on the kosniowski conjecture - math - the university of utahxbli/lu.pdf · 2018. 8. 16. ·...

§1 Kosniowski conjecture§2 Different statements of Kosniowski conjectue§3 Support evidence for Kosniowski conjecture

§4 Some progresses

On the Kosniowski conjecture

Zhi Lü

School of Mathematical SciencesFudan University, Shanghai

2018 Workshop on Algebraic and Geometric TopologySouthwest Jiaotong University

July 28–31, 2018

Zhi Lü On the Kosniowski conjecture


§4 Some progresses

Outline

Kosniowski conjecture

Different statements of Kosniowski conjectue

Support evidence for Kosniowski conjecture

Some progresses



§4 Some progresses

§1 Kosniowski conjecture

Kosniowski conjecture, 1980

Let S1 y M fix isolated points and preserve the unitary structure,where M is a unitary closed manifold.

If M is not a boundary, then the number |MS1 | of isolated points isgreater than f (dimM) where f is some linear function.

Kosniowski conjectured that the most likely function is f (x) = x4 .

So, generally Kosniowski conjecture is stated as follows:

|MS1 | ≥[dimM

4

]+ 1.

Remark. Kosniowski showed that the conjecture is true for|MS1 | ≤ 2. We shall see that dimM is even.



§4 Some progresses

Unitary manifold

Definition

A smooth manifold M is said to be unitary if its stable tangentbundle admits a complex structure, i.e., there is a bundle map

J : TM ⊕ Rl −→ TM ⊕ Rl

satisfying J2 = −id .

In particular, if l = 0, then M is called analmost complex manifold.

Examples:CPnToric varietiesQuasi-toric manifolds



§4 Some progresses

Unitary manifold

Definition

A smooth manifold M is said to be unitary if its stable tangentbundle admits a complex structure, i.e., there is a bundle map

J : TM ⊕ Rl −→ TM ⊕ Rl

satisfying J2 = −id . In particular, if l = 0, then M is called analmost complex manifold.

Examples:CPnToric varietiesQuasi-toric manifolds



§4 Some progresses

§2 Different statements of Kosniowski conjectue

It is well-known

Theorem (e.g., see Bredon’s book or Allday-Puppe’s book)

Let S1 y M be effective smooth action with MS1 6= ∅, where Mis a smooth closed manifold. Then

χ(M) = χ(MS1).

Note: when the fixed point set MS1

just consists of some isolatedpoints,

χ(M) = χ(MS1) = |MS1 | 6= 0

so dimM must be even.



§4 Some progresses

The first restatement

Rewrite

The first restatement of Kosniowski conjecture


If M is not a boundary, then

χ(M) ≥[dimM

4

]+ 1 or 4χ(M) ≥ dimM + 2.

This maybe allow us to modify to the more general case:

The modified conjecture

Let S1 y M preserve the unitary structure with MS1 6= ∅, whereM is a unitary closed manifold.

If M is not a boundary and χ(M) > 0, then 4χ(M) ≥ dimM + 2.



§4 Some progresses

The first restatement

Rewrite

The first restatement of Kosniowski conjecture


If M is not a boundary, then

χ(M) ≥[dimM

4

]+ 1 or 4χ(M) ≥ dimM + 2.

This maybe allow us to modify to the more general case:

The modified conjecture

Let S1 y M preserve the unitary structure with MS1 6= ∅, whereM is a unitary closed manifold.

If M is not a boundary and χ(M) > 0, then 4χ(M) ≥ dimM + 2.Zhi Lü On the Kosniowski conjecture


§4 Some progresses

Case:Almost complex

Lemma

Let S1 y M2n fix isolated points and preserve the almost complexstructure, where M is an almost complex closed manifold.

Then M must not be a boundary.

Proof. This is because 〈cn, [M]〉 = ±χ(M) 6= 0.

In this case, Kosniowski conjecture can be stated as follows:

The Kosniowski conjecture in the setting of almost complexmanifolds

Let S1 y M fix isolated points and preserve the almost complexstructure, where M is an almost complex closed manifold.

Then

χ(M) ≥[dimM

4

]+ 1.



§4 Some progresses

Case:Almost complex

Lemma

Let S1 y M2n fix isolated points and preserve the almost complexstructure, where M is an almost complex closed manifold.

Then M must not be a boundary.

Proof. This is because 〈cn, [M]〉 = ±χ(M) 6= 0.In this case, Kosniowski conjecture can be stated as follows:

The Kosniowski conjecture in the setting of almost complexmanifolds

Let S1 y M fix isolated points and preserve the almost complexstructure, where M is an almost complex closed manifold.

Then

χ(M) ≥[dimM

4

]+ 1.



§4 Some progresses

The second restatement (in terms of analysis)

Let W = (wij) be an m× (n + 1) matrix, where all wij are nonzerointegers and wi ,n+1 = ±1. Write εi = wi ,n+1. Define a complexfunction over variable z

TWx ,y (z) :=m∑i=1

εi∏j

xzwij + y

zwij − 1

where x , y ∈ C are chosen.

The second restatement for Kosniowski conjecture in termsof analysis (posed by Lü-Musin)

Assume that TWx ,y (z) 6= 0 is rigid, i.e., the function TWx ,y (z) doesnot depend upon the choice of z .

Thenm ≥

[n2

]+ 1.



§4 Some progresses


Theorem (Bredon)

Let T k y M be an effective smooth action on an orientedconnected smooth closed manifold. Then there exists a circlesubgroup S < T k such that MT

k= MS .

NOTE: An action of a circle S1 on a manifold may not beextended to an action of T k(k ≥ 2).By Bredon Theorem above, Kosniowski conjecture holds for

toric varieties V 2n (with an action of algebraic torus(C∗)n w T n fixing isolated points)quasitoric manifolds M2n with an action of T n fixing isolatedpoints.



§4 Some progresses


Theorem (Bredon)


k= MS .

NOTE: An action of a circle S1 on a manifold may not beextended to an action of T k(k ≥ 2).

By Bredon Theorem above, Kosniowski conjecture holds for




§4 Some progresses


Theorem (Bredon)


k= MS .

NOTE: An action of a circle S1 on a manifold may not beextended to an action of T k(k ≥ 2).By Bredon Theorem above, Kosniowski conjecture holds for




§4 Some progressesMain tools

Some progresses in the setting of symplecticmanifolds

In some special restricted conditions

Pelayo and Tolman [Ergodic Theory and Dynamical Systems31 (2011), 1237–1247]

Theorem (Pelayo and Tolman)

Let the circle act symplectically on a compact symplectic manifoldM. If the Chern class map is somewhere injective, then the circleaction has at least dimM2 + 1 fixed points.

NOTE: dimM2 + 1 >[

dimM4

]+ 1.




Some progresses in the setting of almost complexmanifolds

Ping Li and Kefeng Liu [Mathematical Research Letters 18(2011), 437–446]

Theorem (Li-Liu)

If M2mn is an almost complex manifold and there exists apartition λ = (λ1, . . . , λr ) of weight m such that the correspondingChern number 〈(cλ1 . . . cλr )n, [M]〉 is nonzero, then forany S1-action on M, it must have at least n + 1 fixed points.

NOTE: If m ≤ 2, then n + 1 ≥[

dimM4

]+ 1 =

[mn2

]+ 1.




Some progresses in the setting of almost complexmanifolds

In 2017, Donghoon Jang showed in [Journal of Geometry andPhysics, 119 (2017), 187-192]

Theorem (Jang)

Let M is an almost complex manifold admitting an action of acircle S1.If the action only fixes a point, then M is a point.If the action exactly fixes two isolated points, then dimM = 2 or 6.If the action exactly fixes three isolated points, then dimM = 4.

Corollary

In the setting of almost complex manifolds, Kosniowski conjectureholds for dimension ≤ 14.




Some progresses in the setting of unitary manifolds

Lü and Tan showed in [Mathematical Research Letters 18(2011), 1319–1327]

Theorem (Lü-Tan)

Let M2n be a unitary manifold with an action of T n. Then Mbounds equivariantly if and only if the equivariant Chern numbers

〈(cT n1 )i (cTn

2 )j , [M]〉 = 0

for all i , j ∈ N, where [M] is the fundamental class of M withrespect to the given orientation.





As an application, Kosniowski conjecture holds for the case of

unitary manifolds M2n with T n-actions

Theorem (Lü-Tan)

Let T n y M2n be an action on a unitary 2n-manifold. If M is nota boundary, then

χ(M) ≥ dn2e+ 1

where dn2e denotes the minimal integer no less thann2 .

Later on, Jun Ma and Shiyun Wen show that

Theorem (Ma-Wen)

Let T n−1 y M2n be an action on a unitary 2n-manifold. If M isnot a boundary, then χ(M) ≥ dn2e+ 1.





Assume that M2n is a unitary closed manifold with an action of S1

fixing isolated points.

Using Equivariant Atiyah–Hirzebruch genus, Lü-Musin reprovedthat

Lemma

If n is odd, then χ(M) = |MS1 | is even.

Theorem (Kosniowski)

If S1 y M is not a boundary with χ(M) = |MS1 | = 2, then n = 1or 3.

Corollary

Kosniowski conjecture holds for n ≤ 7 with n 6= 6.





Assume that M2n is a unitary closed manifold with an action of S1

fixing isolated points.Using Equivariant Atiyah–Hirzebruch genus, Lü-Musin reprovedthat

Lemma

If n is odd, then χ(M) = |MS1 | is even.

Theorem (Kosniowski)

If S1 y M is not a boundary with χ(M) = |MS1 | = 2, then n = 1or 3.

Corollary

Kosniowski conjecture holds for n ≤ 7 with n 6= 6.




Existence of solution: A conjecture

Leta = {a1, ..., a2l} ⊂ Z+b = {b1, ..., b2l} ⊂ Z+c = {c1, ..., c2l} ⊂ Z+

Then1∏2l

i=1 ai+

1∏2li=1 bi

=1∏2l

i=1 ci

and for any z ∈ C∏2li=1(z

ai + 1)∏2li=1(z

ai − 1)+

∏2li=1(z

bi + 1)∏2li=1(z

bi − 1)=

∏2li=1(z

ci + 1)∏2li=1(z

ci − 1)+ 1

if and only if l = 1 and a or b = {k + h, k} or {k + h, h},c = {k , h}




Main tools

Theory of transformation groups

Atiyah–Bott–Berline–Vergne localization formula

ABBV formula

Let M2n be a unitary manifold with an action of T k fixing isolatedpoints. Then

〈f (x1, ..., xn), [M]〉 =∑

p∈MTk

f |pχT k (p)

∈ H∗(BT k) = Z[t1, ..., tk ]

where f (x1, ..., xn) ∈ H∗T k (M), and f |p is the image off (x1, ..., xn) for H

∗T k

(M) −→ HT k (p) induced by inclusion{p} ↪→ M.




Main tools

Equivariant Atiyah–Hirzebruch genus

hG : ΩG∗ ⊗ C −→ K (BG )⊗ CWhen G = S1, K (BS1)⊗ C = C[[u]]. Let M2n be a unitarymanifold with an action of S1 fixing m isolated pointspi (i = 1, ...,m) with weights wi1, ...,win and sign εi = 1 or −1.Then

hS1

(M) =m∑i=1

εi∏j

H(wij u)

wiju

where H(u) = u(xeu(x+y)+y)

eu(x+y)−1 . Set z := e(x+y)u. Then

hS1

(M) =m∑i=1

εi∏j

xzwij + y

zwij − 1.

(Possible) use of GKM theory




Thank You!


§1 Kosniowski conjecture§2 Different statements of Kosniowski conjectue§3 Support evidence for Kosniowski conjecture§4 Some progressesMain tools

on the kosniowski conjecture - math - the university of utahxbli/lu.pdf · 2018. 8. 16. ·...

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