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Slides for Lectures 15 and 16 Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds Complex manifolds and K¨ahlerGeometry Lecture 15 of 16: Introduction to moduli spaces Dominic Joyce, Oxford University September 2019 2019 Nairobi Workshop in Algebraic Geometry These slides available at http://people.maths.ox.ac.uk/∼joyce/ 1 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds Plan of talk: 15 Introduction to moduli spaces 15.1 Moduli spaces in Differential Geometry 15.2 Moduli spaces in Algebraic Geometry 15.3 General questions in moduli space theory 15.4 Deformations of compact complex manifolds 2 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds 15. Introduction to moduli spaces Suppose we want to understand some class C of geometric objects X up to isomorphism (or some weaker kind of equivalence) | for instance, compact complex manifolds diffeomorphic to some fixed real manifold, holomorphic vector bundles on a fixed complex manifold, etc. A common approach is to try and define a moduli space M of such objects X . As a set M = [X ]: X 2 C is just the set of isomorphism classes [X ] of objects X that we want to classify. 3 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds However, usually we want M to have more geometric structure than this. For example, M could be a topological space, or a manifold (real or complex), or some kind of singular complex manifold in algebraic geometry { a variety or scheme { or something worse, e.g. an Artin stack. The rule is that the geometric structure on M must reflect the behaviour of families of the objects X under study. For example, suppose we have a notion of limits in C, that is, when 1 a sequence (Xi )i=1 has Xi ! X1 in C as i ! 1. Then we would require the topology T on M to satisfy [Xi ] ! [X1] in M whenever Xi ! X1 in C, and we could define T to be the strongest topology on M such that this is always true. 4 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds 15.1. Moduli spaces in Differential Geometry Moduli spaces in differential geometry usually involve a quotient of an infinite-dimensional (often singular) \manifold" by an infinite-dimensional group. Here are two examples from complex geometry. Example 15.1 Let (X ; J) be a compact complex manifold, and E ! X a complex vector bundle. We wish to study the moduli space ME of holomorphic vector bundles (F ; @¯F ) on X whose underlying complex vector bundle is isomorphic to E. Choose a fixed @¯-operator @¯0 on E, with (0,2)-curvature 0;2 ¯2 ¯ F0 = @0 . Given a holomorphic vector bundle (F ; @F ), we can choose an isomorphism ι : E ! F , and this identifies @¯F with 1 0;1 @¯0 + A for some A in C End(E) ⊗ Λ X . 5 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds Example (Continued) As (F ; @¯F ) is holomorphic we have 0;2 0;2 F = F + @¯0A + A ^ A = 0: @¯0+A 0 Different choices ι, ι0 of ι yield A; A0 in the same orbit of the infinite-dimensional gauge group G := C 1(Aut(E)). It follows that ∼ 1 0;1 ME = A 2 C End(E) ⊗ Λ X : (15.1) 0;2 ¯ F0 + @0A + A ^ A = 0 =G: From (15.1) we can define a topology on ME , and some kind of singular manifold structure. 6 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds Example 15.2 Let X be a fixed compact 2n-manifold. We wish to study the moduli space MX of complex manifolds (Y ; J) such that Y is diffeomorphic to X . For such (Y ; J), we can choose a diffeomorphism ι : X ! Y , and then I = ι∗(J) is a complex structure on X , that is, I 2 C 1(TX ⊗ T ∗X ) with I 2 = − id and Nijenhuis tensor NI ≡ 0. Different choices ι, ι0 of ι yield I ; I 0 in the same orbit of the infinite-dimensional diffeomorphism group Diff(X ). Hence ∼ 1 ∗ 2 MX = I 2 C TX ⊗ T X : I = − id; NI = 0 = Diff(X ): For example, if X is an oriented 2-manifold of genus g > 1 then MX is Riemann's moduli space Rg . It is a manifold of dimension 3g − 3, with mild singularities (in fact, it is a nonsingular orbifold). 7 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds 15.2. Moduli spaces in Algebraic Geometry In algebraic geometry, one takes a different approach. Here, we study a moduli space M of algebraic objects, such as complex projective manifolds, and the goal is to give M an algebraic geometric structure { most often that of a scheme. I'm not going to define schemes. Loosely, a C-scheme is a geometric space locally n modelled on the zeroes of finitely many polynomials in C . They form a category SchC. Smooth C-schemes are complex manifolds. It is usually not feasible to write algebro-geometric moduli spaces as quotients by infinite-dimensional gauge groups (though often, in a more complicated way, they are written as quotients by finite-dimensional algebraic groups). Instead, moduli spaces are defined to satisfy a universal property expressed in terms of the category of C-schemes SchC. 8 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds Let C be a category of algebraic objects E we want to form a moduli space for. Then we must define a notion of family of objects ES in C over a base scheme S, thought of as Es 2 C for points s 2 S, varying algebraically with s. For example, if C is holomorphic vector bundles over a fixed complex projective manifold X , then a family of objects in C over S is a vector bundle over X × S. Such families form a category CS . If φ : S ! T is a morphism of schemes, then we should have a ∗ pullback functor φ : CT !CS . 9 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds The nicest kind of moduli space, a fine moduli scheme for C, is a scheme M such that for any C-scheme S, there is a 1-1 correspondence between isomorphism classes [ES ] of objects in CS and morphisms [ES ] : S !M in SchC, such that if φ : S ! T is ∗ a morphism in SchC and ET 2 CT then [φ (ET )] = [ET ] ◦ φ. In this case, id : M!M corresponds to a universal family UM in CM, a family of objects Um in C for m 2 M, such that every E in C is isomorphic to Um for unique m 2 M. Coarse moduli schemes satisfy a weaker universal property. 10 / 44 Dominic Joyce, Oxford University Lecture 15: Introduction to moduli spaces Moduli spaces in Differential Geometry Introduction to moduli spaces Moduli spaces in Algebraic Geometry Deformation theory for compact complex manifolds General questions in moduli space theory Deformations of compact complex manifolds 15.3. General questions in moduli space theory In any moduli problem, one can ask a number of general questions: (A) Existence. Does a moduli space exist in some class of spaces, e.g. is there a fine moduli scheme? (B) Local properties. Fix an object E in C. What does M look like in a small neighbourhood of [E], e.g. is it a manifold of some dimension? The study of (B) is called deformation theory. (C) Global properties. Is the moduli space M compact, Hausdorff, etc.? (D) Compactification. If M is not compact, is there a natural compactification M? If so, what do points of M n M parametrize? (E) Explicit description. Can we describe M completely, as a topological space/scheme/manifold? In a few very nice cases { moduli spaces of Riemann surfaces or K3 surfaces, for instance { we can answer all these questions.
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