By various, Huai-Dong Cao (Lehigh University), Shing-Tung Yau (Harvard University)

ISBN-10: 1571461388

ISBN-13: 9781571461384

Contents exact Lagrangian fibrations, wall-crossing, and replicate symmetry (Denis Auroux) Sphere theorems in geometry (Simon Brendle and Richard Schoen) Geometric Langlands and non-Abelian Hodge conception (Ron Donagi and Tony Pantev) advancements round confident sectional curvature (Karsten Grove) Einstein metrics, four-manifolds, and conformally KÃ¤hler geometry (Claude LeBrun) lifestyles of Faddeev knots (Fengbo hold, Fanghua Lin, and Yisong Yang) Milnor K2 and box homomorphisms (Fedor Bogomolov and Yuri Tschinkel) Arakelov inequalities (Eckart Viehweg) A survey of Calabi-Yau manifolds (Shing-Tung Yau)

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**Additional resources for Surveys in Differential Geometry (Volume 13): Geometry, Analysis and Algebraic Geometry**

**Sample text**

The formulation which is the most relevant to us is the one discussed by Abouzaid in [1]: in this version, one considers Lagrangian sub manifolds of XV with boundary contained in the given reference fiber D V = W-1(e t ), and which near the reference fiber are mapped by W to an embedded curve C. 3. +. Floer theory is then defined by choosing a specific set of Hamiltonian perturbations, which amounts to deforming the given admissible Lagrangians so that their phases are in increasing order, and ignoring boundary intersections.

EVk and pushing forward their product by integration along the fibers of evo. This setup allows us to "smudge" incidence conditions by replacing the integration current on a submanifold Ci by a smooth differential form supported in a tubular neighborhood. Given bE CPl(L, L), Fukaya-Oh-Ohta-Ono [14] deform the Aoo-algebra structure on the Floer complex by setting We will actually restrict our attention to the case where b is a cycle, representing a class [b] E Hl(L) (or, dually, in H n -l(L)). 2:0, Ai --+ +oo}.

Denote by (XV, W) the mirror to X, and by D V the mirror to D, which we identify symplectically with a fiber of W, say D V = {W = et } C XV for fixed t » O. First we need to briefly describe the Fukaya category of the LandauGinzburg model W : XV -t C. The general idea, which goes back to Kontsevich [27] and Hori-Iqbal-Vafa [22], is to allow as objects admissible Lagrangian submanifolds of Xv; these can be described either as potentially non-compact Lagrangian submanifolds which, outside of a compact subset, are invariant under the gradient flow of -Re(W), or, truncating, as compact Lagrangian submanifolds with (possibly empty) boundary contained inside a fixed reference fiber of W (and satisfying an additional condition).

### Surveys in Differential Geometry (Volume 13): Geometry, Analysis and Algebraic Geometry by various, Huai-Dong Cao (Lehigh University), Shing-Tung Yau (Harvard University)

by Edward

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