pdf-slides - Geometry and Topology (Russian)

Geometric structures on moment-angle
manifolds
Taras Panov
Moscow State University
based on joint works with with Victor Buchstaber,
Andrey Mironov and Yuri Ustinovsky
Toric Topology and Automorphic Functions.
Khabarovsk, 5-10 Septeber 2011.

Topology of moment-angle manifolds and complexes
(joint with Victor Buchstaber).
A convex polyhedron in R n obtained by intersecting m halfspaces:
Define an affine map
P = { x ∈ R n : ⟨a i , x⟩ + b i 0 for i = 1, . . . , m } .
i P : R n → R m , i P (x) = ( ⟨a 1 , x⟩ + b 1 , . . . , ⟨a m , x⟩ + b m
)
.
If P has a vertex, then i P is monomorphic, and i P (P ) is the intersection
of an n-plane with R m = {y = (y 1, . . . , y m ): y i 0}.
Define the space Z P from the diagram
Z P
⏐
↓
i Z
−→ C m (z 1 , . . . , z m )
⏐
↓ µ
⏐ ⏐⏐↓
P
i P
−→ R m (|z 1 | 2 , . . . , |z m | 2 )
Z P has a T m -action, Z P /T m = P , and i Z is a T m -equivariant inclusion.
2

Proposition 1. If P is a simple polytope (more generally, if the presentation
of P by inequalities is generic), then Z P is a smooth manifold
of dimension m + n.
Proof. Write i P (R n ) by m − n linear equations in (y 1 , . . . , y m ) ∈ R m .
Replace y k by |z k | 2 to obtain a presentation of Z P by quadrics.
Z P : polytopal moment-angle manifold corresponding to P .
Similarly, by considering the projection µ: R m → R m instead of
µ: C m → R m we obtain the real moment-angle manifold R P ⊂ R m .
Example 1. P = {(x 1 , x 2 ) ∈ R 2 : x 1 0, x 2 0, −γ 1 x 1 −γ 2 x 2 +1 0},
γ 1 , γ 2 > 0 (a 2-simplex). Then
Z P = {(z 1 , z 2 , z 3 ) ∈ C 3 : γ 1 |z 1 | 2 + γ 2 |z 2 | 2 + |z 3 | 2 ) = 1} (a 5-sphere),
R P = {(u 1 , u 2 , u 3 ) ∈ R 3 : γ 1 |u 1 | 2 + γ 2 |u 2 | 2 + |u 3 | 2 ) = 1} (a 2-sphere).
3

K an (abstract) simplicial complex on the set [m] = {1, . . . , m}.
I = {i 1 , . . . , i k } ∈ K a simplex. Always assume ∅ ∈ K.
Consider the unit polydisc in C m ,
Given I ⊂ [m], set
D m = { (z 1 , . . . , z m ) ∈ C m : |z i | 1, i = 1, . . . , m } .
B I := { (z 1 , . . . , z m ) ∈ D m : |z j | = 1 for j /∈ I } ∼ =
∏
The moment-angle complex
i∈I
D 2 × ∏ i/∈I
Z K := ∪
B I = ∪ ∏
D
I∈K I∈K( 2 × ∏ )
S 1 ⊂ D m
i∈I i/∈I
It is invariant under the coordinatewise action of the torus T m .
Example 2. K = 2 points, then Z K = D 2 × S 1 ∪ S 1 × D 2 ∼ = S 3 .
K = ∆, then Z K = (D 2 ×D 2 ×S 1 )∪(D 2 ×S 1 ×D 2 )∪(S 1 ×D 2 ×D 2 ) ∼ = S 5 .
S 1 .
4

More generally, let X a space, and A ⊂ X. Given I ⊂ [m], set
(X, A) I = { (x 1 , . . . , x m ) ∈
m∏
i=1
X : x j ∈ A for j /∈ I } ∼ =
∏
The K-polyhedral product of (X, A) is
Z K (X, A) = ∪
(X, A) I ⊂ X m .
I∈K
i∈I
X × ∏ i/∈I
Another important example is the complement of the coordinate subspace
arrangement corresponding to K:
U(K) = C m \
namely,
where C × = C \ {0}.
∪
{i 1 ,...,i k }/∈K
{z ∈ C m : z i1 = . . . = z ik = 0},
U(K) = Z K (C, C × ),
A.
Theorem 1.
Z K ⊂ U(K) is a T m -deformation retract of U(K).
5

Theorem 2. If P is a simple polytope, K P = ∂(P ∗ ) (the dual triangulation),
then Z KP
∼ = ZP (T m -equivariantly homeomorphic).
In particular, Z KP
is a manifold. More generally,
Proposition 2. Assume |K| ∼ = S n−1 (a sphere triangulation with m
vertices). Then Z K is a closed manifold of dimension m + n.
6

The face ring (the Stanley–Reisner ring) of K is
Z[K] = Z[v 1 , . . . , v m ] /( v i1· · · v ik : {i 1 , . . . , i k } /∈ K ) , deg v i = 2.
Theorem 3. There is an isomorphism of (bi)graded algebras
H ∗ (Z K ; Z)
= ∼ Tor ∗,∗ (Z[K], Z)
Z[v 1 ,...,v m ]
=
∼ H [ Λ[u 1 , . . . , u m ] ⊗ Z[K]; d ] ,
where du i = v i , dv i = 0 for 1 i m. In particular,
H p (Z K ) ∼ =
∑
−i+2j=p
Tor −i,2j
Z[v 1 ,...,v m ] (Z[K], Z). 7

Corollary 1. H k (Z K ) ∼ = ⊕
I⊂[m]
˜H k−|I|−1 (K I ),
where K I is the restriction of K to the subset I ⊂ {1, . . . , m}.
If K = K P , then can rewrite the above in terms of P instead of K:
Corollary 2. H k (Z P ) ∼ =
⊕
I⊂[m]
˜H k−|I|−1 (P I ),
where P I is the union of facets F i of P with i ∈ I.
Remark 1. Integral version of Theorem 3 was proved independently
by [Baskakov–Buchstaber–P] and [Franz].
2. The product in H ∗ (Z K ) given by Theorem 3 can be also described
in terms of full subcomplexes K I of Corollary 1 [Baskakov].
3. There is the stable decomposition ΣZ K ≃ ∨ I⊂[m] Σ|I|+2 |K I | behind
the isomorphism of Corollary 1 [Bahri–Bendersky–Cohen–Gitler].
8

Geometric structures I. Lagrangian submanifolds.
(joint with Andrey Mironov).
(M, ω) a symplectic Riemannian 2n-manifold.
An immersion i: N M of an n-manifold N is Lagrangian if i ∗ (ω) = 0.
If i is an embedding, then i(N) is a Lagrangian submanifold of M.
A vector field ξ on M is Hamiltonian if the 1-form ω( · , ξ) is exact.
A Lagrangian immersion i: N M is Hamiltonian minimal
(H-minimal) if the variations of the volume of i(N) along all Hamiltonian
vector fields with compact support are zero, i.e.
d
dt vol(i t(N)) ∣ = 0, t=0
where i 0 (N) = i(N), i t (N) is a Hamiltonian deformation of i(N), and
vol(i t (N)) is the volume of the deformed part of i t (N).
∣
9

Recall: P a simple polytope
P = { x ∈ R n : ⟨a i , x⟩ + b i 0 for i = 1, . . . , m } .
The polytopal moment-angle manifold Z P ,
Z P
⏐
↓
i Z
−→ C m (z 1 , . . . , z m )
⏐
↓ µ
⏐ ⏐⏐↓
P
i P
−→ R m (|z 1 | 2 , . . . , |z m | 2 )
can be written as the intersection of m − n real quadrics,
Z P =
{
z = (z 1 , . . . , z m ) ∈ C m :
m∑
k=1
γ jk |z k | 2 = c j ,
}
for 1 j m − n .
10

Also have the real moment-angle manifold,
{
R P = u = (u 1 , . . . , u m ) ∈ R m m∑
: γ jk u 2 k = c j,
k=1
Set γ k = (γ 1k , . . . , γ m−n,k ) ∈ R m−n for 1 k m.
}
for 1 j m − n .
Assume that the polytope P is rational. Then have two lattices:
Λ = Z⟨a 1 , . . . , a m ⟩ ⊂ R n and L = Z⟨γ 1 , . . . , γ m ⟩ ⊂ R m−n .
Consider the (m − n)-torus
T P = {( e 2πi⟨γ1,φ⟩ , . . . , e 2πi⟨γ m,φ⟩ ) ∈ T m} ,
i.e. T P = R m−n /L ∗ , and set
D P = 1 2 L∗ /L ∗ = ∼ (Z/2) m−n .
Proposition 3. The (m − n)-torus T P acts on Z P almost freely.
11

Consider the map
f : R P × T P −→ C m ,
(u, φ) ↦→ u · φ = (u 1 e 2πi⟨γ1,φ⟩ , . . . , u m e 2πi⟨γm,φ⟩ ).
Note f(R P × T P ) ⊂ Z P is the set of T P -orbits through R P ⊂ C m .
Have an m-dimensional manifold
N P = R P × DP T P .
Lemma 1. f : R P × T P → C m induces an immersion j : N P C m .
Theorem 4 (Mironov). The immersion i Γ : N Γ C m is H-minimal
Lagrangian.
When it is an embedding?
12

A simple rational polytope P is Delzant if for any vertex v ∈ P the
set of vectors a i1 , . . . , a in normal to the facets meeting at v forms a
basis of the lattice Λ = Z⟨a 1 , . . . , a m ⟩:
Z⟨a 1 , . . . , a m ⟩ = Z⟨a i1 , . . . , a in ⟩ for any v = F i1 ∩ · · · ∩ F in .
Theorem 5. The following conditions are equivalent:
1) j : N P → C m is an embedding of an H-minimal Lagrangian submanifold;
2) the (m − n)-torus T P acts on Z P freely.
3) P is a Delzant polytope.
Explicit constructions of families of Delzant polytopes are known in
toric geometry and topology:
- simplices and cubes in all dimensions;
- products and face cuts;
- associahedra (Stasheff ptopes), permutahedra, and generalisations.
13

Example 3 (one quadric). Let P = ∆ m−1 (a simplex), i.e. m − n = 1
and R ∆ m−1 is given by a single quadric
with γ i > 0, i.e. R ∆ m−1 ∼ = S m−1 . Then
N ∼ = S m−1 × Z/2 S 1 ∼ =
⎧
⎨
⎩
γ 1 u 2 1 + · · · + γ mu 2 m = c (1)
S m−1 × S 1 if τ preserves the orient. of S m−1 ,
K m if τ reverses the orient. of S m−1 ,
where τ is the involution and K m is an m-dimensional Klein bottle.
Proposition 4. We obtain an H-minimal Lagrangian embedding of
N ∆ m−1 ∼ = S n−1 × Z/2 S 1 in C m if and only if γ 1 = · · · = γ m in (1). The
topological type of N ∆ m−1 = N(m) depends only on the parity of m:
N(m) = ∼ S m−1 × S 1
N(m) = ∼ K m
if m is even,
if m is odd.
The Klein bottle K m with even m does not admit Lagrangian embeddings
in C m [Nemirovsky, Shevchishin].
14

Example 4 (two quadrics).
Theorem 6. Let m − n = 2, i.e. P ≃ ∆ p−1 × ∆ q−1 .
(a) R P is diffeomorphic to R(p, q) ∼ = S p−1 × S q−1 given by
u 2 1 + . . . + u2 k + u2 k+1 + · · · + u2 p = 1,
u 2 1 + . . . + u2 k +u 2 p+1 + · · · + u2 m = 2,
where p + q = m, 0 < p < m and 0 k p.
(b) If N P → C m is an embedding, then N P is diffeomorphic to
N k (p, q) = R(p, q) × Z/2×Z/2 (S 1 × S 1 ),
where the two involutions act on R(p, q) by
ψ 1 : (u 1 , . . . , u m ) ↦→ (−u 1 , . . . , −u k , −u k+1 , . . . , −u p , u p+1 , . . . , u m ),
ψ 2 : (u 1 , . . . , u m ) ↦→ (−u 1 , . . . , −u k , u k+1 , . . . , u p , −u p+1 , . . . , −u m ).
(2)
There is a fibration N k (p, q) → S q−1 × Z/2 S 1 = N(q) with fibre N(p)
(the manifold from the previous example), which is trivial for k = 0.
15

Example 5 (three quadrics).
In the case m − n = 3 the topology of compact manifolds R P and
Z P was fully described by [Lopez de Medrano]. Each manifold is
diffeomorphic to a product of three spheres, or to a connected sum
of products of spheres, with two spheres in each product.
The simplest P with m − n = 3 is a (Delzant) pentagon, e.g.
P = { (x 1 , x 2 ) ∈ R 2 : x 1 0, x 2 0, −x 1 +2 0, −x 2 +2 0, −x 1 −x 2 +3 0 } .
In this case R P is an oriented surface of genus 5, and Z P is diffeomorphic
to a connected sum of 5 copies of S 3 × S 4 .
Get an H-minimal Lagrangian submanifold N P ⊂ C 5 which is the total
space of a bundle over T 3 with fibre a surface of genus 5.
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Proposition 5. Let P be an m-gon. Then R P is an orientable surface
S g of genus g = 1 + 2 m−3 (m − 4).
Get an H-minimal Lagrangian submanifold N P ⊂ C m which is the
total space of a bundle over T m−2 with fibre S g . It is an aspherical
manifold (for m 4) whose fundamental group enters into the short
exact sequence
1 −→ π 1 (S g ) −→ π 1 (N) −→ Z m−2 −→ 1.
For n > 2 and m − n > 3 the topology of R P and Z P is even more
complicated.
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Geometric structures II. Non-Kähler complex structures.
(joint with Yuri Ustinovskiy).
Recall: if K = K P is the dual triangulation of a simple convex polytope
P , then Z P = Z KP has a canonical smooth structure (e.g. as a
nondegenerate intersection of Hermitian quadrics in C m ).
Let K be a sphere triangulation, i.e. |K| ∼ = S n−1 .
A realisation |K| ⊂ R n is starshaped if there is a point x /∈ |K| such
that any ray from x intersects |K| in exactly one point.
A convex triangulation K P is starshaped, but not vice versa!
K has a starshaped realisation if and only if it is the underlying complexes
of a complete simplicial fan Σ.
Also recall U(K) = C m \ ∪ {i 1 ,...,i k }/∈K {z ∈ Cm : z i1 = . . . = z ik = 0}.
18

Let a 1 , . . . , a m ∈ R n be the generators of the 1-dimensional cones
of Σ. Consider the linear map
Λ R : R m → R n , e i ↦→ a i ,
where e 1 , . . . , e m is the standard basis of R m . Define
R Σ := exp(Ker Λ R ) = { (y 1 , . . . , y m ) ∈ R m > :
m ∏
i=1
y ⟨a i,u⟩
i
= 1 for all u ∈ R n} ,
R Σ ⊂ R m > acts on U(K Σ) ⊂ C m by coordinatewise multiplications.
Theorem 7. Let K be the underlying complex of a complete simplicial
fan Σ. Then
(a) R Σ acts on U(K) freely and properly, so the quotient U(K)/R Σ
has a canonical structure of a smooth (m + n)-manifold;
(b) U(K)/R Σ is T m -equivariantly homeomorphic to Z K .
Therefore, Z K can be smoothed canonically.
19

Assume m − n is even and set l = m−n
2 .
Choose a linear map Ψ: C l → C m satisfying the two conditions:
(a) Re ◦Ψ: C l → R m is a monomorphism.
(b) Λ R ◦ Re ◦Ψ = 0.
Now set
C Ψ,Σ = exp Ψ(C l ) = {( e ⟨ψ 1,w ⟩ , . . . , e ⟨ψ m,w ⟩ ) ∈ (C × ) m} .
Then C Ψ,Σ
∼ = C l is a complex-analytic (but not algebraic) subgroup
of (C × ) m . It acts on U(K) by holomorphic transformations.
Theorem 8. Let K be as before. Then
(a) C Ψ,Σ acts on U(K) freely and properly, so the quotient U(K)/C Ψ,Σ
is a compact complex manifold of complex dimension m − l;
(b) there is a T m -equivariant diffeomorphism U(K)/C Ψ,Σ
∼ = ZK defining
a complex structure on Z K in which T m acts holomorphically.
20

Theorem 9. Let K be the underlying complex of a complete rational
regular simplicial fan. Let k be the number of ghost vertices in K.
Then the Hodge numbers h p,q = h p,q (Z K ) satisfy
(a) ( k−l
p
(b) h 0,q = ( l
q
)
h p,0 ( )
[k/2] for p 0;
(c) h 1,q = (l − k) ( l
q−1
p
)
for q 0;
)
+ h
1,0 ( )
l+1 for q 1;
q
(d) l(3l+1)
2
− h 2 (K) − lk + (l + 1)h 2,0 h 2,1 l(3l+1)
2
− lk + (l + 1)h 2,0 .
At most one ghost vertex is required to make dim Z K = m + n even.
Note that k 1 implies h p,0 (Z K ) = 0, so that Z K does not have
holomorphic forms of any degree in this case.
If m = k = 2l, then Z K is a torus, and h 1,0 (Z K ) = h 0,1 (Z K ) = l.
Otherwise Theorem 9 implies that h 1,0 (Z K ) < h 0,1 (Z K ), and therefore
Z K is not Kähler.
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[1] В. М. Бухштабер, Т. Е. Панов. Действия тора в топологии и
комбинаторике. Москва, изд-во МЦНМО, 2004.
[2] Andrey Mironov and Taras Panov. Intersections of quadrics,
moment-angle manifolds, and Hamiltonian-minimal Lagrangian
embeddings. Preprint (2011); arXiv:1103.4970.
[3] Taras Panov. Moment-angle manifolds and complexes. Trends
in Mathematics - New Series. Information Center for Mathematical
Sciences, KAIST. Vol. 12 (2010), no. 1, pp. 43–69;
arXiv:1008.5047.
[4] Taras Panov and Yuri Ustinovsky. Complex-analytic structures on
moment-angle manifolds. Moscow Math. J. (2011), to appear
(2010); arXiv:1008.4764.
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