Symplectic geometry can be thought of as an "interface" between topics. Historically, comes from Hamiltonian dynamics; also ties to differential topology, algebraic geometry, representation theory Representation theorists look at very special symplectic manifolds (with lots of symmetry); motivation comes from quantum theory Manifold topologists are mostly interested in weird manifolds which carry symplectic structures. We'll be looking at algebraic geometry... Chapter I: Introduction 1. Toric manifolds and amoebas 2. Stein manifolds and plurisubharmonic functions 3. Lefschetz pencils Postscript: using 1 and 3, SYZ fibrations Textbooks: McDuff+Salamon, Introduction to Symplectic Topology Griffiths+Harris, Principles of Algebraic Geometry More specific stuff... Let $p(x,y)=\sum_{m+n \leq d} a_{mn} x^m y^n$ be a 2-variable polynomial with generic coefficients. Consider $M=\{(x,y) \in \C^2 | p(x,y)=0\}$. What is its topology? If $d=1$, we have a sphere with a point deleted If $d=2$, a sphere with 2 points deleted (consider the equation $xy-1=0$) If $d=3$, a torus with 3 points deleted In general, $M$ is diffeomorphic to $\Sigma_{g,n}$, a genus $g$ surface with $d$ deleted points. $n=d$, $g=(d-1)(d-2)/2$. To prove this, use the adjunction formula. Consider $X=\overline M \subset \C P^2$; then $X=\{\sum_{m,n} a_{mn} x^my^nx^{d-m-n}=0\}$ Generically, $X$ is smooth, and intersects the line at $\infty$ transversely. Why is this a generic condition? It defines a Zariski-open (and hence dense) set. Now, we use the adjunction formula. We have a sequence of vector bundles $0 \to TX \to T\CP^2|_X \to \nu_X \to 0$. To compute the Euler characteristic of $X$, we note $\chi(X)=\int_X c_1(TX)=\int_X c_1(T \CP^1|_X)-c_1(\nu_X) =3d-d^2$ So we can compute the genus. The number of missing points is the number of points at which it intersects the line at $\infty$, and hence the degree. Fact: $X$ is connected, and hence $M$ is also. This determines $M$. This is not the point of view we'll be taking. Why? It fails in higher dimensions. Additionally, it fails miserably for symplectic manifolds, where we don't have any sort of classification theorem. Alternative approaches, avoiding the classification theorem: Realize that the topology of $M$ is a priori independent of the choice of coefficients, as long as they are generic. Why? The space of generic things is connected (as the hypersurface of non-generic things has real codimension 2). This applies to $X$, which is closed. $M$ is $X$ with points deleted; those points vary smoothly. Note that we're using Ehrissman's theorem (the triviality of differentiable fibrations over the interval) and the fact that you can "move points around" (a trivial case of the Thom isotopy theorem). There are analogues of these theorems in the symplectic context... Why is this important? We could pick a specific surface, but that also gets harder as we go up in dimension. Hurwitz' theory of branched covers. Consider the projection $\pi_M:M \to \C$. This extends to a map $\pi_X:X \to \C P^1$, sending the missing points to $\infty$. Generically (but less generically than before), $\pi_X$ will be a simple branched cover: that is to say, it will look locally like either $z \mapsto z+c$ (regular) or $z \mapsto z^2+c$ (critical); moreover, each fiber $\pi^{-1}(c)$ contains at most one critical point; the fiber at $\infty$ contains only regular points. Why can we assure this? The critical points are those points where $f(x,y)=0$ and $\frac{\partial f}{\partial y}=0$. This is a pair of polynomial equations, of degrees $d$ and $d-1$; generically, there are $d(d-1)$ points of this sort. If a higher-order thing were to be zero, that would be a third equation in two unknowns, and so generically it doesn't happen. What's the preimage of a point under $\pi_M$? Generically, a point has $d$ preimages. At the critical points, two points coincide. So both $\pi_X$ and $\pi_M$ are $d$-sheeted branched covers, with $d(d-1)$ branch points. From this, we can easily compute the Euler characteristic. But we don't want to do that! Instead, we can analyze the topology of a branched covering as follows: $\pi_M:M \to \C$, with some number of critical values (namely d(d-1)). Choose a base point $z_0$, identify $\pi_M^{-1}(z_0)$ with $\{1,2,\dots,d\}$. If we choose a path that collides with a critical value, 2 points will coincide; if we choose a loop near that path that just goes around that critical value, it interchanges two points, and hence corresponds to a transposition $\tau \in S_d$. Choose a "basis of paths" from $z_0$ to the critical values; then we get transpositions $\tau_1,\dots,\tau_{\kappa}$. If we compose all of them, we get the identity (since it's the same as going in a big loop around all values, i.e., a small loop around the regular value $\infty$). Def'n: A Hurwitz word is a word $\id=\tau_1\tau_2\dots\tau_{\kappa} \in S_d$. A priori, we don't know the transpositions... Changing the paths gives you a so-called Hurwitz move. The standard Hurwitz move: $\tau_1\dots\tau_i \tau_{i+1} \dots \tau_k \to \tau_1 \dots (\tau_i\tau_{i+1}\tau_i)\tau_i \dots \tau_k$ (so we're exchanging $\tau_i$ and $\tau_{i+1}$ and conjugating $\tau_{i+1}$ by $\tau_i$). This corresponds, geometrically, to switching the paths and making the new $i^{\rm th}$ path go the other way around the new $(i+1)^{\rm st}$ point. Additionally, we can globally conjugate; geometrically, we're just relabeling the points. The classification of branched covers therefore reduces to the classification of Hurwitz words up to Hurwitz morphism. Thm: Hurwitz By Hurwitz moves and global conjugation, every (transitive; i.e., the transpositions can exchange any 2 sheets) Hurwitz word can be brought into the {\em Hurwitz normal form} (12)(23)...(d-1,d)(d-1,d)...(12) [(12)]^{2k} for some $k$. (First bit: the basic part; second bit: the tail) Claim: this actually determines the topology of $M$. How? If we choose our paths appropriately, we can divide our manifold into a bit corresponding to the basic piece and $k$ bits corresponding to the tail. The bit corresponding to the basic piece is a $d$-sheeted cover with $2d-2$ critical points; it therefore suffices to find some such branched cover. (since by Hurwitz's theorem they're all equivalent) To do so, map $\C P^1$ to itself by a polynomial of degree $d$ (and restrict your life to a large ball). So the piece over the basic part is a sphere with $d$ discs removed (the boundary of each disc is the boundary of the entire large ball). The piece over each bit of the tail consists of $d-2$ trivial sheets and 2 sheets that combine to form an annulus Glue things together; the trivial sheets don't do anything topologically, the other ones get pieces of their boundaries glued to them, etc... This gives a complete topological description of $M$. Thm: "something I find truly bizarre, and I still haven't fully recovered from the shock of this" Recall $p(x,y)=\sum a_{mn} x^m y^n$. So we can think of each term as lying at the integral points... Let $a_{mn}=t^{w(m,n)}$, where we think of $t$ as being small. $w(m,n)$ is given by a table like 25 16 9 ... 4 1 3 7 0 1 4 9 16 25 extending over a simplex. Consider $M=M_t$, $\mu:M_t \to \R^2$, by $(x,y) \mapsto (\frac{|x|^2}{1+|x|^2+|y|^2}, \frac{|y|^2}{1+|x|^2+|y|^2})$. Then $\mu(M)$ lies in the basic simplex in $\R^2$. Moreover, if we look at the $d$-fold subdivision of this simplex, the intersection of $\mu(M)$ with each of the $d^2$ subsimplices will for sufficiently small $t$ be a "blob" intersecting all 3 sides; the preimage of this blob will be a "pair of pants". What happens along the edges? We cap the pants off along the horizontal and vertical edges; along the diagonal edge, we're going to $\infty$, and so the pants are not capped off.