1

1(M) = V(∆M) ∪ {1}. If, moreover, b1(M) ≥ 4, then ∆M(1) = 0,

and so W1

1(M) = V(∆M).

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CHARACTERISTIC VARIETIES OF 3-MANIFOLDS A TANGENT CONE THEOREM FOR 3-MANIFOLDS

A TANGENT CONE THEOREM FOR 3-MANIFOLDS

Let M be a closed, orientable, 3-manifold, and set n = b1(M). THEOREM

1

If either n ≤ 1, or n is odd, n ≥ 3, and µM is BP-generic, then TC1(V1

1(M)) = R1 1(M).

2

If n is even, n ≥ 2, then R1(M) = H1(M, C). Moreover, TC1(V1

1(M)) = R1 1(M) ⇐

⇒ ∆M = 0. REMARK In case (2), the equality R1(M) = H1(M, C) was first proved in [Dimca–S, 2009], where it was used to show that the only 3-manifold groups which are also Kähler groups are the finite subgroups of O(4).

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CHARACTERISTIC VARIETIES OF 3-MANIFOLDS A TANGENT CONE THEOREM FOR 3-MANIFOLDS

THEOREM

1

If n ≤ 1, then M is formal, and has the rational homotopy type of S3 or S1 × S2.

2

If n is even, n ≥ 2, and ∆M = 0, then M is not 1-formal.

3

If ∆M = 0, yet ∆M(1) = 0 and TC1(V(∆M)) is not a finite union of Q-linear subspaces, then M admits no 1-finite 1-model. EXAMPLE Let M = S1 × S2#S1 × S2; then ∆M = 0, and so TC1(V1

1(M)) = R1 1(M) = C2. In fact, M is formal.

EXAMPLE Let M be the Heisenberg 3-d nilmanifold; then ∆M = 1 and µM = 0, and so TC1(V1

1(M)) = {0}, whereas R1 1(M) = C2.

M admits a finite model, namely, A = (a, b, c) with d a = d b = 0 and d c = ab, but M is not 1-formal.

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CHARACTERISTIC VARIETIES OF 3-MANIFOLDS A TANGENT CONE THEOREM FOR 3-MANIFOLDS

EXAMPLE Let M be a 3-manifold with ∆M = (t1 + t2)(t1t2 + 1) − 4t1t2. Then {0} = τ1(V1

1(M)) TC1(V1 1(M)) = {x2 1 + x2 2 = 0}.

The latter variety decomposes as the union of two lines defined over C, but not over Q. Hence, M admits no 1-finite 1-model. The 3d Tangent Cone theorem does not hold in higher depth. EXAMPLE Let M be a 3-manifold with b1(M) = 10 and intersection 3-form µM = e1e2e5 + e1e3e6 + e2e3e7 + e1e4e8 + e2e4e9 + e3e4e10. R1

7(M) ∼

= {z ∈ C6 | z1z6 − z2z5 + z3z4 = 0}, an irreducible quadric with an isolated singular point at 0. V1

k (M) ⊆ {1}, for all k ≥ 1.

Thus, TC1(V1

7(M)) = R1 7(M), and so M is not 1-formal.

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REFERENCES

REFERENCES

A.I. Suciu, Poincaré duality and resonance varieties

• Proc. Roy. Soc. Edinburgh Sect. A. (to appear)

doi:10.1017/prm.2019.55, arXiv:1809.01801 A.I. Suciu, Cohomology jump loci of 3-manifolds arXiv:1901.01419

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