We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a "near-symplectic" structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

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We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface H in a toric variety V we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of V × C along H × 0, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to H. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

Abstract. We study homological mirror symmetry for Del Pezzo surfaces and their mirror LandauGinzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X k obtained by blowing up CP 2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W k : M k → C with k +3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X k , and give an explicit correspondence between the deformation parameters for X k and the cohomology class [B + iω] ∈ H 2 (M k , C).