The author obtains a complete classification of the symplectic fillings of the lens space \((L(p, q), \bar\xi_{st})\) up to orientation preserving diffeomorphisms, where \(\bar\xi_{st}\) is the contact structure induced from the standard structure \(\xi_{st}\) on the \(3\)-sphere \(S^3\). We assume that \(p>q\geq1\) are coprime integers. The rational number \(p/(p-q)\) is uniquely expressed as a continued fraction \([b_1,\dots, b_k]\) with \(b_1,\dots, b_k\geq2\). Let \(\mathbb Z_{p,q}\subset\mathbb Z^k\) be the set of \(k\)-tuples of non-negative integers \(\mathbf n=(n_1,\dots, n_k)\) such that each of the denominators appearing in the continued fraction \([n_1,\dots, n_k]\) is positive and \([n_1,\dots, n_k]=0\). Let \(N(\mathbf n)\) be the closed oriented \(3\)-manifold given by surgery on \(S^3\) along a certain framed link \(L=\bigcup L_i\) of \(k\) components with Euler numbers \(n_1,\dots, n_k\) respectively. By the condition \([n_1,\dots, n_k]=0\), there exists an orientation preserving diffeomorphism \(\varphi: N(\mathbf n) \to S^1\times S^2\). Let \(\mathbf L\subset N(\mathbf n)\) be the thick framed link obtained by adding \(b_i-n_i\) meridian framed links of Euler number \(-1\) to each component \(L_i\) of \(L\). Define \(W_{p,q}(\mathbf n)\) to be the smooth \(4\)-manifold with boundary obtained by attaching \(2\)-handles to \(S^1\times D^3\) along the framed link \(\varphi(\mathbf L)\subset S^1\times S^3\).
The main results are summarized as follows (Theorem 1.1): (1) Let \((W, \omega)\) be a symplectic filling of the contact \(3\)-manifold \((L(p, q), \bar\xi_{st})\). Then there exists \(\mathbf n\in\mathbb Z_{p,q}\) such that \(W\) is orientation preserving diffeomorphic to a smooth blowup of \(W_{p,q}(\mathbf n)\). (2) For every \(\mathbf n\in\mathbb Z_{p,q}\), the \(4\)-manifold \(W_{p,q}(\mathbf n)\) carries a symplectic form \(\omega\) such that \((W_{p,q}(\mathbf n), \omega)\) is a symplectic filling of the contact \(3\)-manifold \((L(p, q), \bar\xi_{st})\). Moreover, there are no classes in \(H_2(W_{p,q}(\mathbf n); \mathbb Z)\) with self-intersection equal to \(-1\). (3) Let \(\mathbf n\in\mathbb Z_{p,q}\) and \(\mathbf n^\prime\in \mathbb Z_{p^\prime,q^\prime}\). Then \(W_{p,q}(\mathbf n)\sharp r\overline{\mathbb{CP}}^2\) is orientation preserving diffeomorphic to \(W_{p^\prime, q^\prime}(\mathbf n^\prime)\sharp s\overline{\mathbb{CP}}^2\) if and only if: (a) \((p^\prime, s)=(p, r)\) and \(4(q^\prime, \mathbf n^\prime)= (q, \mathbf n)\), or (b) \((p^\prime, s)=(p, r)\) and \((q^\prime, \mathbf n^\prime) =(\overline q, \overline{\mathbf n})\), where \(\overline q\) denotes the only integer satisfying \(p>\overline q\geq1, mod\;p\) and \(\overline{\mathbf n}= (n_k,\dots,n_1)\) for \(\mathbf n=(n_1,\dots,n_k)\).
The arguments of the proof are based on Kirby calculus for surgeries. The proof (1) uses a result (Lemma 3.1) of D. McDuff [J. Am. Math. Soc. 3, 679–712 (1990; Zbl 0723.53019)] on the existence of J-holomorphic curves. The proof of (2) uses a Legendrian band connected sum by R. E. Gompf [Ann. Math. 148, 619–693 (1998; Zbl 0919.57012)]. The proof of (3) uses an invariant of spin structures on a closed oriented \(3\)-manifold of R. E. Gompf [loc. cit.] above, and non-existence of \(2\)-spheres of self-intersection \(-1\) in a Stein \(4\)-manifold by the author and G. Matić [Topology Appl. 88, 55–66 (1998; Zbl 0978.53122)].
In the light of the results obtained in this paper, the author conjectures that a Stein filling of \((L(p, q), \bar\xi_{st})\) is diffeomorphic to \(W_{p,q}(\mathbf n)\).