New Construction of Special LagrangianFibrations

Yu-Shen Lin

Boston University

Seoul National UniversityJun 27, 2019

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Outline of the Talk

Calabi-Yau Manifolds and Strominger-Yau-Zaslow Conjecture

Main Theorems and Applications

Sketch of the Proofs

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Calabi-Yau Manifolds

Calabi-Yau manifold.

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Calabi-Yau Manifolds

Calabi-Yau n-fold X=higher dimension analogue of elliptic curves.=complex manifold X with

1 nowhere vanishing holomorphic n-form Ω2 d-closed non-degenerate positive (1, 1)-form ω such thatωn = cΩ ∧ Ω.

Examples:

degree 5 hypersurface in P4 (quintic 3-fold).(Tian-Yau) complement of a smooth anti-canonical divisor in aFano manifold

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Calabi-Yau Manifolds

Calabi-Yau n-fold X=higher dimension analogue of elliptic curves.=complex manifold X with

1 nowhere vanishing holomorphic n-form Ω2 d-closed non-degenerate positive (1, 1)-form ω such thatωn = cΩ ∧ Ω.

Examples:

degree 5 hypersurface in P4 (quintic 3-fold).(Tian-Yau) complement of a smooth anti-canonical divisor in aFano manifold

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Mirror Symmetry

Physicists found out that each Calabi-Yau manifold X admits amysterious partner X such that

1 A(X ) = B(X ),A(X ) = B(X ),

2 where A/B denote some invariants of symplectic/complexgeometry.

The Calabi-Yau manifold X is called the mirror of X .

Question

How do we find the mirror of X?

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Mirror Symmetry

Physicists found out that each Calabi-Yau manifold X admits amysterious partner X such that

1 A(X ) = B(X ),A(X ) = B(X ),

2 where A/B denote some invariants of symplectic/complexgeometry.

The Calabi-Yau manifold X is called the mirror of X .

Question

How do we find the mirror of X?

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Strominger-Yau-Zaslow Conjecture

Conjecture (Strominger-Yau-Zaslow ’96)

A Calabi-Yau X admits special Lagrangian torus fibration.

The mirror X is the dual torus fibration.

The Ricci-flat metric of X need instanton correction fromholomorphic discs with boundary on the torus fibres.

L ⊆ X is special Lagrangian if ω|L = 0, Ω|L = vol|L.

We know the existence of the Ricci-flat metric for 40 yearsbut don’t know much of the description of it.

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Difficuclties of SYZ Conjecture

The three problems form an iron triangle:

Ricci-flat metric

**SYZ fibration

55

oo instanton correction/holo. discsoo

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Existing Examples

Almost special Lagrangian fibration (not wrt Ricci-flat metric)

1 (Gross ’00) toric Calabi-Yau manifolds

2 (Goldstein ’02) Borisov-Voison Calabi-Yau 3-folds

True special Lagrangian fibration

1 Complex tori with flat metric or

2 hyperKahler rotation of holomorphic Lagrangian fibration in ahyperKahler manifold.

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Ricci-Flat Metrics on Log Calabi–Yau Manifolds

Y = projective manifold of dimension n with

D = s−1(0) smooth effective anticanonical divisor, wheres ∈ H0(Y ,−KY ) and no curves disjoint from D can berealized as linear combination of curves support in D.

Then Y \D admits a non-vanishing holomorphic volume form Ω

Theorem (Tian–Yau ’90)

There exists a complete Ricci-flat metric ωTY on X = Y \D withasymptotics near D

ωTY ∼√−1

2π∂∂(− log ‖ s ‖)(n+1)/n.

Question: Is there a SYZ fibration on X?

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Main Result 1: New Special Lagrangian Fibrations

Theorem 1 (Collins-Jacob-L. ’19)

Y=weak del Pezzo surface, D ∈ | − KY | smooth.Then X = Y \D admits a special Lagrangian fibration with aspecial Lagrangian section with respect to the Tian-Yau metric.

This solves conjectures of Yau and Auroux ’08.

Probably the only non-trivial example so far.

(Collins-Jacob-L.) Y=rational elliptic surface, D=Id typesingular fibre.

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Main Result 2: Application to Mirror Symmetry

Theorem 2 (Collins-Jacob-L. ’19)

Let X be a suitable hypKahler rotation of X . Then X is thefibrewise compactification of the Landau-Ginburg mirror of X .

HyperKahler rotation gives the mirror!

X can be compactified to be a rational elliptic surface of withan Id fibre adding at infinity, d = (−KY )2.

(Auroux-Kartzarkov-Orlov ’05) showed that the above is thecompactification of the Landau-Ginzburg mirror of X .

We don’t have a Floer theoretic explanation of thisphenomenon yet.

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Relation with Gross-Siebert Program

SYZ conjecture is served as a guiding principle for mirrorsymmetry. A lot of implications are proved.

To avoid the analysis difficulties, Kontsevich-Soibelman,Gross-Siebert developed the algebraic alternative for SYZ dualfibration constructing the mirror.

Theorem (Lau-Lee-L.)

The complex affine structure of the SYZ fibration of P2\Ecoincides with the one of Gross-Siebert program.

This lays out the foundation of the comparison of family Floermirror with the mirror constructed in Gross-Siebert program.

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Why SYZ Fibrations?

(Hitchin) ∃ integral affine structure on B0 with integral affinecoordinates

fi (u) =∫γi

Im(e−iϑΩ)

Lemma

If there exists a family of special Lagrangian torus Lt boundingholomorphic discs in relative class γ ∈ H2(X , Lt) in X , the Lt sitover an affine line.

Better control of locus of Lagrangian fibres bounding holomorphicdiscs if the fibration is special.

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Application of SYZ Fibrations in Enumerative Geometry

Usually, it is hard to compute family Floer mirror explictly.

Theorem (Cheung-L.)

An explicit calculation of family Floer mirror for certain HK surface.

For geometric interpretation of the slab functions in GS programs:

Theorem (L.-)

Equivalence of open Gromov-Witten invariants and weighted countof tropical discs counting for HK surfaces with SYZ fibration.

Similar spirit is used to proved

Theorem (Hong-L.-Zhao ’18)

Tropical/holomorphic correspondence for discs with interior bulkinsertions and quantum period theorem for toric Fano surfaces.

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General Existence Theorem of SYZ Fibration

Theorem 3 (Collins-Jacob-L.- ’19)

X=complete hyperKhler surface with

1 bounded sectional curvature and injectivity radius decay mildly

2 χ(X ) <∞3 L ⊆ X smooth or immersed special Lagrangian torus with

[L]2 = 0 and [L] ∈ H2(X ,Z) primitive.

Then X admits a special Lagrangian fibration with L a smoothfibre. Moreover, the singular fibres are those classified by Kodaira.

So existence of SYZ fibration for a HK surface reduce to theexistence of a single special Lagrangian torus.

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Remarks on the new Existence Theorem

The theorem is automatic from Riemann-Roch theorem whenX is a K3 surface.

Minimal use of the hyperKahler condition.

The main advantage of the theorem is the existence of specialLagrangian fibration in a (log) Calabi-Yau surface with explicitequation.

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Existence of Special Lagrangian Submanifolds

Theorem 4 (Collins-Jacob-L.- ’19)

Y=Fano manifold with D ∈ | − KY | smoothL ⊆ D=special Lagrangian submanifold then X = Y \D contains aspecial Lagrangian submanifold with topology the same as L× S1.

This produces a lot of new examples of special Lagrangians inlog Calabi-Yau manifolds. For instance,

Theorem

Every log Calabi-Yau 3-folds contains infinitely many specialLagrangian tori.

Theorem 3 + Theorem 4 ⇒ Theorem 1.

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Proof of Theorem 4

Ansatz Special Lagrangians

Lagrangian Mean Curvature Flow

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Geometric Input

Y=RES, then ωTY is asymptotically cylindrical relatively easy.

Y=del Pezzo surface,then first eigenvalue λ1 and injectivity radius are degenerating.

Need qualitative version of all estimates.

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Calabi Ansatz

D=projective Calabi-Yau of dimension n − 1.

XC=neighborhood of zero section of ND/Y , π : XC → D.

ΩC = f (z)w dz1 ∧ · · · ∧ dzn−1 ∧ dw .

ωC =√−1∂∂ n

n+1 (− log |ξ|2h)n+1n , where ωD =

√−1∂∂h.

Set l0 =(− log |ξ|2h

) 12n . Then

1 |∇kRm| ≤ Ck l−(k+2)0 has good control and

2 C−1ι l1−n

0 ≤ inj ≤ Cιl1−n0 degenerates.

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Ansatz Special Lagrangian Tori

L special Lagrangian in D.

Lε = π−1(L) ∩ |ξ|2h = ε topologically L× S1.

ΩC |Lε =√−1π∗ΩD ∧ dθ|Lε

ωC =√−1n (− log |ξ|2h)

1n−1∂l0 ∧ ∂l0 + (− log |ξ|2h)

1nπ∗ωD .

In particular, Lε is a special Lagrangian wrt (ωC ,ΩC).

Special Lagrangian fibration D implies special Lagrangianfibration on XC .

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Some Geometric Quantities of Lε

Induced metric gC |Lε = (− log ε)1n−1 1

ndθ2 + (− log ε)

1nπ∗gD |L.

Second fundamental form

|A|2 ≤ C (L, n)(− log ε)−1n

where C (L, n) depend only on n and the second fundamentalform of L ⊆ (D, gD).

mean curvature vector H = 0

Lε is κ-non-collapsing at scale rε for

κ =√nκL

2n−1 and rε = 2π√n

(− log ε)1−n2n .

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First Eigenvalue Estimate of Lε

Recall the Rayleigh quotient for first eigenvalue

λ1 = inff ∈C∞(L)

∫L |∇f |

2VolL∫L |f |2VolL

.

|dπ∗f |2gC = (− log (ε))−1/nπ∗|df |2gD⇒ λ1(Lε, gC) ≤ λ1(L,gD)

(− log (ε))1/n .

(Li-Yau) Mn compact Ricci-flat manifold without boundary.Then λ1 ≥ C/(n − 1)d2, where d = diam(M) and C = C (n).

From the expression gC |Lε and Li-Yau gives the other directionof inequality.

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Notion of Bounded Geoemtry

Definition

L has (C ,K , δ′)-bounded geometry if

C−1K < l0|L < CK .

|A|2 ≤ CK−2.

|H|2 ≤ Ce−δ′K2n

.

C−1 ≤ volL ≤ C .

C−1K−2 ≤ λ1 ≤ CK−2.

gTY |L is κ0-non-collapsing on scall r0 with κ0 ≥ C−1,r0 ≤ C−1K 1−n.

Lemma

Lε has (C ,K , δ′)-bounded geometry, with K = (− log ε)1n .

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Moser’s Trick

Lemma (HSVZ ’18)

∃ compact set K ⊆ X ,KC ⊆ XC and diffeomorphismΦ : XC\KC → X\K such that

|∇gC(Φ∗ωTY − ωC)|gC = O(e−δl2n0 ),

for some δ > 0. Same for J,Ω, g .

ωTY − (Φ−1)∗ωC = dβ

Run Moser’s trick with the geometric quantity controls arepreserved. Lε is an almost special Lagrangian wrt ωTY andhas (C ,K , δ′)-bounded geometry.

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Lagrangian Mean Curvature Flow

Let L be a graded Lagrangian submanifold in X . Then thephase θ : L→ R is the function such that

Ω|L = e iθvolL.

L is a special Lagrangian if θ is a constant.

The mean curvature ~H = J∇θ and the mean curvature flow isgiven by evolving family of immersions Ft : L→ X with

∂

∂tFt = ~H.

(Smoczyk) Maslov zero Lagrangian condition is preservedunder mean curvature flow in Kahler–Einstein manifolds.

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Lagrangian Mean Curavature Flow continued

Lagrangian mean curvature flow also preserves theHamiltonian isotopy class.

(Thomas–Yau ’01) There exists at most one specialLagrangians in a given Hamiltonian isotopy class.Thomas–Yau proposed to use mean curvature flow to find theunique special Lagrangian representative analogue of stablebundles in the B-side.

(Neves ’07) Example of Lagrangian mean curvature flow candevelop finite time singularities in dimension four.

(Joyce ’14) Proposed to use Lagrangian mean curvature flowto study the Bridgeland stability conditions on Fukayacategories.

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Some Notations for LMCF

F : L→ X Lagrangian immersion w/ (x1, · · · , xn) local coor. on L.

Fi = ∂F∂xi

and νi = JFi .

gij = 〈Fi ,Fj〉.hijk = −〈νi ,∇Fj

Fk〉.αH = Hidx

i , where Hi := gklhikl .

|A|2 = g ijgpqg lmhiplhjqm.

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Evoluation Equations of LMCF

∂∂t θ = ∆θ.

∂∂tαH = dd∗αH . In particular, αH stay in the samecohomology class. Zero Maslov index condition is preserved.

∂∂t volL = −|H|2volL.

∂∂tHj = ∇j∇iHi .

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L2-Estimates of the Mean Curvature

Lemma (Li ’09)

Assume along the flow |A(t)|2 ≤ Λ, |H(t)|2 ≤ ε and λ1 ≥ δ. Then

∂

∂t

∫Lt

|H|2Vol ≤ −(δ − 2Λε)

∫Lt

|H|2Vol.

As a corollary,∫Lt|H|2vol 0 exponentially.

∂∂t

∫Lt|H|2vol ≤ 2

∫g jlHl∇j∇iHi + 2|A||H|3 − |H|4vol.

∂∂t

∫Lt|H|2vol ≤ −2

∫Lt|∇iHi |2 + 2 supLt |A||H|

∫Lt|H|2vol.

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Let fi are the eigenvector of ∆ with eigenvalue λi .∫|∇iHi |2 =

∫|∆θ|2 ≥

∑i

λ2i

∫f 2i

≥ λ1

∑i

λi

∫f 2i = −λ1

∫θ∆θ = −λ1

∫|∇θ|2.

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From L2 to Pointwise Estimates

Lemma

(M, g) Riemannian, S tensor with

1 (L2-estimates)∫M |S |

2 ≤ ε2 (gradient estimate) |∇S | ≤ C

3 (non-collapsing)∫Br|S | ≥ min|S | · κ0r

n, for r < r0.

Then supM |S | ≤(

1κ0

+ C)ε

1n+2 , if ε < rn+2

0 .

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Conservation of Pointwise Estimates

∂∂t |A|

2 ≤ ∆|A|+ 8|A|2 + 20|Rm||A|+ 4|∇Rm|.

Λ := supL0|A(0)|, then from comparison principle

|A(t)| ≤ Λ + CA(l0)t. A priori can be out of control

∂∂t |H| ≤ ∆|H|+ 2|A|2|H|+ |Rm||H|.

Again from comparison principle, |H(t)| ≤ |H(0)|eCH(l0)t .

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First Eigenvlaue Estimates under LMCF

Recall the Rayleigh quotient for first eigenvalue

λ1(t) = inff ∈C∞(L)

∫L |∇f |

2gt Volt∫

L |ft |2Volt.

Set ft = f − 1volLt

∫L f volt .

| ∂∂t∫L |∇f |

2gt volt | ≤ 2 supL |∂tg |gt

( ∫L |∇ft |

2gt volt

).

| ∂∂t∫L f

2t volt | ≤ supL |∂tgt |gt

∫L f

2t volt .

Set µ(t) =∫ t

0 supL |∂sgs |gsds, then by comparison principle

e−3µ(t)λ1(0) ≤ λ1(t) ≤ e3µ(t)λ1(0).

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Non-Collapsing Estimates

Lemma

Bg0(x , r) ⊆ L geodesic ball with Vol(Bg0(x , r)) ≥ κrn, r < R.⇒ Vol(Bgt (x , r)) ≥ κ′rn

∂∂t ηij = −2aij = −2Hphpkl .

E (t) :=∫ t

0 (supLt |A|. supLu |H|)ds

⇒ e−E(t)dg0(p, q) ≤ dgt (p, q) ≤ e−E(t)dg0(p, q) and

VolLt ≥ e−E(t)VolL0 .∫Bgt (x ,r) VolLt ≥

∫Bg0 (x ′,e−E(t)r) e

−E(t)VolL0 ≥ κe−(n+1)E(t)rn.

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Bounded Geometry under LMCF

Putting together the estimates for second fundamental form, meancurvature, first eigenvalue and non-collapsing constants underLMCF:

Lemma

L=Maslov zero Lagrangian w/ (C ,K , δ′)-bounded geometry.Then LMCF of L preserves ((1 + δ)C ,K , δ′)-bounded geometry fort ∈ [0, αK 2), α = α(C , δ).

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Smoothing Estimates

From estimates of second fundamental form to its higherderivatives:

Lemma

Assume that Lε has (C ,K , δ′)-bounded geometry for t ∈ [0, αK 2).Then

|∇lA|2 ≤ C (l)K−2

t l.

We will only need the case l = 1, 2, 3 for estimates of |∇lH|.

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Improved Estimates for A under LMCF

(Smoczyk) ∂∂t |A|

2 ≤ 100(|A||∇2H|+ |A|3|H|+ |Rm||H|).

To estimate |∇2H|,∫|∇2H|2 ≤

∫|H||∇4H| ≤

√C4K

−1e− δ′

2(n+2)2 K2n− a

2(n+2)t.

Together with the estimate |∇3H|,one has |∇2H| ∼ O

(e− δ′

2(n+2)K2n− a

2(n+2)t)

.

Feedback to the top inequality, one has∂∂t |A|

2 ∼ O(e− δ′

2(n+2)K2n− a

2(n+2)t)

.

Integrating t, one has |A(t)|2 ≤ 2CK−2 + O(e− δ′

2(n+2)K2n

).

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Lemma

For K 0, then the LMCF of Lε with (C ,K , δ′)-boundedgeometry converges smoothly with (4C ,K , δ′

n+2 )-boundedgeometry.

Assume that T is the maximal time of LMCF preserving(4C ,K , δ′

n+2 )-bounded geometry.

Claim: Lt is of (3C ,K , δ′

n+2 )-bounded geometry.This contradicts to the definition ot T .

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The proof of Theorem 3

HyperKahler rotation

Theory of J-holomorphic curves

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HyperKahler Rotation

Since SU(2) ∼= Sp(1), Calabi-Yau 2-folds are hyperKahler.L: holomorphic curve in X ⇔ Ω|L = 0

⇔ Re(e iϑΩ)|L = 0 = Im(e iϑΩϑ)|L⇔ ωϑ|L = 0 = ImΩϑ|L

L: special Lagrangian in Xϑ, ∀ϑ ∈ S1.

It suffices to prove the holomorphic version of Theorem 3. 41 / 52

Sacks-Ulenbeck-Gromov compactness

Lemma (Sacks-Ulenbeck,Gromov, Sikrov, Groman, CJL,...)

(X , ω, J): complete, bounded sectional curvature, injectivity radiusdecay mildly and J uniformly tamed⇒ The set of J-holo. curves passing through a fixed compactsubset K and bounded area is compact.

Use monotonicity of J-holomorphic curves to bound thediameters of all such J-holomorphic curves.

If ‖ dfi ‖ bounded, then fi converges uniformly byArzela-Ascoli theorem and the limit is holomorphic.

Otherwise, ∃xi s.t. ‖ df (xi ) ‖→ ∞. There will be a spherebubble near x = lim xi .

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Monotonicity of J-Holomorphic Curves

J-holomorphic curves follow the monotonicity.

Lemma

With above assumptions, ∃r0,C > 0 such that for r < r0f : Σ→ B(x , r) and f (∂Σ) ⊆ ∂B(x , r)⇒ Area(f (Σ)) ≥ Cr2

Monotonicity implies the following diameter estimate.

Lemma

With above assumptions, ∃R = R(X ,K ,A) such thatC ∩ K 6= ∅, ∂K ⊆ K and Area(C ) ≤ A⇒ diam(C ) ≤ R.

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Step 1:

(McLean) Deformation of special Lagrangians areunobstructed.

X=suitable HK rotation of X .

∃ tubular neighborhood U ⊆ X of C with U → C ellipticfibration. Fibres are smooth elliptic curves deformable to C .

X1 := union of smooth elliiptic curves deformable to C .Then X1 is open.

X2 := ∂X1, then X1 ∪ X2 is closed in X

Claim: X1 ∪ X2 is open in X .

From compactness theorem, X2 consists of singular ellipticcurves deformable to C .

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Step 2: Classification of Singular Fibres

Assume that Ci → C0, where C0 with components C(k)0 with

multiplicity nk .

[C0]2 = 0 and [C(k)0 ].[C0] = 0

then Q ≤ 0 and KerQ = Z[C0].

Claim: the singular fibres are those classified by Kodaira.

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(Adjunction Formula, McDuff, M-White)

2δ + χ(C(k)0 ) = [C (k)]2 + deg(f ∗TX )

implies that g(C (k)) ≤ 1.

g(C (k)) = 1, then only one component and embedding (I0).

g(C (k)) = 0, then either1 [C (k)]2 = 0, only one components and C0 is a nodal rational

curve (I1) or a cuspidal curve (II ).2 [C k ]2 < 0, then [C

(k)0 ] = −2 for all k .

[C k0 ].[C (k′)] = 2⇒ exactly two components (I2, III )

[C(k)0 ].[C

(k′)0 ] = 0⇒ configurations are affine Dynkin diagrams

An (In, IV ), Dn (I ∗n ), E6 (IV ∗), E7 (III ∗), E8 (II ∗).

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Step 3: X1 ∪ X2 = X

Claim: X1\X2 is path connected.

Singular curves of the 2nd kind is rigid.

Singular curves of the 1st kind is rigid.

Thus, only singular curves of the 1st kind can converges tothe singular curves of the 2nd kind

This can’t happen since χ(X ) = χ(X ) <∞.

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Step 4: Fibration Structures

B := X/ ∼, where x ∼ y if there exists an elliptic curvecontains both x , y .

(Hithin, Voisin) complex structure near b ∈ B0 correspondingto a smooth elliptic curve.

Extend the complex structure to b in discriminant locus.

X → B is continuous, holo. over B0, locally bounded⇒ X → B holo., genus one fibration.

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Step 5: More Structures

[L] ∈ H2(X ,Z) primitive ⇒ no multiple fibre.

paths in B locally liftable and connected fibres⇒ π1(X ) π1(B)

In particular, π1(X ) torsion implies π1(B) torsion.

B open Riemann surface ⇒ deformation retract to CW1-complex. Thus, B simply connected.

(uniformization) B ∼= D2 or C.Former case, the coordinate function pull back to anon-constant bounded holo. function on X .

Yau said NO!

Above all holds for del Pezzo surfaces or rational elliptic surfaces.

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Step 5: More Structures

[L] ∈ H2(X ,Z) primitive ⇒ no multiple fibre.

paths in B locally liftable and connected fibres⇒ π1(X ) π1(B)

In particular, π1(X ) torsion implies π1(B) torsion.

B open Riemann surface ⇒ deformation retract to CW1-complex. Thus, B simply connected.

(uniformization) B ∼= D2 or C.Former case, the coordinate function pull back to anon-constant bounded holo. function on X . Yau said NO!

Above all holds for del Pezzo surfaces or rational elliptic surfaces.

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Compactification of X

D∗∞ ⊆ B neighborhood of ∞.⇒ There exists local section of X → B over D∗∞.

D∗∞ → M1,1j−→ C

j : D∗∞ → C unbounded calculated on the model(Schwartz lemma) ⇒ no essential singularity.

j : D∗∞ has a pole and extends to D∞ → M1,1j−→ P1.

The monodromy near infinity is

(1 d0 1

), where d = (−KY )2.

Glue the universal family to X → B to get Y .

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From classification of minimal surfaces ⇒ Y is not minimal.

(canonical bundle formula)

KY∼= π∗(KP1 ⊗OP1(k)), k ≥ 0.

(−1)-curves are sections ⇒ k = 1.

(Castulnuvo’s criterion for rationality)p2(Y ) = q(Y ) = 0⇒ Y is rational.canonical bundle formula ⇒ p2(Y ) = 0 andh1(Y ) = 0⇒ q(Y ) = 0.

Thus, Y is a rational elliptic surface.

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THANK YOU!

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