Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits

Transcription

1 Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits Mark McLean arxiv:

2 Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits Mark McLean arxiv:

3 A C N affine variety of dimension n with an isolated singularity at 0. L A = A S ɛ where S ɛ = {z C N z = ɛ}. Here L A is a real 2n 1 dimensional C manifold called the link of A at 0 for ɛ small enough. A C manifold diffeomorphic to L A is said to be Milnor fillable by A.

4 Examples: L C n = S 2n 1 L {x 2 +y 2 +z 2 =0} = RP 3. A is called differentiably smooth if L A is diffeomorphic to L C n = S 2n 1. Question: Which singularities are differentiably smooth?

5 Theorem (Mumford) Let A be a normal surface singularity with link diffeomorphic to S 3. Then A is smooth at 0. This is false in higher dimensions (Brieskorn): L {x 2 +y 2 +z 2 +w 3 =0} = S 5 Need more structure on the link.

6 Introduction to Contact Geometry Let C be a real 2n 1 dimensional C manifold and let ξ TC be a hyperplane distribution. For simplicity, assume ξ = ker(α) for some 1-form α on M. (Frobenious Integrability Theorem): ξ is the tangent space to a foliation iff dα ξ = 0.

7 Definition: ξ is a Contact structure if dα ξ is a non-degenerate 2-form at every point. A contact structure is the opposite of a Foliation! Equivalently: ξ is a Contact structure iff α (dα) n 1 0 at every point.

8 We will call (C, ξ) a contact manifold. Any 1-form α satisfying Ker(α) = ξ is a contact form associated to ξ. Two contact manifolds are contactomorphic if there is a diffeomorphism preserving the respective hyperplane distributions. (Gray s stability theorem). If I have a smooth family of contact structures on a compact manifold, then they are all contactomorphic.

9 Example: n 1 (R 2n 1, ker(dz y j dx j )) where (x 1, y 1,, x n 1, y n 1, z) are the natural coordinates. j=1

10 The Reeb vector field of α is the unique vector field R on C satisfying i R dα = 0, i R α = 1. Intuition: Think of C as the level set of a Hamiltonian, and R is the Hamiltonian flow inside that level set. I.e. some dynamical system in some fixed energy level. R is uniquely determined by α, but R is not an invariant of ξ. If I replace α with f α for some f : C R \ {0}, the associated Reeb vector field changes a lot. A periodic Reeb orbit of period L is a map R/LZ C tangent to R.

11 Example: Reeb vector field of n 1 dz y j dx j j=1 is z.

12 Let A C N have an isolated singularity at 0 with link L A = A S ɛ as before. Let i : T (A \ {0}) T (A \ {0}) be complex multiplication. Define: ξ A := TL A itl A. Lemma (Varchenko): For all ɛ > 0 small enough, (L A, ξ A ) is a contact manifold and is an invariant of the germ of A at 0 up to contactomorphism.

13 Conjecture (Seidel) If A is normal and (L A, ξ A ) is contactomorphic to (L C n, ξ C n) then A is smooth at 0. Seidel observed that this is true for hypersurface singularities using work by Eliashberg,Gromov,McDuff.

14 Definition of the Conley-Zehnder index Let (C, ξ) be a general contact manifold with ξ = ker(α). Choose a complex structure J on the bundle ξ compatible with the symplectic form dα ξ. We define c 1 (ξ) := c 1 (ξ, J). We will assume H 1 (C; Q) = 0, c 1 (ξ) = 0.

15 These topological conditions tell us that for each periodic Reeb orbit γ, we get an index: CZ(γ) Q called the Conley-Zehnder index. Intuition: CZ(γ) describes how many times the Reeb flow wraps around γ.

16 Let φ t : C C be the Flow of the Reeb vector field R of α. This flow preserves ξ (i.e. Dφ t (ξ) = ξ). The linearized return map of γ : R/LZ C is the natural map Dφ L ξγ(0) : ξ γ(0) ξ γ(l) = ξ γ(0).

17 For simplicity, we will define CZ(γ) under the following conditions: 1. Dφ t ξ is J holomorphic for some compatible almost complex structure J on ξ. 2. Dφ L ξγ(0) = id. 3. c 1 (ξ) = 0. Choose a trivialization of the complex vector bundle γ ξ with complex structure J.

18 Using this trivialization and the above properties, the map t (φ t (ξ)γ(0) ) is viewed as a map from Q : R/LZ U(n 1). We define CZ(γ) to be twice the degree of the map det(q) : R/LZ U(1).

19 Define lsft(γ) := CZ(γ) 1 2 dim ker(dφ L ξ γ(0) id) + (n 3). For any α such that ker(α) = ξ, define the minimal index of α as mi(α) := inf(lsft(γ)). Define the highest minimal index hmi(c, ξ) := sup α mi(α) where the supremum is taken over all α such that ker(α) = ξ.

20 Minimal discrepancy Recall: A is an isolated singularity and L A is its link. Assume c 1 (TA LA ) is torsion. Fact: c 1 (TA LA ) = c 1 (ξ A ). Such a singularity is called numerically Q-Gorenstein. Fix some resolution π : Ã A so that π 1 (0) has smooth normal crossing exceptional divisors E 1,, E l.

21 Define: B ɛ := { z ɛ}, A ɛ := B ɛ A and Ãɛ := π 1 (A ɛ ). Note: Ãɛ = A ɛ = L A.

22 0 c 1 (Ãɛ, L A ; Q) c 1 (Ãɛ; Q) c 1 (L A ; Q) H 1 (L A ; Q) 0 H 2 (Ãɛ, L A ; Q) H 2 (Ãɛ; Q) H 2 (L A ; Q) H 2n 2 (Ã; Q) freely generated by [E j ] So c 1 (Ãɛ, L A ; Q) = i a i[e i ] for unique a i Q.

23 Define a j to be the discrepancy of E j. Define Minimal discrepancy to be { min(aj ) if min(a md(a) = j ) 1 0 otherwise. Minimal discrepancy measures how singular A is at 0. Examples: 1. md(c n ) = n md({x 2 + y 2 + z 2 + w 3 = 0}) = md({x 7 + y 11 + z 13 + w 17 = 0}) =.

24 Theorem: If A is numerically Q-Gorenstein (i.e. c 1 (ξ A ) is torsion) and H 1 (L A ; Q) = 0 then: { 2md(A) if md(a) 0 hmi(l A, ξ A ) = < 0 otherwise.

25 Shokurov s Conjecture (Combined with work from: Boucksom, de Fernex, Favre, Urbinati): If A is numerically Q-Gorenstein with md(a) = n 1 then A is smooth at 0. Corollary. Suppose that Shokurov s Conjecture is true. If A is normal and (L A, ξ A ) = (L C n, ξ C n) then A is smooth at 0. cont. (Markushevich, Reid, Kawamata), Shokurov s conjecture is true in dimension 3. Corollary. For all n 3, if A is normal and (L A, ξ A ) = (L C n, ξ C n) then A is smooth at 0. cont.

26 Proof Easier part: Find some contact form α A associated to ξ A so that: mi(α A ) = 2md(A) This gives us a lower bound form hmi(ξ). Hard part: For every compatible contact form, find a Reeb orbit γ so that: { 2md(A) if md(a) 0 lsft(γ) < 0 otherwise. This gives us a upper bound form hmi(ξ).

27 Proof in the case of cone singularities. Assume A is the cone over a smooth projective X CP N 1. E.g. X = CP n 1, A = C n. Ã = Bl 0 A and let π : Ã A be the blowdown map. We also have the O( 1) bundle P : Ã X. We identify X with the zero section of P.

28 Easier Part: A C N. Define α A := j x jdy j y j dx j LA where z j = x j + iy j. P : Ã X is a Hermitian line bundle O X ( 1) with Hermitain form coming from the standard symplectic form on C N. The Reeb flow uniformly rotates the fibers of P. I.e. φ t (z) = e it (z) (up to a time reparameterization).

29 So through each point p in L A there are Reeb orbits of period 2kπ wrapping k times around X. The lsft index of such an orbit is 2k(a 1 + 1) 2 where a 1 is the discrepancy of X Ã. Hence mi(α A ) = 2a 1 = 2md(A).

30 Sketch of Proof of Hard Part We now start with any contact form α associated to ξ A. We wish to find an orbit γ with the right bound on its index. Compactify à to A = P(C O X ( 1)) and let π : A X be the natural map. Let [F ] H 2 (A) be the class of the fiber of π, then GW [F ],0 ([pt]) 0. Hence for any compatible almost complex structure J on A, there is a J-holomorphic curve: u J : P 1 A representing [F ].

31 We now deform the symplectic form on A through symplectic forms to a new symplectic form ω so that we have an embedding ι : C A so that ι ω = dα. We now choose a family J i s compatible with ω which stretch along C. The associated u Ji s break and their ends converge to Reeb orbits γ 1,, γ k. Simple Example: X is a point, so A = CP 1. Our degeneration is {x 2 + y 2 = t} CP 2 as t 0. The complex structure here stretches along the equator RP 1.

32

33 Schematic Picture

34 The space of such broken maps u converging to γ 1,, γ k has dimension given by a formula involving the discrepancy and Reeb orbits. This gives us an inequality: 2a 1 j lsft(γ j ) 0 proving the hard part of the theorem.

35 Further directions What other parts of the resolution can we recover? E.g. Information from the dual graph? other invariants such as Log Canonical Threshold? Some of the holomorphic curves involved look like arcs. What is the relationship between these curves and the (short) arc space? Secretly our proof is showing that a group called Contact Homology has lowest non-zero degree equal to md(a) or is < 0 depending on the sign of md(a). What is the relationship between this group and the singularity?