\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: Symmetric products in symplectic geometry

Symmetric products in symplectic geometry

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definition 0.0.1

Let \(X\) be a topological space. The \(k\)-symmetric product is the space \[\Sym^k(X):=X^k/S_k\] where \(S_n\) is the symmetric group on \(k\)-generators which acts by permutation on the coordinates of \(\Sym^k(X)\). In general, the symmetric product of a manifold does not have the structure of a manifold (as the symmetric group does not act freely on the diagonal). However, when \(X\) is a complex curve (real surface) we are able to equip \(\Sym^k(X)\) with the structure of a smooth complex manifold.

theorem 0.0.2

Let \((\Sigma, \jmath)\) be a complex curve. Then \(\Sym^n(\Sigma)\) is a complex manifold, and the map \(\Sigma^k\to \Sym^k(\Sigma)\) is holomorphic. We prove this in the setting where \(\Sigma=\CC\) (this will serve as a local model for the general setting). Consider the space of polynomials of degree \(n\) with leading coefficient 1, \[\CC[z]_n:=\{z^n+a_{n-1}z^{n-1}+\cdots + a_1z+a_0\st a_i\in \CC\}\] There is a map from \(\CC[z]_n\to \Sym^n(\CC)\) which sends each polynomial to its (unordered) set of zeros. This map is a bijection as every set of \(n\) points (with possible repetition) in \(\CC\) uniquely determines a degree \(n\) polynomial with leading coefficient 1. The complex structure comes from identifying \(\CC[z]_n = \CC^n\).

example 0.0.3

We identify the symmetric product \(\Sym^2(\CP^1)\) with \(\CP^2\). To each point in \(\CP^2\) we can associate a degree 2 homogenous polynomial in 2-variables: \[(z_0:z_1:z_2)\mapsto z_0 s^2 + z_1 st + z_2 t^2\] We then factor the polynomial as \[( z_0 s^2 + z_1 st + z_2 t^2)=(y_0 s + y_1 t)\cdot (x_0 s + x_1 t)\] This gives us a bijection between the points of \(\CP^2\) and \(\Sym^2(\CP^1)\). \[(z_0:z_1:z_2)\mapsto [(x_0:x_1), (y_0,y_1)]\] In figure 0.0.4 the moment polytope of \(\CP^1\times \CP^1\) is the square given by \[\text{Convex Hull}((0,0), (0,1), (1, 0), (1, 1)),\] while the moment polytope of \(\CP^2\) is given by the convex hull of \[\text{Convex Hull}((0,0), (0,1), (1,1)).\] There is map from the first moment polytope to the second (given by ``folding'' along the diagonal) which is 2-to-1 away from the diagonal, allowing us to see the symmetric product on the level of moment polytopes. Along the diagonal, we cannot define a relation between the symplectic form on \(\CP^1\times \CP^1\) and \(\CP^2\).
figure 0.0.4:\(\CP^2\) can be identified with the 2-fold symmetric product of \(\CP^1\). Away from the diagonal, we have symplectic maps.
While \(\Sym^k(\Sigma)\) is a complex manifold it does not come with a canonical choice of symplectic structure, so we are leaving some of the tools that we use to study Lagrangian intersection Floer cohomology behind.