 
Summary: LUTTINGER SURGERY ALONG LAGRANGIAN TORI AND
NONISOTOPY FOR SINGULAR SYMPLECTIC PLANE CURVES
D. AUROUX, S. K. DONALDSON, AND L. KATZARKOV
Abstract. We discuss the properties of a certain type of Dehn surgery along
a Lagrangian torus in a symplectic 4manifold, known as Luttinger's surgery,
and use this construction to provide a purely topological interpretation of a
nonisotopy result for symplectic plane curves with cusp and node singularities
due to Moishezon [9].
1. Introduction
It is an important open question in symplectic topology to determine whether,
in a given symplectic manifold, all (connected) symplectic submanifolds realizing a
given homology class are mutually isotopic. In the case where the ambient manifold
is Kšahler or complex projective, one may in particular ask whether symplectic
submanifolds are always isotopic to complex submanifolds.
The isotopy results known so far rely heavily on the theory of pseudoholomorphic
curves and on the Gromov compactness theorem [7]. The best currently known re
sult for smooth curves is due to Siebert and Tian [11], who have proved that smooth
connected symplectic curves of degree at most 17 in CP2
, or realizing homology
classes with intersection pairing at most 7 with the fiber class in a S2
