New Construction of Special Lagrangian Fibrations Yu-Shen Lin - - PowerPoint PPT Presentation

New Construction of Special Lagrangian Fibrations

Yu-Shen Lin

Boston University

Seoul National University Jun 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 Ω

2

d-closed non-degenerate positive (1, 1)-form ω such that ωn = cΩ ∧ ¯ Ω. 4 / 52

Calabi-Yau Manifolds

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

1

nowhere vanishing holomorphic n-form Ω

2

d-closed non-degenerate positive (1, 1)-form ω such that ωn = cΩ ∧ ¯ Ω.

Examples:

degree 5 hypersurface in P4 (quintic 3-fold). (Tian-Yau) complement of a smooth anti-canonical divisor in a Fano manifold 4 / 52

Mirror Symmetry

Physicists found out that each Calabi-Yau manifold X admits a mysterious partner ˇ X such that

1 A(X) = B( ˇ

X), A( ˇ X) = B(X),

2 where A/B denote some invariants of symplectic/complex

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

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Mirror Symmetry

Physicists found out that each Calabi-Yau manifold X admits a mysterious partner ˇ X such that

1 A(X) = B( ˇ

X), A( ˇ X) = B(X),

2 where A/B denote some invariants of symplectic/complex

geometry. The Calabi-Yau manifold ˇ 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 ˇ X is the dual torus fibration. The Ricci-flat metric of ˇ X need instanton correction from holomorphic 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 years but 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
• instanton correction/holo. discs
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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

hyperK¨ ahler rotation of holomorphic Lagrangian fibration in a hyperK¨ ahler manifold. 8 / 52

Ricci-Flat Metrics on Log Calabi–Yau Manifolds

Y = projective manifold of dimension n with D = s−1(0) smooth effective anticanonical divisor, where s ∈ H0(Y , −KY ) and no curves disjoint from D can be realized 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 with asymptotics 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 a special 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 type singular fibre.

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Main Result 2: Application to Mirror Symmetry

Theorem 2 (Collins-Jacob-L. ’19) Let ˇ X be a suitable hypK¨ ahler rotation of X. Then ˇ X is the fibrewise compactification of the Landau-Ginburg mirror of X. HyperK¨ ahler rotation gives the mirror! ˇ X can be compactified to be a rational elliptic surface of with an Id fibre adding at infinity, d = (−KY )2. (Auroux-Kartzarkov-Orlov ’05) showed that the above is the compactification of the Landau-Ginzburg mirror of X. We don’t have a Floer theoretic explanation of this phenomenon yet.

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Relation with Gross-Siebert Program

SYZ conjecture is served as a guiding principle for mirror

• symmetry. A lot of implications are proved.

To avoid the analysis difficulties, Kontsevich-Soibelman, Gross-Siebert developed the algebraic alternative for SYZ dual fibration constructing the mirror. Theorem (Lau-Lee-L.) The complex affine structure of the SYZ fibration of P2\E coincides with the one of Gross-Siebert program. This lays out the foundation of the comparison of family Floer mirror with the mirror constructed in Gross-Siebert program.

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Why SYZ Fibrations?

(Hitchin) ∃ integral affine structure on B0 with integral affine coordinates fi(u) =

• γi Im(e−iϑΩ)

Lemma If there exists a family of special Lagrangian torus Lt bounding holomorphic discs in relative class γ ∈ H2(X, Lt) in X, the Lt sit

• ver an affine line.

Better control of locus of Lagrangian fibres bounding holomorphic discs 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 count

• f 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 bulk insertions 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 hyperK¨ hler 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 smooth

• fibre. Moreover, the singular fibres are those classified by Kodaira.

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

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Remarks on the new Existence Theorem

The theorem is automatic from Riemann-Roch theorem when X is a K3 surface. Minimal use of the hyperK¨ ahler condition. The main advantage of the theorem is the existence of special Lagrangian fibration in a (log) Calabi-Yau surface with explicit equation.

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Existence of Special Lagrangian Submanifolds

Theorem 4 (Collins-Jacob-L.- ’19) Y =Fano manifold with D ∈ | − KY | smooth L ⊆ D=special Lagrangian submanifold then X = Y \D contains a special Lagrangian submanifold with topology the same as L × S1. This produces a lot of new examples of special Lagrangians in log Calabi-Yau manifolds. For instance, Theorem Every log Calabi-Yau 3-folds contains infinitely many special Lagrangian 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|2 h)

n+1 n , where ωD = √−1∂ ¯

∂h. Set l0 =

• − log |ξ|2

h

1

2n . Then 1

|∇kRm| ≤ Ckl−(k+2) has good control and

2

C −1

ι

l1−n ≤ inj ≤ Cιl1−n degenerates. 20 / 52

Ansatz Special Lagrangian Tori

L special Lagrangian in D. Lǫ = π−1(L) ∩ {|ξ|2

h = ǫ} topologically L × S1.

ΩC|Lǫ = √−1π∗ΩD ∧ dθ|Lǫ ωC =

√−1 n (− log |ξ|2 h)

1 n −1∂l0 ∧ ¯

∂l0 + (− log |ξ|2

h)

1 n π∗ωD.

In particular, Lǫ is a special Lagrangian wrt (ωC, ΩC). Special Lagrangian fibration D implies special Lagrangian fibration on XC.

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Some Geometric Quantities of Lǫ

Induced metric gC|Lǫ = (− log ǫ)

1 n −1 1

ndθ2 + (− log ǫ)

1 n π∗gD|L.

Second fundamental form |A|2 ≤ C(L, n)(− log ǫ)− 1

n

where C(L, n) depend only on n and the second fundamental form of L ⊆ (D, gD). mean curvature vector H = 0 Lǫ is κ-non-collapsing at scale rǫ for κ =

√nκL 2n−1 and rǫ = 2π √n(− log ǫ)

1−n 2n .

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First Eigenvalue Estimate of Lǫ

Recall the Rayleigh quotient for first eigenvalue λ1 = inf

f ∈C ∞(L)

• L |∇f |2VolL
• L |f |2VolL

. |dπ∗f |2

gC = (− log (ǫ))−1/nπ∗|df |2 gD

⇒ λ1(Lǫ, gC) ≤

λ1(L,gD) (− log (ǫ))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|Lǫ and Li-Yau gives the other direction

• f 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′K 2n. 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 Lǫ has (C, K, δ′)-bounded geometry, with K = (− log ǫ)

1 n .

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Moser’s Trick

Lemma (HSVZ ’18) ∃ compact set K ⊆ X, KC ⊆ XC and diffeomorphism Φ : XC\KC → X\K such that |∇gC(Φ∗ωTY − ωC)|gC = O(e−δl2n

0 ),

for some δ > 0. Same for J, Ω, g. ωTY − (Φ−1)∗ωC = dβ Run Moser’s trick with the geometric quantity controls are

• preserved. Lǫ is an almost special Lagrangian wrt ωTY and

has (C, K, δ′)-bounded geometry.

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Lagrangian Mean Curvature Flow

Let L be a graded Lagrangian submanifold in X. Then the phase θ : L → R is the function such that Ω|L = eiθvolL. L is a special Lagrangian if θ is a constant. The mean curvature H = J∇θ and the mean curvature flow is given by evolving family of immersions Ft : L → X with ∂ ∂t Ft = H. (Smoczyk) Maslov zero Lagrangian condition is preserved under mean curvature flow in K¨ ahler–Einstein manifolds.

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Lagrangian Mean Curavature Flow continued

Lagrangian mean curvature flow also preserves the Hamiltonian isotopy class. (Thomas–Yau ’01) There exists at most one special Lagrangians in a given Hamiltonian isotopy class. Thomas–Yau proposed to use mean curvature flow to find the unique special Lagrangian representative analogue of stable bundles in the B-side. (Neves ’07) Example of Lagrangian mean curvature flow can develop finite time singularities in dimension four. (Joyce ’14) Proposed to use Lagrangian mean curvature flow to study the Bridgeland stability conditions on Fukaya categories.

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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, ∇FjFk. αH = Hidxi, where Hi := gklhikl. |A|2 = gijgpqglmhiplhjqm.

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Evoluation Equations of LMCF

∂ ∂t θ = ∆θ. ∂ ∂t αH = dd∗αH. In particular, αH stay in the same

cohomology class. Zero Maslov index condition is preserved.

∂ ∂t volL = −|H|2volL. ∂ ∂t Hj = ∇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 ≤ ǫ and λ1 ≥ δ. Then ∂ ∂t

• Lt

|H|2Vol ≤ −(δ − 2Λǫ)

• Lt

|H|2Vol. As a corollary,

• Lt |H|2vol ց 0 exponentially.

∂ ∂t

• Lt |H|2vol ≤ 2
• gjlHl∇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

λ2

i

• f 2

i

≥ λ1

• i

λi

• f 2

i = −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 ≤ ǫ

2 (gradient estimate) |∇S| ≤ C 3 (non-collapsing)

• Br |S| ≥ min|S| · κ0rn, for r < r0.

Then supM |S| ≤ 1

κ0 + C

• ǫ

1 n+2 , if ǫ < rn+2

.

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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) = inf

f ∈C ∞(L)

• L |∇f |2

gtVolt

• L |ft|2Volt

. Set ft = f −

1

volLt

• L f volt.

| ∂

∂t

• L |∇f |2

gtvolt| ≤ 2 supL |∂tg|gt L |∇ft|2 gtvolt

• .

| ∂

∂t

• L f 2

t volt| ≤ supL |∂tgt|gt

• L f 2

t 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, mean curvature, first eigenvalue and non-collapsing constants under LMCF: Lemma L=Maslov zero Lagrangian w/ (C, K, δ′)-bounded geometry. Then LMCF of L preserves ((1 + δ)C, K, δ′)-bounded geometry for t ∈ [0, αK 2), α = α(C, δ).

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Smoothing Estimates

From estimates of second fundamental form to its higher derivatives: Lemma Assume that Lǫ has (C, K, δ′)-bounded geometry for t ∈ [0, αK 2). Then |∇lA|2 ≤ C(l)K −2 tl . 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 K 2n− a 2(n+2) t.

Together with the estimate |∇3H|,

• ne has |∇2H| ∼ O
• e−

δ′ 2(n+2) K 2n− a 2(n+2) t

. Feedback to the top inequality, one has

∂ ∂t |A|2 ∼ O

• e−

δ′ 2(n+2) K 2n− a 2(n+2) t

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

δ′ 2(n+2) K 2n).

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Lemma For K ≫ 0, then the LMCF of Lǫ with (C, K, δ′)-bounded geometry converges smoothly with (4C, K,

δ′ n+2)-bounded

geometry. 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

HyperK¨ ahler rotation Theory of J-holomorphic curves

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HyperK¨ ahler Rotation

Since SU(2) ∼ = Sp(1), Calabi-Yau 2-folds are hyperK¨ ahler. L: holomorphic curve in X ⇔ Ω|L = 0 ⇔ Re(eiϑΩ)|L = 0 = Im(eiϑΩϑ)|L ⇔ ωϑ|L = 0 = ImΩϑ|L L: special Lagrangian in Xϑ, ∀ϑ ∈ S1. It suffices to prove the holomorphic version of Theorem 3.

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Sacks-Ulenbeck-Gromov compactness

Lemma (Sacks-Ulenbeck,Gromov, Sikrov, Groman, CJL,...) (X, ω, J): complete, bounded sectional curvature, injectivity radius decay mildly and J uniformly tamed ⇒ The set of J-holo. curves passing through a fixed compact subset K and bounded area is compact. Use monotonicity of J-holomorphic curves to bound the diameters of all such J-holomorphic curves. If dfi bounded, then fi converges uniformly by Arzela-Ascoli theorem and the limit is holomorphic. Otherwise, ∃xi s.t. df (xi) → ∞. There will be a sphere bubble 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 < r0 f : 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 that C ∩ K = ∅, ∂K ⊆ K and Area(C) ≤ A ⇒ diam(C) ≤ R.

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Step 1:

(McLean) Deformation of special Lagrangians are unobstructed. ˇ X=suitable HK rotation of X. ∃ tubular neighborhood U ⊆ ˇ X of C with U → C elliptic

• fibration. 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 ˇ X Claim: X1 ∪ X2 is open in ˇ X. From compactness theorem, X2 consists of singular elliptic curves deformable to C.

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Step 2: Classification of Singular Fibres

Assume that Ci → C0, where C0 with components C (k) with multiplicity nk. [C0]2 = 0 and [C (k) ].[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) ) = [C (k)]2 + deg(f ∗T ˇ X) implies that g(C (k)) ≤ 1. g(C (k)) = 1, then only one component and embedding (I0). g(C (k)) = 0, then either

1

[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 k

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

[C (k)

0 ].[C (k′)

] = 0 ⇒ configurations are affine Dynkin diagrams ˜ An (In, IV ), ˜ Dn (I ∗

n ), ˜

E6 (IV ∗), ˜ E7 (III ∗), ˜ E8 (II ∗). 46 / 52

Step 3: X1 ∪ X2 = ˇ 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 to the singular curves of the 2nd kind This can’t happen since χ( ˇ X) = χ(X) < ∞.

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Step 4: Fibration Structures

B := ˇ X/ ∼, where x ∼ y if there exists an elliptic curve contains both x, y. (Hithin, Voisin) complex structure near b ∈ B0 corresponding to a smooth elliptic curve. Extend the complex structure to b in discriminant locus. ˇ X → B is continuous, holo. over B0, locally bounded ⇒ ˇ 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 CW 1-complex. Thus, B simply connected. (uniformization) B ∼ = D2 or C. Former case, the coordinate function pull back to a non-constant bounded holo. function on ˇ X.

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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 CW 1-complex. Thus, B simply connected. (uniformization) B ∼ = D2 or C. Former case, the coordinate function pull back to a non-constant bounded holo. function on ˇ

• X. Yau said NO!

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

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Compactification of ˇ X

D∗

∞ ⊆ B neighborhood of ∞.

⇒ There exists local section of ˇ X → B over D∗

∞.

D∗

∞ → M1,1 j

− → C j : D∗

∞ → C unbounded calculated on the model

(Schwartz lemma) ⇒ no essential singularity. j : D∗

∞ has a pole and extends to D∞ → ¯

M1,1

j

− → P1. The monodromy near infinity is 1 d 1

• , where d = (−KY )2.

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

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From classification of minimal surfaces ⇒ ˇ Y is not minimal. (canonical bundle formula) K ˇ

Y ∼

= π∗(KP1 ⊗ OP1(k)), k ≥ 0. (−1)-curves are sections ⇒ k = 1. (Castulnuvo’s criterion for rationality) p2( ˇ Y ) = q( ˇ Y ) = 0 ⇒ ˇ Y is rational. canonical bundle formula ⇒ p2( ˇ Y ) = 0 and h1( ˇ Y ) = 0 ⇒ q( ˇ Y ) = 0. Thus, ˇ Y is a rational elliptic surface.

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THANK YOU!

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