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Denis Auroux studies symplectic geometry, with work linking geometric constructions to topology and algebraic geometry. His Harvard research profile identifies the specialization.^[ref]

His lectures with Ivan Smith develop descriptions of symplectic manifolds through Lefschetz pencils and branched covers of the projective plane. The resulting invariants combine the algebra of monodromy with geometric information from pseudoholomorphic curves. This creates a bridge between concrete maps and the classification of manifolds.^[ref]

Ivan Smith works on symplectic topology. His Cambridge profile identifies him as a geometer working in that subject.^[ref]

With Denis Auroux, he wrote lectures connecting Lefschetz pencils, branched covers and symplectic invariants. A Lefschetz pencil describes a manifold by a family of surfaces with controlled singular members; blowing up the base points produces a fibration. The approach makes a global geometric object accessible through fibres, singularities and the way the fibres fit together.^[ref]

A Lefschetz fibration is a proper map from an oriented smooth manifold $X^{2n}$ to an oriented surface, with isolated critical points modeled in orientation-compatible complex coordinates byAway from the critical values it is a proper submersion, hence a smooth fiber bundle by the Ehresmann fibration theorem. Properness is needed for this conclusion; the local critical-point model alone cannot ensure that fibers do not escape to infinity.^[ref]

In real dimension four, a regular fiber is a compact surface. Near a critical value, a circle in that surface shrinks to a node. Blowing up the base points of a Lefschetz pencil supplies a central family of examples: each exceptional sphere becomes a section of the resolved fibration.

On a closed oriented smooth four-manifold $X$, a Lefschetz pencil is a smooth map $f:X\setminus B\to {\mathbb{CP}}^1$ with finite base locus $B$ and isolated critical points. Orientation-compatible complex local coordinates give the modelsA regular fiber closes to a surface through the base points. Blowing up those points resolves the local ratio and gives a Lefschetz fibration, with an exceptional section for each base point.^[ref]

For a pencil generated by homogeneous polynomials $F_0,F_1$ on a smooth projective surface, $f=[F_0:F_1]$ and $B$ is their common zero set. Generic choices in sufficiently positive linear systems give the isolated complex Morse singularities required here. The local models are stronger conditions than merely having a rational map to the projective line.

Let $X$ be a smooth projective surface over the complex numbers, and let $H$ be an ample divisor. For sufficiently large integers $m$, a general pencil in $|mH|$ is a Lefschetz pencil. Its smooth fiber represents $mH$, so the arithmetic adjunction formula on a smooth surface givesIf $m=4k$, then $m^2{H}^2/2=8k^2{H}^2$ and $m{K}_X\cdot H/2=2k{K}_X\cdot H$ are both even integers. Therefore $g$ is odd. Positivity is needed to obtain a generic Lefschetz pencil; divisibility by four is the separate parity argument. The genericity assertion here uses complex algebraic geometry, rather than an unqualified positive-characteristic version of the complex Morse local model.^[ref]

Suppose symplectic manifolds $(M_1^{2n},{\omega }_1)$ and $(M_2^{2n},{\omega }_2)$ contain compact codimension-two symplectic submanifolds $V_1,V_2$. A symplectomorphism $V_1\to V_2$ identifies their restricted forms. To form their symplectic sum, their symplectic normal bundles, viewed as complex line bundles, must have opposite First Chern classes under this identification. Choosing a fiber-reversing normal-bundle identification then permits removal of tubular neighborhoods and gluing of their circle boundaries.^[ref]

The local smoothing resembles $xy=\epsilon$ in the two normal directions. After adjusting the forms near the gluing locus, the joined manifold carries a symplectic form. The identification is part of the construction, so the result is not determined solely by the two ambient manifolds.^[ref]

In real dimension four, the surfaces must have matching genus and symplectic area, and their self-intersection numbers must sum to zero. For square-zero surfaces the normal bundles are trivial, giving the symplectic fiber sum along a square-zero surface.