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Symplectic vector space
About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.
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TL;DR: In this article, the authors follow an analogous approach for compact complex Lagrangian submanifolds of a complex symplectic manifold, where the natural geometric structure on the moduli space is a special K\"ahler metric.
Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.
89 citations
TL;DR: In this paper, it was shown that the symplectic 4-manifold Z is minimal unless either one of the Xi contains a ( 1)-sphere disjoint from Fi, or one of them admits a ruling with Fi as a section.
Abstract: Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X1 and X2 along the Fi. Using relative Gromov-Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the Xi contains a ( 1)-sphere disjoint from Fi; or one of the Xi admits a ruling with Fi as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal. Let (X1,!1), (X2,!2) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy (F1) 2 + (F2) 2 = 0, with the !i normalized to give equal area to the surfaces Fi. For i = 1,2, a neighborhood of Fi is symplectically identified by Weinstein's symplectic neighborhood theorem (19) with the disc normal bundlei of Fi in Xi. Choose a smooth isomorphismof the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2. According to (2) (and independently (11)), the symplectic sum
88 citations
TL;DR: In this article, it was shown that the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves, which is a symplectic form.
88 citations
87 citations
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TL;DR: In this article, the authors follow an analogous approach for compact complex Lagrangian submanifolds of a complex symplectic manifold, where the natural geometric structure on the moduli space is a special K\"ahler metric.
Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.
87 citations