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Small contractions of symplectic 4-folds. (English) Zbl 1036.14007

From the introduction: Definition 1. A symplectic manifold is defined to be a complex algebraic or analytic manifold \(X\) of dimension \(2n\) which carries a (holomorphic) symplectic 2-form \(\sigma\in H^0(X, \Omega^2_X)\); that is, \(\sigma\) is closed and nondegenerate, which means that \(d\sigma=0\) and \(\sigma^{\wedge n}\) vanishes nowhere.
We want to understand the local structure of birational morphisms of projective symplectic manifolds. For this purpose we introduce:
Definition 2. Let \(\varphi:X\to Y\) be a birational projective morphism (of complex algebraic varieties or complex analytic spaces), where \(X\) is a symplectic manifold and \(Y\) is normal. We say that \(\varphi:X\to Y\) is a symplectic contraction.
We note that \(K_X= \varphi^*K_Y\), so \(\varphi\) is a crepant contraction in the sense of the minimal model program.
Let \(X\) be a symplectic manifold of dimension 4, and suppose that \(X\) contains a \(\mathbb{P}^2\). Let \(\beta:Z\to X\) be the blow-up of \(X\) along \(\mathbb{P}^2\). In an analytic neighborhood of the exceptional divisor of \(\beta\) the manifold \(Z\) is isomorphic to the cotangent bundle of \(\mathbb{P}^2\) blown up along the zero section. Thus there exists another blow-down map \(\beta':Z\to X'\), where \(X'\) is again a symplectic manifold with a symplectic form \(\sigma'\) that coincides with \(\sigma \) outside the exceptional locus of the transformation. We note that although the above arguments are performed in the analytic category, their algebraic counterpart holds whenever one assumes that the \(\mathbb{P}^2\) in question is an isolated positive-dimensional fiber of a symplectic contraction \(\varphi:X\to Y\).
Definition 3. The birational map \(\varphi:X\to X'\) constructed above is called the Mukai flop.
Main result:
Theorem 1.1. Let \(\varphi:X\to Y\) be a symplectic contraction with \(Y\) quasiprojective. Suppose that \(\dim X=4\) and that \(\varphi\) is small (i.e., that it does not contract a divisor). Then \(\varphi \) is locally analytically isomorphic to the collapsing of the zero section in the cotangent bundle of \(\mathbb{P}^2\), and therefore it admits a Mukai flop.
Using theorem 1.1 and the minimal model program and standard arguments providing termination of log-flips, one gets the following theorem:
Theorem 1.2. Let \(\varphi:X\to X'\) be a birational map of two smooth projective 4-folds which is an isomorphism in codimension 1. Suppose that \(X\) is symplectic. Then \(X'\) is symplectic as well and \(\varphi\) can be factorised into a finite sequence of Mukai flops.

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps
32Q57 Classification theorems for complex manifolds
14J35 \(4\)-folds
53D35 Global theory of symplectic and contact manifolds
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