On the Laplace-Beltrami operator on compact complex spaces › ... › 83 › 90 ›...

On the Laplace-Beltrami operator on compactcomplex spaces

Francesco Bei

Index Theory and Singular StructuresMathematics Institute of Toulouse, 29 May–02 June 2017

June 1, 2017

Francesco Bei On the Laplace-Beltrami operator on compact complex spaces

OUTLINECheeger-Goresky-MacPherson’s conjecture andMacPherson’s conjecture.

Setting and main problem.

The Laplace-Beltrami operator on compact complexspaces.

Hodge-Kodaira Laplacian on possibly singular surfaces.

Cheeger-Goresky-MacPherson’s Conjecture, 1982

Let V ⊂ CPn be a projective variety and let g be the Kahlermetric induced on reg(V ), the regular part of V , by theFubini-Study metric of CPn.

Then:dk ,max = dk ,min

Hk2 (reg(V ),g) ∼= ImHk (V ,R)

and the previous isomorphism induces a Hodge decompositionof ImHk (V ,R) in terms of L2-Dolbeault-cohomology groups.

This conjecture is still largely open.

Let V ⊂ CPn be a projective variety and let g be the Kahlermetric induced on reg(V ), the regular part of V , by theFubini-Study metric of CPn.Then:

dk ,max = dk ,min

MacPherson’s conjecture, 1983

Conjecture

Let V ⊂ CPn be a complex projective variety, π : V −→ V aresolution of V and let g be the Kahler metric on reg(V )induced by the Fubini-Study metric.Then:

χ2(reg(V ),g) = χ(OV )

whereχ2(reg(V ),g)) =

∑(−1)q dim(H0,q

2,∂(reg(V ),g))

χ(V ) =∑

(−1)q dim(H0,q∂

Solved by Pardon and Stern in 1991 proving a stronger result:

H0,q2,∂min

(reg(V ),g) ∼= H0,q∂

(V ), q = 0, ..., v

Conjecture

∑(−1)q dim(H0,q

2,∂(reg(V ),g))

χ(V ) =∑

(−1)q dim(H0,q∂

H0,q2,∂min

(reg(V ),g) ∼= H0,q∂

(V ), q = 0, ..., v

Conjecture

∑(−1)q dim(H0,q

2,∂(reg(V ),g))

χ(V ) =∑

(−1)q dim(H0,q∂

H0,q2,∂min

(reg(V ),g) ∼= H0,q∂

(V ), q = 0, ..., v

Related problems

Existence of self-adjoint extensions of ∆k and ∆∂,p,q withdiscrete sprectrum

Estimates for the growth of the eigenvalues

Properties of the heat operator: trace class, estimates forthe trace etc

Index formulas

Related problems

Index formulas

Related problems

Index formulas

Related problems

Index formulas

Related problems

Index formulas

The setting

(X ,h) irreducible Hermitian complex space.

h is a Hermitian metric on reg(X ), such that for each p ∈ X∃ U ⊂ X open, p ∈ U, ψ : U → BN ⊂ CN proper holomorphicembedding, β Hermitian metric on BN such that on U ∩ reg(X )

h = ψ∗β.

Examples:(M,g) complex Hermitian manifold, X ⊂ M analyticsub-variety and h := g|reg(X)

V ⊂ CPn complex projective variety endowed with themetric induced by Fubini-Study metric.

Consider now a compact and irreducible Hermitian complexspace (X ,h) of complex dimension m.

The setting

h = ψ∗β.

The setting

h = ψ∗β.

The setting

h = ψ∗β.

The problem

We are interested in the following operators

∆ : L2(reg(X ),h)→ L2(reg(X ),h) (0.1)

∆∂ : L2(reg(X ),h)→ L2(reg(X ),h) (0.2)

with domain given by C∞c (reg(X )).

∆ := Laplace-Beltrami operator

∆∂ = ∂t ∂ := Hodge-Kodaira Laplacian (acting on functions)

Questions

Existence of self-adjoint extensions for ∆ and ∆∂ withdiscrete sprectrum

The problem

∆ : L2(reg(X ),h)→ L2(reg(X ),h) (0.1)

∆∂ : L2(reg(X ),h)→ L2(reg(X ),h) (0.2)

Questions

The problem

∆ : L2(reg(X ),h)→ L2(reg(X ),h) (0.1)

∆∂ : L2(reg(X ),h)→ L2(reg(X ),h) (0.2)

Questions

First result

The first result toward these questions is the following:

TheoremLet (X ,h) be a compact and irreducible Hermitian complexspace of complex dimension v. Then we have the followingproperties:

D(dmax) = D(dmin) ( L2-Stokes theorem on functions)Assume that v > 1. We have a continuous inclusion

D(dmax) → L2v

v−1 (reg(X ),h).

Assume that v > 1. We have a compact inclusion

D(dmax) → L2(reg(X ),h).

First result

D(dmax) → L2v

v−1 (reg(X ),h).

First result

D(dmax) → L2v

v−1 (reg(X ),h).

Sketch of the proof

D(dmax) = D(dmin)

This follows as a consequence of the following properties:

(reg(X ),h) is parabolic, ∃ φn ⊂ Lipc(reg(X ),h) such that0 ≤ φn ≤ 1, φn → 1, ‖dφn‖L2Ω1(reg(X),h) → 0.

Density result: L∞(reg(X ),h) ∩ C∞(reg(X )) ∩ D(dmax) isdense in D(dmax)

D(dmax) → L2v

v−1 (reg(X ),h)

This follows by the existence of an open cover V1, ...,Vmsuch that

Local Sobolev embedding holds on (reg(Vi),h|reg(Vi ))

D(dmax) → L2v

v−1 (reg(Vi),h|reg(Vi ))

Sketch of the proof

D(dmax) = D(dmin)

D(dmax) → L2v

v−1 (reg(X ),h)

D(dmax) → L2v

Sketch of the proof

D(dmax) = D(dmin)

D(dmax) → L2v

v−1 (reg(X ),h)

D(dmax) → L2v

Sketch of the proof

D(dmax) = D(dmin)

D(dmax) → L2v

v−1 (reg(X ),h)

D(dmax) → L2v

The LSE follows by using a local quasi-isometric Kahler modelof h|reg(Vi ) and the Michael-Simon’s Sobolev inequality:(∫

mm−1 dvolg

)m−1m

≤ C(∫

M|H||f |dvolg +

∫M|df |gdvolg

Continuous partition of unity γ1, ..., γm subordinated toV1, ...,Vm such that

γi |reg(Vi ) is smooth and ‖d(γi |reg(Vi ))‖L∞Ω1(reg(X),h) <∞

D(dmax) → L2(reg(X ),h)

This follows using the following properties:

D(dmax) → L2v

v−1 (reg(X ),h) (Sobolev embedding)

Volh(reg(X )) <∞ (finite volume)

mm−1 dvolg

)m−1m

≤ C(∫

M|H||f |dvolg +

∫M|df |gdvolg

D(dmax) → L2v

Volh(reg(X )) <∞ (finite volume)

mm−1 dvolg

)m−1m

≤ C(∫

M|H||f |dvolg +

∫M|df |gdvolg

D(dmax) → L2v

Volh(reg(X )) <∞ (finite volume)Francesco Bei On the Laplace-Beltrami operator on compact complex spaces

Application to the Laplace-Beltrami operator

TheoremThe Friedrich extension

∆F : L2(reg(X ),h)→ L2(reg(X ),h) (0.3)

of the Laplace-Beltrami operator has discrete spectrum.

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

be the eigenvalues of (0.3). Then

lim infλkk−1v > 0 (0.4)

as k →∞. Finally e−t∆F : L2(reg(X ),h)→ L2(reg(X ),h) istrace class and for some C > 0 and 0 < t ≤ 1 its trace satisfies

Tr(e−t∆F ) ≤ Ct−v (0.5)

∆F : L2(reg(X ),h)→ L2(reg(X ),h) (0.3)

of the Laplace-Beltrami operator has discrete spectrum. Let

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

as k →∞.

Finally e−t∆F : L2(reg(X ),h)→ L2(reg(X ),h) istrace class and for some C > 0 and 0 < t ≤ 1 its trace satisfies

Tr(e−t∆F ) ≤ Ct−v (0.5)

∆F : L2(reg(X ),h)→ L2(reg(X ),h) (0.3)

of the Laplace-Beltrami operator has discrete spectrum. Let

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

as k →∞. Finally e−t∆F : L2(reg(X ),h)→ L2(reg(X ),h) istrace class and for some C > 0 and 0 < t ≤ 1 its trace satisfies

Tr(e−t∆F ) ≤ Ct−v (0.5)

Sketch of the proof

Let k(t , x , y) be the heat kernel corresponding to e−t∆F . Then:

D(dmax) → L2v

v−1 (reg(X ),h) =⇒ k(t , x , y) ≤ Bt−v

for (x , y) ∈ reg(X )× reg(Y ), 0 ≤ t ≤ 1, B > 0,

k(t , x , y) ≤ Bt−v and volh(reg(X )) <∞ =⇒ e−t∆F is traceclass and Tr(e−t∆F ) ≤ Ct−v for 0 ≤ t ≤ 1 and C > 0,

=⇒∆F has discrete spectrum and lim infλkk1v > 0 as k →∞.

Note: ∆F has discrete spectrum follows directly by the compactinclusion D(dmax) → L2(reg(X ),h).

Other consequences: e−t∆F is ultracontractive for 0 ≤ t ≤ 1, feigenfunction of ∆F then f ∈ L∞(reg(X ),h).

Sketch of the proof

D(dmax) → L2v

v−1 (reg(X ),h) =⇒ k(t , x , y) ≤ Bt−v

Sketch of the proof

D(dmax) → L2v

v−1 (reg(X ),h) =⇒ k(t , x , y) ≤ Bt−v

Application to the Hodge-Kodaira Laplacian

TheoremConsider the Friedrich extension

∆F∂

: L2(reg(X ),h)→ L2(reg(X ),h)

of the Hodge-Kodaira Laplacian. Then ∆F∂

has discretespectrum.

Sketch of the proof:

1) ∆F∂

= ∂tmax ∂min

2) Continuous inclusion: D(∆F∂

) → D(∂min)

‖∂minf‖2L2Ω0,1(reg(X),h) ≤ ‖f‖2L2(reg(X),h) + ‖∆F

∂f‖2L2(reg(X),h)

3) Compact inclusion: D(∂min) → L2(reg(X ),h)

Application to the Hodge-Kodaira Laplacian

TheoremConsider the Friedrich extension

∆F∂

: L2(reg(X ),h)→ L2(reg(X ),h)

of the Hodge-Kodaira Laplacian. Then ∆F∂

has discretespectrum.

Sketch of the proof:

1) ∆F∂

= ∂tmax ∂min

2) Continuous inclusion: D(∆F∂

) → D(∂min)

‖∂minf‖2L2Ω0,1(reg(X),h) ≤ ‖f‖2L2(reg(X),h) + ‖∆F

∂f‖2L2(reg(X),h)

3) Compact inclusion: D(∂min) → L2(reg(X ),h)

Some remarks

h is not required to be Kahler,No assumptions on sing(X ),No assumptions on dim(X )

If h is Kahler then the eigenvalues λk of ∆F∂

lim infλkk−1v > 0

and e−t∆F∂ : L2(reg(X ),h)→ L2(reg(X ),h) is trace class with

Tr(e−t∆F∂ ) ≤ Bt−v

for 0 ≤ t ≤ 1. This follows simply by the fact that ∆ = 2∆∂ onC∞(reg(X )) and therefore

∆F = 2∆F∂

on L2(reg(X ),h).

Some remarks

for 0 ≤ t ≤ 1.

This follows simply by the fact that ∆ = 2∆∂ onC∞(reg(X )) and therefore

∆F = 2∆F∂

on L2(reg(X ),h).

Some remarks

for 0 ≤ t ≤ 1. This follows simply by the fact that ∆ = 2∆∂ onC∞(reg(X )) and therefore

∆F = 2∆F∂

on L2(reg(X ),h).Francesco Bei On the Laplace-Beltrami operator on compact complex spaces

Application to possibly singular complex surfaces

Let (X ,h) be a compact and irreducible Hermitian complexspace of complex dimension 2.

Let q = 0, ...,2 and consider the Hodge-Kodaira Laplacian

∆∂,2,q : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

Let us introduce the following self-adjoint extension, called theabsolute extension

∆∂,2,q,abs : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

∆∂,2,q,abs := ∂t2,q,min ∂2,q,max + ∂2,q−1,max ∂

t2,q−1,min

with domain D(∆∂,2,q,abs) := ω ∈ D(∂2,q,max) ∩ D(∂t2,q−1,min) :

∂t2,q−1,minω ∈ D(∂2,q−1,max), ∂2,q,maxω ∈ D(∂

t2,q,min).

Let (X ,h) be a compact and irreducible Hermitian complexspace of complex dimension 2.Let q = 0, ...,2 and consider the Hodge-Kodaira Laplacian

∆∂,2,q : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

t2,q−1,min

t2,q,min).

∆∂,2,q : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

t2,q−1,min

t2,q,min).

∆∂,2,q : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

t2,q−1,min

t2,q,min).

∆∂,2,q : Ω2,q(reg(X ))→ Ω2,q(reg(X ))

t2,q−1,min

t2,q,min).

∆∂,2,q,abs are the Laplacians associated to the Hilbert complex

(L2Ω2,q(reg(V ),h), ∂2,q,max)

that is the maximal extension of the Dolbeault complex

(Ω2,qc (reg(V ), ∂2,q)

Combining our previous results with a theorem proved by J.Ruppenthal in 2014 we have the following result:

(Ω2,qc (reg(V ), ∂2,q)

TheoremLet (X ,h) be a compact and irreducible Hermitian complexspace of complex dimension 2. Then, for each q = 0, ...,2, theoperator

∆∂,2,q,abs : L2Ω2,q(reg(V ),h)→ L2Ω2,q(reg(V ),h)

has discrete spectrum.

Assume now that h is Kahler. Let

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

be the eigenvalues of ∆∂,2,q,abs. Then we have the followingasymptotic inequality

lim infλkk−12 > 0

as k →∞.

has discrete spectrum. Assume now that h is Kahler. Let

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

be the eigenvalues of ∆∂,2,q,abs.

Then we have the followingasymptotic inequality

as k →∞.

has discrete spectrum. Assume now that h is Kahler. Let

0 ≤ λ1 ≤ λ2 ≤ ... ≤ λk ≤ ...

be the eigenvalues of ∆∂,2,q,abs. Then we have the followingasymptotic inequality

as k →∞.

Sketch of the proof

The case (2,0) follows by F. B. 2016.

The case (2,2) follows by the results previously described.In particular

∆∂,2,2,abs ∗ c = ∗ c ∆F∂

The case (2,1) follows in this way:

Thanks to Ruppenthal we know that ker(∆∂,2,1,abs) is finitedimensional and that im(∆∂,2,1,abs) is closed.Therefore ∆∂,2,1,abs has discrete spectrum if and only if theinclusion

D(∆∂,2,1,abs) ∩ im(∆∂,2,1,abs) → L2Ω2,1(reg(X ),h)

is compact. Now this last point follows as a consequence of thenext proposition

Sketch of the proof

The case (2,0) follows by F. B. 2016.The case (2,2) follows by the results previously described.In particular

∆∂,2,2,abs ∗ c = ∗ c ∆F∂

Sketch of the proof

∆∂,2,2,abs ∗ c = ∗ c ∆F∂

Thanks to Ruppenthal we know that ker(∆∂,2,1,abs) is finitedimensional and that im(∆∂,2,1,abs) is closed.

Therefore ∆∂,2,1,abs has discrete spectrum if and only if theinclusion

Sketch of the proof

∆∂,2,2,abs ∗ c = ∗ c ∆F∂

PropositionLet Hk , k = 1,2,3 be separable Hilbert spaces and letTk : Hk → Hk+1 be unbounded, densely defined and closedoperators such that

im(T1) ⊂ ker(T2)

im(Tk ) is closed, k = 1,2.Let

∆T := T2∗ T2 + T1 T ∗1 .

Then the following properties are equivalent:D(T ∗1 T1) ∩ im(T ∗1 T1) → H1 andD(T2 T ∗2 ) ∩ im(T2 T ∗2 ) → H3 are both compactinclusions.

D(∆T ) ∩ im(∆T ) → H2 is a compact inclusion

Since we know that both ∆∂,2,0,abs and ∆∂,2,2,abs have discretespectrum the proof is complete.

im(T1) ⊂ ker(T2)

∆T := T2∗ T2 + T1 T ∗1 .

im(T1) ⊂ ker(T2)

im(Tk ) is closed, k = 1,2.

Let∆T := T2∗ T2 + T1 T ∗1 .

im(T1) ⊂ ker(T2)

∆T := T2∗ T2 + T1 T ∗1 .

Then the following properties are equivalent:

D(T ∗1 T1) ∩ im(T ∗1 T1) → H1 andD(T2 T ∗2 ) ∩ im(T2 T ∗2 ) → H3 are both compactinclusions.

im(T1) ⊂ ker(T2)

∆T := T2∗ T2 + T1 T ∗1 .

im(T1) ⊂ ker(T2)

∆T := T2∗ T2 + T1 T ∗1 .

im(T1) ⊂ ker(T2)

∆T := T2∗ T2 + T1 T ∗1 .

Mckean-Singer formula

Corollary

χ(M,KM) = ind((∂2,max + ∂t2,min)+) =

2∑q=0

(−1)q Tr(e−t∆∂,2,q,abs),

χ(M,OM) = ind((∂0,min + ∂t0,max)+) =

2∑q=0

(−1)q Tr(e−t∆∂,0,q,rel).

whereπ : M → X is any resolution of X ,KM is the sheaf of holomorphic (2,0)-forms on M,χ(M,KM) =

∑2q=0(−1)q dim(Hq(M,KM)),

χ(M,OM) =∑2

q=0(−1)q dim(H0,q∂

Corollary

χ(M,KM) = ind((∂2,max + ∂t2,min)+) =

2∑q=0

(−1)q Tr(e−t∆∂,2,q,abs),

χ(M,OM) = ind((∂0,min + ∂t0,max)+) =

2∑q=0

(−1)q Tr(e−t∆∂,0,q,rel).

χ(M,OM) =∑2

q=0(−1)q dim(H0,q∂

Corollary

χ(M,KM) = ind((∂2,max + ∂t2,min)+) =

2∑q=0

(−1)q Tr(e−t∆∂,2,q,abs),

χ(M,OM) = ind((∂0,min + ∂t0,max)+) =

2∑q=0

(−1)q Tr(e−t∆∂,0,q,rel).

χ(M,OM) =∑2

q=0(−1)q dim(H0,q∂

Corollary

χ(M,KM) = ind((∂2,max + ∂t2,min)+) =

2∑q=0

(−1)q Tr(e−t∆∂,2,q,abs),

χ(M,OM) = ind((∂0,min + ∂t0,max)+) =

2∑q=0

(−1)q Tr(e−t∆∂,0,q,rel).

χ(M,OM) =∑2

q=0(−1)q dim(H0,q∂

Thanks for your attention

On the Laplace-Beltrami operator on compact complex spaces › ... › 83 › 90 ›...

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