Symplectic log Calabi–Yau surface: deformation class

Abstract

We study the symplectic analogue of log Calabi-Yau surfaces and show that the symplectic deformation classes of these surfaces are completely determined by the homological information.Theorem 1.4. Let (X i , D i , ω i ) be symplectic log Calabi-Yau surfaces for i = 1, 2. Then (X 1 , D 1 , ω 1 ) is (resp. strictly) symplectic deformation equivalent to (X 2 , D 2 , ω 2 ) if and only if they are (resp. strictly) homological equivalent.Moreover, the symplectomorphism in the (resp. strict) symplectic deformation equivalence has same homological effect as the (resp. strict) homological equivalence.We remark that when D is a smooth divisor, the relative Kodaira dimension κ(X, D, ω) was introduced in [14] and it was noted there that this notion could be extended to nodal divisors. With this extension understood, symplectic Calabi-Yau surfaces have relative Kodaira dimension κ = 0 (cf. Theorem 3.28 in [14]). Moreover, Theorem 1.4 is also valid when κ(X, D, ω) = −∞. This will be treated in the sequel. Coupled with the techniques developed in [11], [12], some applications to Stein fillings will also be treated in the sequel.The paper is organized as follows. In Section 2 we introduce marked divisors and establish the invariance of their deformation class under blow-up/down in Proposition 2.10. This reduces Theorem 1.4 to the minimal cases. In Section 3, we classify the deformation classes of minimal models and finish the proof of Theorem 1.4.The authors benefit from discussions with Mark Gross, Paul Hacking and Sean Keel.

Geometry of symplectic log Calabi–Yau pairs

“…In this section we review some homological facts about topological divisors, especially cycles of spheres, and we refer to [20], [6] and [11] for details. We first introduce a pair of basic operations for topological divisors.…”

“…[16]). The following is the main result in [14]. Let us explain the various equivalence notions in the theorem (See [27] for a thorough discussion of equivalence notions for symplectic manifolds).…”

“…A sequence (s i ) of integers is said to be anti-canonical if it is realized as S(D) for a symplectic log Calabi-Yau pair (X, D, ω). Combined with Theorem 3.1 in [4], we obtain There is an algorithm to write down the anti-canonical sequences, starting from the list of minimal pairs and reverse the minimal reduction process in [14]. It is interesting to compare anti-canonical sequences with spherical circular sequences.…”

“…An anti-canonical sequence (s i ) is said to be rigid if, for any cycle of symplectic spheres D ⊂ (X, ω) with S(D) = (s i ) and (X − D, ω) minimal, (X, D, ω) is a symplectic log Calabi-Yau pair. Theorem 1.5 (Anti-canonical sequences, [11]). Each spherical circular sequence with b + = 1 is anti-canonical, and each anti-canonical sequence with b + = 1 is rigid.…”

“…Theorem 1.6 (Contact trichotomy, [11]). Let (X, D, ω) be a symplectic log Calabi-Yau pair, Q D the intersection matrix of D and (s i ) the self intersection sequence.…”