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87

Nov 14, 2013
11/13

by
Denis Auroux

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Goal of the class: this is not always going to be the most rigorous class, but the goal is to tell the story of mirror symmetry.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1961

55
55

Sep 18, 2013
09/13

by
Denis Auroux

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The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their...

Source: http://arxiv.org/abs/0803.2734v1

72
72

Nov 14, 2013
11/13

by
Denis Auroux

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1. Lagrangian Floer Homology (contd) Recall �rst our approaches to CF _(L, L) with the A� algebraic structure:

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/1967

88
88

Nov 14, 2013
11/13

by
Denis Auroux

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1. The Quintic (contd.) Recall that we had a quintic mirror family...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1962

58
58

Nov 14, 2013
11/13

by
Denis Auroux

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Last time, we say that a deformation of (X, J) is given by...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1979

283
283

Nov 14, 2013
11/13

by
Denis Auroux

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An important application of the higher partial derivatives is that they are used in partial di_erential equations to express some laws of physics which are basic to most science and engineering subjects. In this section, we will give examples of a few such equations. The reason is partly cultural, so you meet these equations early and learn to recognize them, and partly technical: to give you a little more practice with the chain rule and computing higher derivatives.

Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...

Source: http://www.flooved.com/reader/1811

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216

Nov 14, 2013
11/13

by
Denis Auroux

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However in real-world applications this is frequently not so. Computing partial derivatives then becomes confusing, but it is better to face these complications now while you are still in a calculus course, than wait to be hit with them at the same time that you are struggling to cope with the thermodynamics or economics or whatever else is involved.

Topics: Maths, Analysis and Calculus, Analysis, Calculus, Differentiation from ? m to ??, Differentials,...

Source: http://www.flooved.com/reader/1810

80
80

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1963

125
125

Nov 14, 2013
11/13

by
Denis Auroux

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1. Hodge Theory - Theorem 1 (Hodge)....

Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Differential Equations (ODEs &...

Source: http://www.flooved.com/reader/2109

80
80

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2127

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93

Nov 14, 2013
11/13

by
Denis Auroux

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We now return to the complex K�hler case. Let (M, _, J) be a complex K�hler manifold...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2113

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114

Nov 14, 2013
11/13

by
Denis Auroux

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1. Branched Covers...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2118

294
294

Nov 14, 2013
11/13

by
Denis Auroux

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As we pointed out in the introduction, vectors will be used throughout the course. The basic concepts are straight forward, but you will have to master some new terminology. Another important point we made earlier is that we can view vectors in two di_erent ways: geometrically and algebraically. We will start with the geometric view and introduce terminology along the way.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1826

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102

Nov 14, 2013
11/13

by
Denis Auroux

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1. Homological Mirror Symmetry - Conjecture 1. X, X_ are mirror Calabi-Yau varieties

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1971

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73

Nov 14, 2013
11/13

by
Denis Auroux

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1. Homology and Cohomology - Recall from last time that, for M a smooth manifold, we produced a graded di_erential algebra...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2114

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98

Nov 14, 2013
11/13

by
Denis Auroux

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Last time, we were considering CP1 mirror to ... for ... the latter object is a Landau-Ginzburg model, i.e. a K�hler manifold with a holomorphic function called the �superpotential�. Homological mirror symmetry gave...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1978

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72

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2124

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128

Nov 14, 2013
11/13

by
Denis Auroux

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We now generalize to 3-space the normal form of Green�s theorem (Section V4).

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1814

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98

Nov 14, 2013
11/13

by
Denis Auroux

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De�nition. A domain D in 3-space is simply-connected if each closed curve in it can be shrunk to a point without ever getting outside of D during the shrinking.

Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Topology and Metric Spaces, Geometry,...

Source: http://www.flooved.com/reader/1820

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140

Nov 14, 2013
11/13

by
Denis Auroux

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Linear algebra is essentially about solving systems of linear equations, an important application of mathematics to real-world problems in engineering, business, and science, especially the social sciences. Here we will just stick to the most important case, where the system is square, i.e., there are as many variables as there are equations.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1808

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53

Sep 20, 2013
09/13

by
Denis Auroux

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We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.

Source: http://arxiv.org/abs/math/0204286v1

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102

Nov 14, 2013
11/13

by
Denis Auroux

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In this section and the next we give a di_erent way of looking at Green�s theorem which both shows its signi�cance for �ow �elds and allows us to give an intuitive physical meaning for this rather mysterious equality between integrals.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1823

111
111

Nov 14, 2013
11/13

by
Denis Auroux

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Recall from last time the statement of the following lemma: given L a holomorphic line bundle with curvature...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2115

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97

Nov 14, 2013
11/13

by
Denis Auroux

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1. Existence of Almost-Complex Structures - Let (M, _) be a symplectic manifold....

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2106

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271

Nov 14, 2013
11/13

by
Denis Auroux

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Proof of Stokes' Theorem. We will prove Stokes� theorem for a vector �eld of the form P (x, y, z) k . That is, we will show, with the usual notations, ...

Topics: Maths, Analysis and Calculus, Calculus, Integration Theorems, Stokes�s Theorem, Mathematics

Source: http://www.flooved.com/reader/1819

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64

Nov 14, 2013
11/13

by
Denis Auroux

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1. Derived Fukaya Category - Last time: derived categories for abelian categories (e.g. DbCoh(X)). This time: the derived Fukaya category.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1970

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169

Nov 14, 2013
11/13

by
Denis Auroux

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The criterion for gradient �elds: The curl in space. We seek now to generalize to space our earlier criterion (Section V2) for gradient �elds in the plane.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1817

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123

Nov 14, 2013
11/13

by
Denis Auroux

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Theorems about homogeneous and inhomogeneous systems: On the basis of our work so far, we can formulate a few general results about square systems of linear equations. They are the theorems most frequently referred to in the applications.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1809

76
76

Nov 14, 2013
11/13

by
Denis Auroux

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1. Spin Structures...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2120

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40

Sep 21, 2013
09/13

by
Denis Auroux

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In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.

Source: http://arxiv.org/abs/0902.1595v1

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57

Sep 23, 2013
09/13

by
Denis Auroux

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We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we hope that it can serve as an introduction to the subject, and will stimulate interest in some of the open questions mentioned in the final section.

Source: http://arxiv.org/abs/math/0411233v1

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84

Nov 14, 2013
11/13

by
Denis Auroux

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1. Symplectic Manifolds...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2121

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76

Sep 20, 2013
09/13

by
Denis Auroux

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Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

Source: http://arxiv.org/abs/math/0204285v1

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248

Nov 14, 2013
11/13

by
Denis Auroux

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1. K�hler Geometry

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2108

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68

Nov 14, 2013
11/13

by
Denis Auroux

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1. Pseudoholomorphic Curves ...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1980

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105

Nov 14, 2013
11/13

by
Denis Auroux

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1. Homeomorphism Classification of Simply Connected Compact 4-Manifolds...

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Geometry, Geometry of Manifolds, Complex...

Source: http://www.flooved.com/reader/2119

76
76

Nov 14, 2013
11/13

by
Denis Auroux

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Let (M, _, J) be a compact K�hler manifold, .... Then we can �nd a line bundle L_M with �rst Chern class ...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2112

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60

Sep 24, 2013
09/13

by
Denis Auroux

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We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with f^0_g they become isomorphic. As a consequence, any two...

Source: http://arxiv.org/abs/math/0412120v2

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4.0

Jun 30, 2018
06/18

by
Denis Auroux

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We construct infinitely many families of monotone Lagrangian tori in $\mathbb{R}^6$, no two of which are related by Hamiltonian isotopies (or symplectomorphisms). These families are distinguished by the (arbitrarily large) numbers of families of Maslov index 2 pseudo-holomorphic discs that they bound.

Topics: Symplectic Geometry, Mathematics

Source: http://arxiv.org/abs/1407.3725

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158

Nov 14, 2013
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by
Denis Auroux

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In this section, we consider the most common case � �nding a line which goes approximately through a set of data points.

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1806

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71

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds intersecting transversely.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1965

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89

Nov 14, 2013
11/13

by
Denis Auroux

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1. Deformations of Complex Structures An (almost) complex structure (X, J) splits the complexi�ed tangent and (wedge powers of) cotangent bundles as...

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1972

81
81

Nov 14, 2013
11/13

by
Denis Auroux

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1. Integrability of Almost-Complex Structures...

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2107

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94

Nov 14, 2013
11/13

by
Denis Auroux

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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2126

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102

Nov 14, 2013
11/13

by
Denis Auroux

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The Hodge decomposition stated last time places strong constraints on H* of K�hler manifolds, e.g. dim H^k is even for k odd because C conjugation gives isomorphisms ... (note that this is false for symplectic manifolds in general).

Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics

Source: http://www.flooved.com/reader/2111

82
82

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds intersecting transversely.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1966

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48

Sep 19, 2013
09/13

by
Denis Auroux

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This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the...

Source: http://arxiv.org/abs/math/0304113v1

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73

Sep 17, 2013
09/13

by
Denis Auroux

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After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

Source: http://arxiv.org/abs/math/0007130v1

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146

Nov 14, 2013
11/13

by
Denis Auroux

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Polar Coordinates Polar coordinates are a di_erent way of describing points in the plane. The polar coordinates (r, _) are related to the usual rectangular coordinates (x, y) by by x = r cos _, y = r sin _

Topics: Maths, Analysis and Calculus, Calculus, Mathematics

Source: http://www.flooved.com/reader/1805

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99

Nov 14, 2013
11/13

by
Denis Auroux

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0.1. Coherent Sheaves on a Complex Manifold (contd.) - Let X be a com_plex manifold, OX the sheaf of holomorphic functions on X.

Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics

Source: http://www.flooved.com/reader/1968