## 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

### Keywords:

symplectic manifold; symplectic contraction; Mukai flops
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### References:

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