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**Standard surfaces and nodal curves in symplectic 4-manifolds.**
*(English)*
Zbl 1151.53073

This paper extends the author’s earlier result that relates the Donaldson-Smith invariants to the Gromov-Taubes invariants for the case of symplectic 4-manifolds.

For a closed symplectic 4-manifold \(( X,\omega ) \) with \([ \omega ] \in H^{2}( X,\mathbb{Z}) \), Donaldson showed that by blowing up along the common zeros of two suitable sections of \(L^{\otimes k}\) for \(k\) large enough, where \(L\) is a line bundle over \(X\) with Chern class \( [ \omega ] \), a symplectic Lefschetz fibration \(f:X^{\prime }\to \mathbb{C}\mathbb{P}^{1}\) is obtained. Donaldson and Smith constructed the relative Hilbert scheme \(X_{r}( f) \), a symplectic manifold with a map \(F:X_{r}( f) \to \mathbb{C}\mathbb{P}^{1}\) whose fiber over a regular value \(t\) of \(f\) is the symmetric product \(S^{r}f^{-1}( t) \). Roughly speaking, \(j\)-holomorphic curves \(C\) (containing no fiber components) in \(X^{\prime }\) for an almost complex structure \(j\) on \( X^{\prime }\) (making \(f\) pseudoholomorphic) give rise to \(\mathbb{J}_{j}\) -holomorphic sections \(s_{C}\) (with \(s_{C}( t) =C\cap f^{-1}( t) \)) of \(X_{r}( f) \) for a corresponding almost complex structure \(\mathbb{J}_{j}\) on \(X_{r}( f) \), and vice versa. The author showed earlier that the Gromov-Taubes invariant \(Gr( \alpha ) \), counting pseudoholomorphic submanifolds of \(X^{\prime }\) which are Poincaré dual to \(\alpha \), agrees with the Donaldson-Smith invariant \(\mathcal{DS}_{f}( \alpha ) \), counting \(J\)-holomorphic sections \(s\) of \(X_{r}( f) \) whose corresponding sets \(C_{s}\) are Poincaré dual to \(\alpha \in H^{2}( X^{\prime },\mathbb{Z}) \) and pass through a generic set of \(d( \alpha ) =\frac{1}{2}( \alpha ^{2}-\kappa _{X^{\prime }}\cdot \alpha ) \) points for a generic almost complex structure \(J\) on \(X_{r}( f) \), where \(\kappa _{X^{\prime }}=c_{1}( T^{\ast }X^{\prime }) \).

In this paper, the author constructs a finer Donaldson-Smith invariant \( \widetilde{\mathcal{DS}}_{f}( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) counting sections \(s\) whose corresponding curves satisfy \(C_{s}=\cup C_{s}^{i}\) with each \(C_{s}^{i}\) Poincaré dual to \(\alpha _{i}\), where \( \alpha =\sum_{i=1}^{n}\alpha _{i}\) for \(\alpha _{i}\) satisfying certain condition. It is shown that when the degree of the fibration is large enough, \(\widetilde{\mathcal{DS}}_{f}( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) equals \(( \sum d( \alpha _{i}) ) !/\prod ( d( \alpha _{i}) !) \) times \(Gr( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) which counts reducible curves with smooth irreducible components and a total of \(\sum \alpha _{i}\cdot \alpha _{j}\) nodes arising from intersections between these components. Furthermore the author introduces an invariant \(\mathcal{FDS}_{f}( \alpha ) \), called the family Donaldson-Smith invariant, and shows that for \(\alpha \) satisfying certain conditions, \(\mathcal{FDS}_{f}( \alpha -2\sum e_{i}) \) equals \(n!\) times \(Gr_{n}( \alpha ) \), which counts, roughly speaking, \(j\)-holomorphic curves Poincaré dual to \( \alpha \) with \(n\) transverse double points and passing through a generic set of \(d( \alpha ) -n\) points. The author also gives a proof of the fact that if \(\pi :X^{\prime }\to X\) is the blowup of an almost complex manifold \(( X,J) \) at a point \(p\in X\), then there is a Lipschitz continuous almost complex structure \(J^{\prime }\) on \(X\) such that \(\pi \) is \(( J^{\prime },J) \)-holomorphic.

For a closed symplectic 4-manifold \(( X,\omega ) \) with \([ \omega ] \in H^{2}( X,\mathbb{Z}) \), Donaldson showed that by blowing up along the common zeros of two suitable sections of \(L^{\otimes k}\) for \(k\) large enough, where \(L\) is a line bundle over \(X\) with Chern class \( [ \omega ] \), a symplectic Lefschetz fibration \(f:X^{\prime }\to \mathbb{C}\mathbb{P}^{1}\) is obtained. Donaldson and Smith constructed the relative Hilbert scheme \(X_{r}( f) \), a symplectic manifold with a map \(F:X_{r}( f) \to \mathbb{C}\mathbb{P}^{1}\) whose fiber over a regular value \(t\) of \(f\) is the symmetric product \(S^{r}f^{-1}( t) \). Roughly speaking, \(j\)-holomorphic curves \(C\) (containing no fiber components) in \(X^{\prime }\) for an almost complex structure \(j\) on \( X^{\prime }\) (making \(f\) pseudoholomorphic) give rise to \(\mathbb{J}_{j}\) -holomorphic sections \(s_{C}\) (with \(s_{C}( t) =C\cap f^{-1}( t) \)) of \(X_{r}( f) \) for a corresponding almost complex structure \(\mathbb{J}_{j}\) on \(X_{r}( f) \), and vice versa. The author showed earlier that the Gromov-Taubes invariant \(Gr( \alpha ) \), counting pseudoholomorphic submanifolds of \(X^{\prime }\) which are Poincaré dual to \(\alpha \), agrees with the Donaldson-Smith invariant \(\mathcal{DS}_{f}( \alpha ) \), counting \(J\)-holomorphic sections \(s\) of \(X_{r}( f) \) whose corresponding sets \(C_{s}\) are Poincaré dual to \(\alpha \in H^{2}( X^{\prime },\mathbb{Z}) \) and pass through a generic set of \(d( \alpha ) =\frac{1}{2}( \alpha ^{2}-\kappa _{X^{\prime }}\cdot \alpha ) \) points for a generic almost complex structure \(J\) on \(X_{r}( f) \), where \(\kappa _{X^{\prime }}=c_{1}( T^{\ast }X^{\prime }) \).

In this paper, the author constructs a finer Donaldson-Smith invariant \( \widetilde{\mathcal{DS}}_{f}( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) counting sections \(s\) whose corresponding curves satisfy \(C_{s}=\cup C_{s}^{i}\) with each \(C_{s}^{i}\) Poincaré dual to \(\alpha _{i}\), where \( \alpha =\sum_{i=1}^{n}\alpha _{i}\) for \(\alpha _{i}\) satisfying certain condition. It is shown that when the degree of the fibration is large enough, \(\widetilde{\mathcal{DS}}_{f}( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) equals \(( \sum d( \alpha _{i}) ) !/\prod ( d( \alpha _{i}) !) \) times \(Gr( \alpha ;\alpha _{1},\dots,\alpha _{n}) \) which counts reducible curves with smooth irreducible components and a total of \(\sum \alpha _{i}\cdot \alpha _{j}\) nodes arising from intersections between these components. Furthermore the author introduces an invariant \(\mathcal{FDS}_{f}( \alpha ) \), called the family Donaldson-Smith invariant, and shows that for \(\alpha \) satisfying certain conditions, \(\mathcal{FDS}_{f}( \alpha -2\sum e_{i}) \) equals \(n!\) times \(Gr_{n}( \alpha ) \), which counts, roughly speaking, \(j\)-holomorphic curves Poincaré dual to \( \alpha \) with \(n\) transverse double points and passing through a generic set of \(d( \alpha ) -n\) points. The author also gives a proof of the fact that if \(\pi :X^{\prime }\to X\) is the blowup of an almost complex manifold \(( X,J) \) at a point \(p\in X\), then there is a Lipschitz continuous almost complex structure \(J^{\prime }\) on \(X\) such that \(\pi \) is \(( J^{\prime },J) \)-holomorphic.

Reviewer: Albert Sheu (Lawrence)

### MSC:

53D35 | Global theory of symplectic and contact manifolds |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

32Q65 | Pseudoholomorphic curves |