Part 2: Pseudo-holomorphic Curves
نویسنده
چکیده
1. Properties of J-holomorphic curves 1 1.1. Basic definitions 1 1.2. Unique continuation and critical points 5 1.3. Simple curves 8 1.4. Adjunction inequality 9 2. Gromov compactness 12 2.1. Gromov compactness theorem 12 2.2. Energy estimate and bubbling 15 2.3. The isoperimetric inequality 19 2.4. Bubbles connect 22 3. Moduli spaces of J-holomorphic curves 25 3.1. The Fredholm setup 25 3.2. Transversality 31 4. Gromov-Witten invariants 35 4.1. Stable maps and axioms of Gromov-Witten invariants 35 4.2. Genus zero invariants of semipositive manifolds 42 4.3. Gromov invariants of symplectic 4-manifolds 45 References 48
منابع مشابه
Pseudo-holomorphic and Algebraic Classifications
Almost all known restrictions on the topology of nonsingular real algebraic curves in the projective plane are also valid for a wider class of objects: real pseudo-holomorphic curves. It is still unknown if there exists a nonsingular real pseudo-holomorphic curve not isotopic in the projective plane to a real algebraic curve of the same degree. In this article, we focus our study on symmetric r...
متن کاملPseudo holomorphic curves in symplectic manifolds
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V whic...
متن کاملSW ⇒ Gr: FROM THE SEIBERG-WITTEN EQUATIONS TO PSEUDO-HOLOMORPHIC CURVES
The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of the main theorem in the announcement [T1]. That theorem, Theorem 4.1, ...
متن کاملInvariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry
We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten [3, 15, 10], we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing t...
متن کاملHolomorphic curves in Exploded Torus Fibrations: Compactness
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the type of limits considered in tropical geometry appear smooth.) In this paper, we prove a compactness theorem for (pseudo)holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov’s com...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011