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Symplectic Topology ofPolynomial and Rationally

Convexity

Yakov EliashbergStanford University

September 29, 2014

CMI Conference on Symplectic Topology

Yakov Eliashberg Stanford University Symplectic Topology of Polynomial and Rationally Convexity

This talk discusses symplectic topological meaningof various convexity notions in complex analysis:

pseudoconvexity

holomorphic convexity

polynomial convexity

rational convexity

pseudoconvexity

rational convexity

pseudoconvexity

rational convexity

pseudoconvexity

rational convexity

pseudoconvexity

rational convexity

pseudoconvexity

rational convexity

pseudoconvexity

rational convexity

Euclidean + Symplectic = Hermitian

Consider the complex vector space V = Cn with coordinates(z1 = x1 + iy1, . . . , zn = xn + iyn). We will also view V as thespace R2n with coordinates (x1, y1, . . . , xn, yn), endowed with alinear operator J of multiplication by i =

√−1. J satisfies

J2 = −Id.

Consider the complex vector space V = Cn with coordinates(z1 = x1 + iy1, . . . , zn = xn + iyn). We will also view V as thespace R2n with coordinates (x1, y1, . . . , xn, yn), endowed with alinear operator J of multiplication by i =

√−1. J satisfies

J2 = −Id.

H(Z ,Z ′) =n∑

zjz′j = g(Z ,Z ′)− iω(Z ,Z ′)

g(Z ,Z ′) = 〈Z ,Z ′〉 =n∑

xjx′j + yjy

′j ,

ω(Z ,Z ′) = 〈Z , JZ ′〉 =n∑

xjy′j − yjx

′j (1)

The forms g and ω determine each other:

g(Z ,Z ′) = ω(Z , JZ ′), ω(Z ,Z ′) = g(JZ ,Z ′).

for Z ,Z ′ ∈ V and invariant under J :

ω(Z ,Z ′) = ω(JZ , JZ ′), g(Z ,Z ′) = g(JZ , JZ ′).

H(Z ,Z ′) =n∑

zjz′j = g(Z ,Z ′)− iω(Z ,Z ′)

g(Z ,Z ′) = 〈Z ,Z ′〉 =n∑

xjx′j + yjy

′j ,

ω(Z ,Z ′) = 〈Z , JZ ′〉 =n∑

xjy′j − yjx

′j (1)

The forms g and ω determine each other:

g(Z ,Z ′) = ω(Z , JZ ′), ω(Z ,Z ′) = g(JZ ,Z ′).

for Z ,Z ′ ∈ V and invariant under J :

ω(Z ,Z ′) = ω(JZ , JZ ′), g(Z ,Z ′) = g(JZ , JZ ′).

J-convex functions

Given a complex space (or manifold) (V , J) we associate to asmooth function φ : V → R the 2-form

ωφ := −ddCφ,

where the differential operator dC is defined by

dCφ(X ) := dφ(JX ) for X ∈ TV .

The form ωφ is automatically J-invariant (provided that J isintegrable). Hence, it defines an Hermitian formHφ := gφ − iωφ. In local coordinates:

ωφ = 2i∂∂φ = 2in∑

i ,j=1

∂2φ

∂zi∂zjdzi ∧ dzj .

J-convex functions

ωφ := −ddCφ,

The form ωφ is automatically J-invariant (provided that J isintegrable). Hence, it defines an Hermitian formHφ := gφ − iωφ.

In local coordinates:

ωφ = 2i∂∂φ = 2in∑

i ,j=1

∂2φ

J-convex functions

ωφ := −ddCφ,

The form ωφ is automatically J-invariant (provided that J isintegrable). Hence, it defines an Hermitian formHφ := gφ − iωφ. In local coordinates:

ωφ = 2i∂∂φ = 2in∑

i ,j=1

∂2φ

J-convex functions and hypersurfaces

A function φ is called J-convex (or strictly plurisubharmonic) ifthe Hermitian form Hφ is positive definite: ωφ(X , JX ) > 0 forall tangent vectors X 6= 0. In other words, Hφ = gφ − iωφ is aKahler metric.

Level-sets Σ = {φ = C} of J-convex functions co-oriented by∇φ are called J-convex (or strictly pseudo-convex).

By convexity and J-convexity we always mean strict convexityand J-convexity.

J-convexity of a hypersurface can be also characterized by itsLevi form

J-convexity of a hypersurface can be also characterized by itsLevi form.

Levi form

Each tangent space TpΣ ⊂ TpV , p ∈ Σ, of a regular level setΣ = {φ = C} in a complex manifold (V , J) contains a uniquemaximal complex subspace ξp ⊂ TpΣ which is given byξp = TpΣ ∩ JTpΣ..

For a co-oriented Σ the Hermitian formLΣ = Hφ|ξ is independent of the choice of φ up to a positivescalar factor and is called the Levi form of Σ. The J-convexityof Σ is equivalent to positive definiteness of its Levi form LΣ.

We have LΣ(X ,X ) =12

(II (X ,X ) + II (iX , iX )

other words, the Levi form isthe mean normal curvatureform in complex tangentdirections.

Levi form

Each tangent space TpΣ ⊂ TpV , p ∈ Σ, of a regular level setΣ = {φ = C} in a complex manifold (V , J) contains a uniquemaximal complex subspace ξp ⊂ TpΣ which is given byξp = TpΣ ∩ JTpΣ.. For a co-oriented Σ the Hermitian formLΣ = Hφ|ξ is independent of the choice of φ up to a positivescalar factor and is called the Levi form of Σ.

The J-convexityof Σ is equivalent to positive definiteness of its Levi form LΣ.

(II (X ,X ) + II (iX , iX )

Levi form

Each tangent space TpΣ ⊂ TpV , p ∈ Σ, of a regular level setΣ = {φ = C} in a complex manifold (V , J) contains a uniquemaximal complex subspace ξp ⊂ TpΣ which is given byξp = TpΣ ∩ JTpΣ.. For a co-oriented Σ the Hermitian formLΣ = Hφ|ξ is independent of the choice of φ up to a positivescalar factor and is called the Levi form of Σ. The J-convexityof Σ is equivalent to positive definiteness of its Levi form LΣ.

(II (X ,X ) + II (iX , iX )

Levi form

Each tangent space TpΣ ⊂ TpV , p ∈ Σ, of a regular level setΣ = {φ = C} in a complex manifold (V , J) contains a uniquemaximal complex subspace ξp ⊂ TpΣ which is given byξp = TpΣ ∩ JTpΣ.. For a co-oriented Σ the Hermitian formLΣ = Hφ|ξ is independent of the choice of φ up to a positivescalar factor and is called the Levi form of Σ. The J-convexityof Σ is equivalent to positive definiteness of its Levi form LΣ.

(II (X ,X ) + II (iX , iX )

Examples

1. Quadratic functions. The functionφ(z) :=

∑nk=1 λkx

2k + µky

2k is J-convex if and only if

λk + µk > 0 for all k = 1, . . . , n.

2. Distance to a totally real submanifold. For any totally realsubmanifold L ⊂ V the squared distance function dist2

L isJ-convex in a neighborhood of LHere

distL(x) := inf{dist(x , y) | y ∈ L}.

for any Hermitian metric on V .

Examples

∑nk=1 λkx

2k + µky

λk + µk > 0 for all k = 1, . . . , n.

for any Hermitian metric on V .

Examples

∑nk=1 λkx

2k + µky

λk + µk > 0 for all k = 1, . . . , n.

for any Hermitian metric on V .A submanifold L of an almost complex manifold (V , J) is calledtotally real if it has no complex tangent lines, i.e. J(TL) ∩TL = {0} at every point. This condition implies dimR L ≤ n =dimC V .

Examples

∑nk=1 λkx

2k + µky

λk + µk > 0 for all k = 1, . . . , n.

for any Hermitian metric on V .3. On Cn any convex function is J-convex.

4. Locally for a hypersurface, J-convexity ⇐⇒convexity up to a biholomorphism.

Examples

∑nk=1 λkx

2k + µky

λk + µk > 0 for all k = 1, . . . , n.

for any Hermitian metric on V .3. On Cn any convex function is J-convex.4. Locally for a hypersurface, J-convexity ⇐⇒convexity up to a biholomorphism.

Neighborhood of a totally real submanifold

Index of a critical point of a J-convex function

Summary of symplectic aspects of J-convexity

For a J-convex function φ the form ωφ, is a symplecticform.

The vector field Xφ = ∇gφφ is a Liouville vector field forωφ, i.e. LXφ

ωφ = ωφ. The form λφ = i(Xφ)ωφ) is calledthe Liouville form. We have, equivalently, dλφ = ωφ or(X t

φ)∗ωφ = etωφ.

If the function φ is Morse then the stable manifolds of itscritical points are isotropic. Hence, indices of all criticalpoints are ≤ n.

Level sets {φ = c} are contact submanifolds with thecontact structure ξ given by complex tangencies.

Lagrangian stablemanifold of acritical point of aJ-convex function φintersects a contactlevel set {φ = c}along a Legendriansubmanifold of thelevel sets.

Holomorphic convexity

Given a compact set K ⊂ (V , J) its holomorphic hull

KH = {z ∈ V ; |f (z)| ≤ maxu∈K|f (u)| for all holomorphic f : V → C}.

(V , J) is called holomorphically convex if KH is compact forevery compact K .

Let K be a compact manifold with boundary

E. Levi: If IntK is holomorphically convex then∂K is (weakly) J-convex.

K. Oka – . . . – H. Grauert: If ∂K is (weakly)J-convex then IntK is holomorphically convex.

Holomorphic convexity

Given a compact set K ⊂ (V , J) its holomorphic hull

KH = {z ∈ V ; |f (z)| ≤ maxu∈K|f (u)| for all holomorphic f : V → C}.

(V , J) is called holomorphically convex if KH is compact forevery compact K .

Let K be a compact manifold with boundary.

E. Levi: If IntK is holomorphically convex then∂K is (weakly) J-convex.

K. Oka – . . . – H. Grauert: If ∂K is (weakly)J-convex then IntK is holomorphically convex.

A holomorphically convex manifold compact without analyticsubsets of positive dimension ⇐⇒ Stein.

For a domain U in Cn or a Stein manifold V being a Steinequivalent to holomorphic convexity

Ignoring a subtle difference between J-convexity and weakJ-convexity holomorphically convex (Stein) domains aredomains with J-convex boundary.Equivalently, they are domains admitting a defining J-convexfunction. In particular, they have the homotopy type of aCW-complex of dimension ≤ n.

Theorem (Y.E., 1989)

Any domain W in a complex manifold (V , J), dimC V > 2,which admits a defining Morse function without critical pointsof index > n is isotopic to a J-convex, and henceholomorphically convex domain.

When n = 2 the situation is more subtle. For instance,S2 × D2 does not admit a structure of a Stein domain.

A holomorphically convex manifold compact without analyticsubsets of positive dimension ⇐⇒ Stein.For a domain U in Cn or a Stein manifold V being a Steinequivalent to holomorphic convexity.

Ignoring a subtle difference between J-convexity and weakJ-convexity holomorphically convex (Stein) domains aredomains with J-convex boundary.Equivalently, they are domains admitting a defining J-convexfunction. In particular, they have the homotopy type of aCW-complex of dimension ≤ n.

A holomorphically convex manifold compact without analyticsubsets of positive dimension ⇐⇒ Stein.For a domain U in Cn or a Stein manifold V being a Steinequivalent to holomorphic convexity.Ignoring a subtle difference between J-convexity and weakJ-convexity holomorphically convex (Stein) domains aredomains with J-convex boundary.

Equivalently, they are domains admitting a defining J-convexfunction. In particular, they have the homotopy type of aCW-complex of dimension ≤ n.

A holomorphically convex manifold compact without analyticsubsets of positive dimension ⇐⇒ Stein.For a domain U in Cn or a Stein manifold V being a Steinequivalent to holomorphic convexity.Ignoring a subtle difference between J-convexity and weakJ-convexity holomorphically convex (Stein) domains aredomains with J-convex boundary.Equivalently, they are domains admitting a defining J-convexfunction. In particular, they have the homotopy type of aCW-complex of dimension ≤ n.

Polynomial and Rational Convexity

Let K ⊂ Cn be a compact set. Define

polynomial hull KP = {z ∈ Cn; |P(z)| ≤maxu∈K|P(u)| for all polynomials P};

rational hull KR = {z ∈ Cn; |R(z)| ≤ maxu∈K|R(u)|

for all rational functions R = PQ, Q|K 6= 0};

A compact set K ⊂ Cn is called polynomially (resp. rationally)

convex if KP = K (resp. KR = K ).

Equivalently, these are domains where every holomorphicfunction can be C 0-approximated by polynomial or rationalfunctions. Rational convexity of K is also equivalent toexistence of an analytic hypersurface in Cn \ K through anypoint.

Equivalently, these are domains where every holomorphicfunction can be C 0-approximated by polynomial or rationalfunctions.

Rational convexity of K is also equivalent toexistence of an analytic hypersurface in Cn \ K through anypoint.

Criteria for rational and polynomial convexity

Criterion (Oka)

An i -convex domain W ⊂ Cn is polynomially convex if andonly if there exists an exhausting i -convex functionφ : Cn → R such that W = {φ ≤ 0}.

Criterion (Duval-Sibony,Nemirovski)

An i -convex domain W ⊂ Cn isrationally convex if and only ifthere exists a defining i -convexfunction φ : W → R such thatform −ddCφ on W extends to aKahler form ω on the whole Cn.

Criteria for rational and polynomial convexity

Criterion (Oka)

An i -convex domain W ⊂ Cn is polynomially convex if andonly if there exists an exhausting i -convex functionφ : Cn → R such that W = {φ ≤ 0}.

Criterion (Duval-Sibony,Nemirovski)

An i -convex domain W ⊂ Cn isrationally convex if and only ifthere exists a defining i -convexfunction φ : W → R such thatform −ddCφ on W extends to aKahler form ω on the whole Cn.

Topology of rationally convex domains

Theorem (K. Cieliebak - Y.E.,2013)

Let U ⊂ Cn, n > 2, be acompact domain with a J-convexsmooth boundary. Then thereexists a diffeomorphic to∂U × [0, 1] interior collar of theboundary ∂U such thatU \ IntW is rationally convex.

In particular, any domain in Cn

which admits a defining Morsefunction φ : W → R withoutcritical points of index > n isisotopic to a rationally convexdomain.

Topology of rationally convex domains

Theorem (K. Cieliebak - Y.E.,2013)

Let U ⊂ Cn, n > 2, be acompact domain with a J-convexsmooth boundary. Then thereexists a diffeomorphic to∂U × [0, 1] interior collar of theboundary ∂U such thatU \ IntW is rationally convex.In particular, any domain in Cn

which admits a defining Morsefunction φ : W → R withoutcritical points of index > n isisotopic to a rationally convexdomain.

Rationally convex domains in C2

The situation in C2 is much more subtle. I already mentionedadditional constraints on the topology of 2-dimensional Steindomain, e.g. S2 × D2 cannot be realized as a holomorphicallyconvex domain in C2. It turns out that for rational convexitythere are further constraints.

Recently, S. Nemirovski and K. Siegel completely characterizedall disc bundles over 2-dimensional surfaces which could berealized as rationally convex domains in C2.

The situation in C2 is much more subtle. I already mentionedadditional constraints on the topology of 2-dimensional Steindomain, e.g. S2 × D2 cannot be realized as a holomorphicallyconvex domain in C2. It turns out that for rational convexitythere are further constraints.

Recently, S. Nemirovski and K. Siegel completely characterizedall disc bundles over 2-dimensional surfaces which could berealized as rationally convex domains in C2.

Let Do(χ, e) denote the D2-bundle over anorientable surface of Euler characterfistic χ, withthe Euler number e. The notation Dn−o(χ, e) hasthe same meaning in the case of a non-orientablesurface.

Theorem (K. Siegel, S. Nemirovski)

There exist strictly pseudo-convex rationally convexdomains in C2 diffeomorphic to the followingdisc-bundles:

Do(χ, 0) for ξ 6= 2;

Dn−o(χ, e) for (χ, e) 6= (1,−2) or (0, 0) ande ∈ {2χ− 4, 2χ+ 4, . . . ,−2χ− 4 + 4[χ/4 + 1]}.

Let Do(χ, e) denote the D2-bundle over anorientable surface of Euler characterfistic χ, withthe Euler number e. The notation Dn−o(χ, e) hasthe same meaning in the case of a non-orientablesurface.

Theorem (K. Siegel, S. Nemirovski)

There exist strictly pseudo-convex rationally convexdomains in C2 diffeomorphic to the followingdisc-bundles:

Do(χ, 0) for ξ 6= 2;

Dn−o(χ, e) for (χ, e) 6= (1,−2) or (0, 0) ande ∈ {2χ− 4, 2χ+ 4, . . . ,−2χ− 4 + 4[χ/4 + 1]}.

Singular Lagrangian surfaces in C 2

In the Nemirovski-Siegel list there are two bundles which arerealizable as holomorphically, but not rationally convexdomains: Dn−o(0, 0) and Dn−o(1− 2).

The first corresponds to the cotangent bundle of a Kleinbottle, the second is a neighborhood of a Lagrangianprojective plane with one so-called Whitney umbrella.

Their result gives a complete list of Lagrangian surfaces withconical singularities, and in particular, rules out existence ofembedded in C2 Lagrangian projective plane with one Whitneyumbrella singularity.

Topology of polynomially convex domains

K. Cieliebak– Y.E. : For polynomial convexity indimension n > 2 the following additional (toexistence of a defining function without criticalpoints of index > n) condition is necessary andsufficient: Hn(W ;G ) = 0 for any Abelian group G .

In the not simply-connected case a polynomiallyconvex domain need not to be subcritical.

Conjecture

A simply-connected polynomially convex domain hasto be subcritical.

More generally any polynomially convex

domain in Cn, n > 2 has to be flexible.

Conjecture

A simply-connected polynomially convex domain hasto be subcritical. More generally any polynomially convex

J-convex surrounding

Lemma (Y.E., 1990 and K. Cieliebak - Y.E.,2013)

Let W ⊂ Cn be a domain with aJ-convex boundary and(L, ∂L) ⊂ (Cn \ IntW , ∂W ) be atotally real submanifold with Legendrianboundary. Then W ⊂ L has afundamental system of rationally convexneighborhoods.

If W ⊂ Cn a rationally convex with aJ-convex boundary, and(L, ∂L) ⊂ (Cn \ IntW , ∂W ) aLagrangian with Legendrian boundary,then W ⊂ L has a fundamental systemof rationally convex neighborhoods.

∂−W

Theorem (Y.E., E. Murphy)

Suppose that n > 2. Then any n-dimensionalmanifold L ⊂ Cn \ IntW with ∂L ⊂ ∂W isisotopic through submanifolds in Cn \ IntWwith boundary in ∂W to a Lagrangiansubmanifold with Legendrian boundary.

For n = 2 an analogous statement is wrong: there are noLagrangian 2-discs attached to the round ball in C2 along aLegendrian curve. Indeed, the existence of such aLagrangian disc implies that the Thurston-Bennequin invariantof the Legendrian knot ∂∆ ⊂ S3 is equal to +1. But thiscontradicts to the slice Bennequin’s inequality. (L. Rudolph,Kronheimer-Mrowka).

In fact, this theorem is a special case of a general h-principlefor Lagrangian embeddings with the so-called loose Legendrianboundary.

For n = 2 an analogous statement is wrong: there are noLagrangian 2-discs attached to the round ball in C2 along aLegendrian curve.

Indeed, the existence of such aLagrangian disc implies that the Thurston-Bennequin invariantof the Legendrian knot ∂∆ ⊂ S3 is equal to +1. But thiscontradicts to the slice Bennequin’s inequality. (L. Rudolph,Kronheimer-Mrowka).

Murphy’s theory of loose knots

Any Legendrian knot of dimension > 1 can be made looseby a standard C 0-small bifurcation (called stabilization) ina neighborhood of a point. Moreover, this can be donepreserving the formal isotopy class of the knot.

Any two loose Legendrian knots which are formallyLegendrian isotopic can be connected by a genuineLegendrian isotopy.

A Stein domain W of complex dimension n is called flexible ifthere exists a defining J-convex function φ : W → R whose alln-handles attached along loose Legendrian knots.

Flexible Stein domains

More precisely, A domain (W , J) is flexible if there a definingJ-convex function φ : W → R such that for regular valuesc0 < minφ < · · · < maxφ < cN each cobordism{cj ≤ φ ≤ cj+1} contains just one critical point pj whosestable manifold intersects the contact level {φ = cj} along aloose Legendrian knot.

Flexible Stein manifold are not very interesting symplectically.In particular, they have vanishing symplectic homology.

An outstanding open question is whether the flexibilityproperty is independent on the choice of φ.

A positive answer to this question would imply thatpolynomially convex domains in Cn, n > 2 are flexible.

In fact, this could be true for n = 2 as well: any polynomiallyconvex domains in C2 is subcritical.

Further questions

What is symplectic topology of rationally convex domains.

Do their classifications up to isotopy throughholomorphically convex domains and through rationallyconvex domain coincide?

Replacing Cn with other Stein manifiolds.

Results aboutpolynomial convexity break if the ambient manifold is notsubcritical or flexible.

What is symplectic geometry of rational and polynomialconvexity?

If W ⊂ Cn is a Stein cobordism and ∂+Wbounds a rationally (polynomially) convex domain then sodoes ∂−W , but not the other way around.

Further questions

Replacing Cn with other Stein manifiolds. Results aboutpolynomial convexity break if the ambient manifold is notsubcritical or flexible.

Further questions

What is symplectic geometry of rational and polynomialconvexity? If W ⊂ Cn is a Stein cobordism and ∂+Wbounds a rationally (polynomially) convex domain then sodoes ∂−W , but not the other way around.

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