Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Iterated residue formula of generalized Bottmanifolds and its applications
Hao LiUniversity of Singapore

Toric Topology 2022 in Osaka
Osaka City University (online), March 25, 2022

Hao Li

Iterated residue formula and application

Iterated residue formula and application

1

Motivation

2

Brief introduction to generalized Bott manifolds

3

Iterated residue formula

4

Applications

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Common objects in geometric (topological) study
Product of projective spaces. e.g. CP n × CP m ;Complete intersections in product of projective spaces. e.g.
Milnor hypersurfaces Hn1 ,n2 Poincaré dual to u + v in
CP n1 × CP n2 ;Homogeneous spaces of compact Lie groups. e.g. structuregroups, Grassmannian manifolds;
Principle G bundle, where G is a compact Lie group. e.g.
U(n) bundles;
Toric varieties which are 1-1 corresponding to rational fans.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Common objects in geometric (topological) study
Product of projective spaces. e.g. CP n × CP m ;Complete intersections in product of projective spaces. e.g.
Milnor hypersurfaces Hn1 ,n2 Poincaré dual to u + v in
CP n1 × CP n2 ;Homogeneous spaces of compact Lie groups. e.g. structuregroups, Grassmannian manifolds;
Principle G bundle, where G is a compact Lie group. e.g.
U(n) bundles;
Toric varieties which are 1-1 corresponding to rational fans.
Feature: They admit clear topology——tangential bundles andcohomology rings.
Furthermore, they also admit nice properties: such as complex
structure, symplectic structure, group actions, Riemannian metricof positive (or non negative) sectional (or Ricci, scalar) curvature...
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

QuestionCan quasi-toric manifolds (moment-angle manifolds) and theircomplete intersections play the roles of above examples?

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

QuestionCan quasi-toric manifolds (moment-angle manifolds) and theircomplete intersections play the roles of above examples?
Quasi-toric manifolds
Let (M 2n , Λ) be a quasi-toric manifold over simple polytope P nwith m facets. Then M 2n admits stable complex structureTM ⊕ Cm−n ∼= ρ1 ⊕ · · · ⊕ ρmwhere each ρi is a complex line bundle corresponding to each facet.
H ∗ (M, Z) ∼= Z[v1 , · · · , vm ]/(I + J)
where vi = c1 (ρi ), idea I comes from missing faces of P n , idea J
comes from characteristic matrix Λ.
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Geometric properties

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Geometric properties

Moment-angle manifolds and toric manifolds admit symplectic
structure;

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Geometric properties

Moment-angle manifolds and toric manifolds admit symplectic
structure;
Moment-angle manifolds and quasi-toric manifolds admit a
Riemannian metric of positive scalar curvature;

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Geometric properties

Moment-angle manifolds and toric manifolds admit symplectic
structure;
Moment-angle manifolds and quasi-toric manifolds admit a
Riemannian metric of positive scalar curvature;
Moment-angle manifolds admit trivialized normal bundle, thus
potentially can serve as representatives in framed bordism
group (stable homotopy groups).

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Index theory and rigidity
Index theory build a bridge between geometry and topology. Usually
the geometric property of quasi-toric manifolds is not obvious, one
may use topological index to detect geometric properties.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Index theory and rigidity
Index theory build a bridge between geometry and topology. Usually
the geometric property of quasi-toric manifolds is not obvious, one
may use topological index to detect geometric properties.
Definition
Let R be an integral domain over Q. Then a genus ϕ is a ringhomomorphism ϕ : ΩSO ⊗ Q −→ R.
Every genus corresponds to a power series in R.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Index theory and rigidity
Index theory build a bridge between geometry and topology. Usually
the geometric property of quasi-toric manifolds is not obvious, one
may use topological index to detect geometric properties.
Definition
Let R be an integral domain over Q. Then a genus ϕ is a ringhomomorphism ϕ : ΩSO ⊗ Q −→ R.
Every genus corresponds to a power series in R.
Common index
b genus, Elliptic genus, Witten
L genus (signature), Todd genus, A
genus, Euler characteristic, Pontryagin numbers...

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Riemannian metric is the core concept in differential geometry.
Geometers are interested in three curvatures—sectional curvature,
Ricci curvature and scalar curvature.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Riemannian metric is the core concept in differential geometry.
Geometers are interested in three curvatures—sectional curvature,
Ricci curvature and scalar curvature.
Stolz theorem (positive scalar curvature)
A simply connected, closed and dim ≥ 5 spin manifold carries a
Riemannian metric of positive scalar curvature if and only if its α
b genus.
invariant vanishes, where α invariant is a refined version of A

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Riemannian metric is the core concept in differential geometry.
Geometers are interested in three curvatures—sectional curvature,
Ricci curvature and scalar curvature.
Stolz theorem (positive scalar curvature)
A simply connected, closed and dim ≥ 5 spin manifold carries a
Riemannian metric of positive scalar curvature if and only if its α
b genus.
invariant vanishes, where α invariant is a refined version of A
Stolz conjecture (positive Ricci curvature)
If a compact string manifold M 4k carries a Riemannian metric ofpositive Ricci curvature, then the Witten genus ϕW (M) vanishes.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Riemannian metric is the core concept in differential geometry.
Geometers are interested in three curvatures—sectional curvature,
Ricci curvature and scalar curvature.
Stolz theorem (positive scalar curvature)
A simply connected, closed and dim ≥ 5 spin manifold carries a
Riemannian metric of positive scalar curvature if and only if its α
b genus.
invariant vanishes, where α invariant is a refined version of A
Stolz conjecture (positive Ricci curvature)
If a compact string manifold M 4k carries a Riemannian metric ofpositive Ricci curvature, then the Witten genus ϕW (M) vanishes.
Geometers are looking for simply connected manifolds that admits
a metric of positive scalar curvature but no metric of positive Ricci
curvature.
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Rigidity of group action

Atiyah-Hirzebruch
If a compact spin manifold admits non-trivial compact connectedb genus vanishes.
Lie group (e.g. S 1 ) action, then its A

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Rigidity of group action

Atiyah-Hirzebruch
If a compact spin manifold admits non-trivial compact connectedb genus vanishes.
Lie group (e.g. S 1 ) action, then its A
Kefeng Liu
If a compact string manifold admits non-trivial a semi-simple Liegroup (e.g. S 3 ) action, then its Witten genus vanishes.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Rigidity of group action

Atiyah-Hirzebruch
If a compact spin manifold admits non-trivial compact connectedb genus vanishes.
Lie group (e.g. S 1 ) action, then its A
Kefeng Liu
If a compact string manifold admits non-trivial a semi-simple Liegroup (e.g. S 3 ) action, then its Witten genus vanishes.
Topologists are interested in manifolds that admits non-trivial S 1
action but no non-trivial S 3 action.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Interesting problems
Since the kernel of ring homomorphism ϕW : ΩString −→ MF
precisely consists of ( bordism classes of ) OP 2 (Cayley plane)
bundles with connected structure group.
Problem (Wilderich Tuschmann)
Can every string cobordism class with vanishing Witten genus berepresented by some manifolds with positive Ricci curvature?

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Interesting problems
Since the kernel of ring homomorphism ϕW : ΩString −→ MF
precisely consists of ( bordism classes of ) OP 2 (Cayley plane)
bundles with connected structure group.
Problem (Wilderich Tuschmann)
Can every string cobordism class with vanishing Witten genus berepresented by some manifolds with positive Ricci curvature?
Hirzebruch prize question
Does there exist a 24-dimensional compact, differential stringb ) = 1 and A(Xb , TC ) = 0?
manifolds X with A(XSuch manifold admits action of Monster by diffeomorphism.
Mohowald-Hopkins constructed a 24-dimensional manifold fromhomotopy theory, but it is hard to know its geometry.
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Quasi-toric manifolds can also be chosen as representatives of ncobordism and equivariant cobordism. Such as ΩU , ΩSU , ΩU,T
2n .

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Quasi-toric manifolds can also be chosen as representatives of ncobordism and equivariant cobordism. Such as ΩU , ΩSU , ΩU,T2n .

In summary, one can study the geometric and topologic propertiesof quasi-toric manifolds by calculating all kinds of index andcharacteristic numbers.
In our work, we will focus on the calculation process and give an
explicit formula of genus of generalized Bott manifolds.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Generalised Bott manifolds

Definition
A generalised Bott tower of height n is a tower of projective
bundles
pn

pn−1

p2

Bn −→ Bn−1 −→ · · · −→ B2 −→ B1 −→ ptof complex manifolds, where B1 = CP n1 and each Bk is thecomplex projectivisation of a sum of nk complex line bundles andone trivial line bundle over Bk−1 .
The last stage Bn in a generalised Bott tower is called generalised
Bott manifolds.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Tangential bundle
TBk+1 ⊕ C ∼= p ∗ (TBk ) ⊕ (η̄k ⊗ p ∗ (ξk ⊕ C)).

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Tangential bundle
TBk+1 ⊕ C ∼= p ∗ (TBk ) ⊕ (η̄k ⊗ p ∗ (ξk ⊕ C)).
Suppose the formal Chern roots of ξk are {xk1 , xk2 , · · · , xknk }, let−uk be the first Chern class of tautological line bundle ηk over
Bk = CP(ξk ⊕ C).
Cohomology ringH ∗ (Bn ) ∼= Z[u1 , · · · , un ]/hfi (u1 , · · · , un ) : i = 1, · · · , ni
Q i
where fi (u1 , · · · , un ) = ui · nj=1
(ui + xij ).

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Let X be the submanifold of Bn Poincaré dual to x ∈ H 2 (Bn ; Z).
Suppose ν denote the normal bundle of inclusion i : X ,→ Bn . Thenc1 (ν) = i ∗ (x), i ∗ (TBn ) ∼= TX ⊕ ν. Thenc(X ) · c(ν) = i ∗ (c(Bn )); p(X ) · p(ν) = i ∗ (p(Bn ))Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Let X be the submanifold of Bn Poincaré dual to x ∈ H 2 (Bn ; Z).
Suppose ν denote the normal bundle of inclusion i : X ,→ Bn . Thenc1 (ν) = i ∗ (x), i ∗ (TBn ) ∼= TX ⊕ ν. Thenc(X ) · c(ν) = i ∗ (c(Bn )); p(X ) · p(ν) = i ∗ (p(Bn ))Complete intersection
Let x1 , · · · , xr ∈ H 2 (Bn ; Z), The transversal intersection of
X1 , · · · , Xr is called complete intersections of codimension 2r .

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

For any genus ψ with the charateristic power series Q(x) = x/f (x),we haveψ(X ) = h(

n

n

i=1

j=1

i
Y ui Yui + xij
u1 n1 +1)
· f (x) ·, [Bn ]i.
f (u1 )f (ui )f (ui + xij )Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

For any genus ψ with the charateristic power series Q(x) = x/f (x),we haveψ(X ) = h(

n

n

i=1

j=1

i
Y ui Yui + xij
u1 n1 +1)
· f (x) ·, [Bn ]i.
f (u1 )f (ui )f (ui + xij )The calculation of genus ψ is transferred into the calculation ofdegree dim Bn items in complex analysis.
If Bn is direct products of projective spaces, ψ(X ) can be simplified
into a very neat expression. As to general Bn , the idea fi ’s inH ∗ (Bn ; Z) is the biggest obstacle in calculating genus, the variables
ui ’s are dependent, we have to cope with the relation carefully.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

iterated residue formula
To overcome this obstacle, we introduce the concept "iterated
residue".

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

iterated residue formula
To overcome this obstacle, we introduce the concept "iterated
residue".
Theorem+···+nn (B ; Z), we have
For any F ∈ H n1nh F , [Bn ] i = Res0 Res0 · · ·F
Res0 Qni
du1f (u ,··· ,un )j =1 i 1

Hao Li· · · dun−1 dun

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

iterated residue formula
To overcome this obstacle, we introduce the concept "iterated
residue".
Theorem+···+nn (B ; Z), we have
For any F ∈ H n1nh F , [Bn ] i = Res0 Res0 · · ·F
Res0 Qni
du1f (u ,··· ,un )j =1 i 1· · · dun−1 dun

RemarkIt should be pointed out that the order of ui ’s in the "iterated
residue" can’t be exchanged, otherwise we will get the result zero.
This illustrates why global residue theorem fails.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Main theorem
Let X be the submanifold of generalised Bott manifold Bn Poincaré
dual to x ∈ H 2 (Bn ; Z).
For any genus ψ with the charateristic power series Q(x) = x/f (x),we have n n
o
o
f (x)

ψ(X ) = Res0 · · · Res0 f (u )n1 +1 ·
1

1 Qni
1
i =1 f (ui )j=1 f (ui +xij ) du1

Qn

Hao Li···dun .

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Main theorem
Let X be the submanifold of generalised Bott manifold Bn Poincaré
dual to x ∈ H 2 (Bn ; Z).
For any genus ψ with the charateristic power series Q(x) = x/f (x),we have n n
o
o
f (x)

ψ(X ) = Res0 · · · Res0 f (u )n1 +1 ·
1

1 Qni
1
i =1 f (ui )j=1 f (ui +xij ) du1

Qn

···

dun .

Formula kills the product of Chern roots with only f left. If power
series f admits good property, it is possible to give explicit result of
ψ(X ).

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Main theorem
Let X be the submanifold of generalised Bott manifold Bn Poincaré
dual to x ∈ H 2 (Bn ; Z).
For any genus ψ with the charateristic power series Q(x) = x/f (x),we have n n
o
o
f (x)

ψ(X ) = Res0 · · · Res0 f (u )n1 +1 ·
1

1 Qni
1
i =1 f (ui )j=1 f (ui +xij ) du1

Qn

···

dun .

Formula kills the product of Chern roots with only f left. If power
series f admits good property, it is possible to give explicit result of
ψ(X ).
RemarkIt is a pity that above result can not be generalized to quasi-toric
manifolds. For arbitrary quasi-toric manifolds, the product of its
Chern roots can not kill the ideas in H ∗ (M; Z), thus ψ(X ) contains
both function f and monomials in ui .
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Reformulation of α invariant
With the help of "iterated residue", we can simplify thecomputation of α invariant significantly.
Fn1 ,n2 ,I (d1 , d2 )


n2
I

I
e (d1 u+d2 v )/2 Y
1 2
1
1
n +1
du dv
)
) 1
·
=(
(

2πi
e v /2 − e −v /2 j =1 e (v −ij u)/2 − e −(v −ij u)/2
γ1  γ2 e u/2 − e −u/2

=(

1
2πi

)

2

=(

=(

1
2πi
1
2πi
1
2πi

(n1 +1+d1 +

(

γ1 

 γ2

let t = e

=(




I

I

)

2

u

2

(1 + s)

I

2

I
γ1

v

)

n1 +1

·

− 1, k1 =


k2 I

s

n1 − 1 + d1 +(t + 1)k1s γt n1 +1(1 + s)
n2 +k2 I2

s n2 +1


 γ2

P

2

t n1 +1

2ij )u+(−n2 +1+d2 )v2n2Yev − 1(t + 1)k1

 γ

P

e


k2 I

(1 + s)Iγ1)eu − 1− 1, s = e

γ1

)

1

n2

·

·

ij

j =1 1 − e, k2 =ij
j =1 1 − (1 + t) /(1 + s)n2
Y(t + 1)k1·

dt(1 + s)/s

j =1 1 − {(1 + t)

t n1 +1−n2 − 1 + d2

1

Y

1

ij
− 1}/s2

ij u−v

du






dv





.

ds


dt




ds






n2 n1
Y
X (1 + t)ij − 1 l() j dt ds

s

j =1
lj =0

=······
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Witten genus of string complete intersections
Proposition
Twisted Milnor hypersurfaces HnI 1 ,n2 (d1 , d2 ) can’t be string forn2 ≥ 3.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Witten genus of string complete intersections
Proposition
Twisted Milnor hypersurfaces HnI 1 ,n2 (d1 , d2 ) can’t be string forn2 ≥ 3.
Theoremb genus of string complete intersections in generalised Bott
The A
manifold vanish.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Witten genus of string complete intersections
Proposition
Twisted Milnor hypersurfaces HnI 1 ,n2 (d1 , d2 ) can’t be string forn2 ≥ 3.
Theoremb genus of string complete intersections in generalised Bott
The A
manifold vanish.
Thus we shall seek the string complete intersections
HnI 1 ,n2 (d1 , d2 ; d3 , d4 ) Poincaré dual to(d1 u + d2 v ) · (d3 u + d4 v ) ∈ H 4 (B2 ; Z).

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Witten genus
Consider Jocabi theta function
θ(z, τ ) = 2q 1/8 sin(πz)∞
Y[(1 − q j )(1 − e 2πiz q j )(1 − e −2πiz q j )]

j=1

Let
f (x) = (e x/2 − e −x/2 ) ·∞Y(1 − q n e x )(1 − q n e −x )n=1

(1 − q n )2,θ(z)= f (2πiz), then Witten genus can be reformulated asθ 0 (0)ϕW (HnI 1 ,n2 ) = h(n2v − ij u f (d1 u + d2 v ) f (d3 u + d4 v )u n1 +1 v Y), [HnI 1 ,n2 ]if (u)f (v ) j=1 f (v − ij u) d1 u + d2 v
d3 u + d4 v

n2v − ij u
u n1 +1 v Y)
f (d1 u + d2 v ) · f (d3 u + d4 v ), [V ]if (u)f (v ) j=1 f (v − ij u)()
f (d1 u + d2 v ) · f (d3 u + d4 v )Q 2= Res0 Res0 n +1
du dv
f 1 (u)f (v ) nj=1
f (v − ij u)= h(

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Vanishing result
For string complete intersection HnI 1 , n2 (d1 , d2 ; d3 , d4 ), if
I = (i1 , · · · , in2 ) are relative prime,ϕW (HnI 1 , n2 (d1 , d2 ; d3 , d4 )) = 0.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Vanishing result
For string complete intersection HnI 1 , n2 (d1 , d2 ; d3 , d4 ), if
I = (i1 , · · · , in2 ) are relative prime,ϕW (HnI 1 , n2 (d1 , d2 ; d3 , d4 )) = 0.
It is more valuable to find examples with non vanishing Witten
genus.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Toric forms in toric varieties
Toric variety is a very hot area in both algebraic geometry andgeometric topology. Borisov and Libgober first gave the explicit
formula of elliptic genus of a toric variety by its combinatorial data.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Toric forms in toric varieties
Toric variety is a very hot area in both algebraic geometry andgeometric topology. Borisov and Libgober first gave the explicit
formula of elliptic genus of a toric variety by its combinatorial data.
Let N ∈ Rr be a lattice, M be its dual lattice. For a complete
rational polyhedral fan Σ ⊂ N ⊗ R. A degree function deg : N 7→ C
is a piecewise linear function that is linear on the cones of Σ.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Toric forms in toric varieties
Toric variety is a very hot area in both algebraic geometry andgeometric topology. Borisov and Libgober first gave the explicit
formula of elliptic genus of a toric variety by its combinatorial data.
Let N ∈ Rr be a lattice, M be its dual lattice. For a complete
rational polyhedral fan Σ ⊂ N ⊗ R. A degree function deg : N 7→ C
is a piecewise linear function that is linear on the cones of Σ.
The toric form associated to (N, deg) is the function
fN,deg : H −→ C defined by
X X
X
fN,deg (q) =((−1)codim C a.c.(q m·n e 2πi deg(n) ))m∈M C ∈Σn∈C

here q = e 2πiτ , τ ∈ H, the upper halfplane, a.c. denotes analytic
continuation.
Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

ExampleLet N = Z2 , and Σ be the fan corresponding toric variety is
projective plane CP 2 . Assume that deg takes α, β, γ on thegenerators e1 , e2 , −e1 − e2 .

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

ExampleLet N = Z2 , and Σ be the fan corresponding toric variety is
projective plane CP 2 . Assume that deg takes α, β, γ on thegenerators e1 , e2 , −e1 − e2 .
Then the toric form is
fN,deg (q) =

+

=

1

X
a,b∈Z

(1 − e 2πi α q a )(1 − e 2πi β q b )1(1 − e 2πi α q a )(1 − e 2πi γ q −a−b )

X
a,b∈Z

+

1
(1 − e 2πi β q b )(1 − e 2πi γ q −a−b )−1
1 − e 2πi α q a−

1 − e 2πi (α+β+γ)(1 − e 2πi α q a )(1 − e 2πi β q b )(1 − e 2πi γ q −a−b )Hao Li1
1 − e 2πi β q b−

1
1 − e 2πi γ q −a−b.

Iterated residue formula and application+1

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Borisov, Gunnells gave toric form a topological interpretation byHirzebruch-Riemann-Roch theorem.
Borisov, Gunnells
Assume that the toric variety X is nonsingular, and that αi ∈/ Z forall primitive generator of any 1-cone of Σ. ThenZ Y
(Di /2πi)θ(Di /2πi − αi )θ0 (0)fN,deg (q) =θ(Di /2πi)θ(−αi )Xiwhere Di denote the cohomology class of 1-cone of fan Σ.
Our "iterated residue" can be applied to calculate above Euler
characteristic under certain conditions, thus give numerical
identities regarding to toric forms.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Theorem
Consider toric variety V = CP(η ⊗i1 ⊕ · · · η ⊗in2 ⊕ C) over CP n1 ,n1where η denote the tautological bundle
Pn2 of CP .
If (i1 , · · · , in2 ) are relatively prime, j=0 αn1 +2+j ∈ Z,
Pn1 +1Pn2i=1 αi −j=1 ij αn1 +2+j ∈ Z , then fN,deg (q) = 0.

Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Theorem
Consider toric variety V = CP(η ⊗i1 ⊕ · · · η ⊗in2 ⊕ C) over CP n1 ,n1where η denote the tautological bundle
Pn2 of CP .
If (i1 , · · · , in2 ) are relatively prime, j=0 αn1 +2+j ∈ Z,
Pn1 +1Pn2i=1 αi −j=1 ij αn1 +2+j ∈ Z , then fN,deg (q) = 0.
We take Hirzebruch surface Fk = CP(C ⊕ O(k)) over CP 1 forexample to illustrate our result. Its corresponding fan Σ consists offour two-dimensional cones generated by the pairs of vectors(e1 , e2 ), (e1 , −e2 ), (−e1 + ke2 , −e2 ), (−e1 + ke2 , e2 ),Hao Li

Iterated residue formula and application

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Theorem
Consider toric variety V = CP(η ⊗i1 ⊕ · · · η ⊗in2 ⊕ C) over CP n1 ,n1where η denote the tautological bundle
Pn2 of CP .
If (i1 , · · · , in2 ) are relatively prime, j=0 αn1 +2+j ∈ Z,
Pn1 +1Pn2i=1 αi −j=1 ij αn1 +2+j ∈ Z , then fN,deg (q) = 0.
We take Hirzebruch surface Fk = CP(C ⊕ O(k)) over CP 1 forexample to illustrate our result. Its corresponding fan Σ consists offour two-dimensional cones generated by the pairs of vectors(e1 , e2 ), (e1 , −e2 ), (−e1 + ke2 , −e2 ), (−e1 + ke2 , e2 ),Proposition
If α2 + α4 ∈ Z, α1 + α3 − kα4 ∈ Z, τ ∈ H, q = e 2πiτ ,fN,deg (q) =

X
a,b∈Z

(1 − e 2πi (α1 +α3 ) )(1 − q ka )(1 − e 2πi α1 q a )(1 − e 2πi α3 q −a )(1 − e −2πi α4 q b )(1 − e 2πi α4 q ka−b )Hao Li

Iterated residue formula and application= 0.

Motivation
Brief introduction to generalized Bott manifolds
Iterated residue formula
Applications

Thanks for listening!
Email: 14110840001@fudan.edu.cn

Hao Li

Iterated residue formula and application