# Wednesday, March 03 Find first 5m. :::{.remark} When we considered $\bar{\CP}^2$, we implicitly assumed $T\bar{\CP}^2$ was a complex rank 2 vector bundle with some purported complex structure. ::: :::{.claim} \[ c_1( T\bar{\CP}^2) = \pm 3H ,\] although it's not clear that $c_1(K) \in H^2( \bar{\CP}^2; \ZZ) \cong (\ZZ, [-1] )$. ::: :::{.remark} We had \( \chi(\OO) = {1\over 12} \qty{ K^2 + \chi_\Top} = {1\over 12}(3-n^2) \), and since $3-n^2 \in 12\ZZ$, we have $n^2 \in 3 + 12\ZZ \subset 3 + 4\ZZ$ and this forces $n^2 \equiv 3 \mod 4$. ::: :::{.definition title="Differential Complex"} Let \[ 0 \to \bundle{E}^0 \mapsvia{d_0} \bundle{E}^1 \mapsvia{d_1} \cdots \to \bundle{E}^n \to 0 \] be a complex (so $d^2 = 0$) of smooth vector bundles on a smooth manifold $X\im \Mfd_\RR^{C^\infty}$. Suppose that the $d_i$ are **differential operators**, i.e. in local trivializing charts over $U$ we have \[ \bundle{E}^i \cong \OO^{\oplus r_i} \OO^{\oplus r_{i+1}} \cong \bundle{E}^{i+1} \] where in every matrix coordinate, $d_i$ is of the form $\sum_{\abs I < N} g_I \bd_I$ where $\bd_I \da \bd_{i_1} \cdots \bd_{i_N}$ is a partial derived and the $g_I$ are smooth functions. ::: :::{.example title="?"} For $X\in \Mfd_\RR^{C^ \infty }$, we can take \[ 0 \to \OO \mapsvia{d} \Omega^1 \mapsvia{d} \Omega^2 \mapsvia{d} \cdots .\] In local coordinates, - \( \Omega^1 \) is spanned over $\OO$ by $dx_1, \cdots, dx_n$ where $n = \dim_\RR(X)$ - \( \Omega^2 \) is spanned over $\OO$ by $dx_i \wedge dx_j$ for $1\leq i, j \leq n$. Then the component of $d$ sending $dx_i \to dx_i \wedge dx_j$ is of the form \[ fdx_i &\mapsto -\dd{f}{x_j} dx_i \wedge dx_j .\] ::: :::{.example title="?"} For $X\in \Mfd_\CC$ and \( \bundle{E} \to X \) a holomorphic vector bundle, take \[ \bundle{E} \tensor A^{0,0} \mapsvia{\delbar} \bundle{E} \tensor A^{0, 1} \mapsvia{\delbar} \bundle{E} \tensor A^{0, 2} \to \cdots .\] This is because for $s_i$ local holomorphic sections and $\omega$ a smooth form we have \[ \delbar \qty{ (s_1, \cdots, s_r) \tensor \omega } = \qty{s_1, \cdots, s_r} \tensor \delbar \omega .\] ::: :::{.definition title="Order of an operator"} The maximal $N$ that appears in $\sum_{ \abs I \leq N} g_I \bd_I$ is the **order**. ::: :::{.definition title="Symbol Complex"} The **symbol complex** is a sequence of vector bundles on $T\dual X$. Noting that we have $\pi: T\dual X\to X$, and using pullbacks we can obtain bundles over the cotangent bundle: \[ 0 \to \pi^* \bundle{E}_0 \mapsvia{\sigma(d_0)} \pi^* \bundle{E}_1 \mapsvia{\sigma(d_1)} \cdots \to \pi^* \bundle{E}_n \to 0 .\] The **symbol** of the differential operator $d_i$ is $\sigma(d_i)$. It is defined by replacing $\del_i$ in $\sum_{\abs I {\color{red} =} N } g_I \del_I$ with $y_i$ where \[ y_i: T\dual U \to \RR \] is the coordinate function on the second factor of $T\dual U = U \cross \RR^n$ associated to the local coordinate $i$. Using that $TU = (T\dual)\dual U$, we can view $\del_i$ as functions on the cotangent bundle, $\sigma(d_i)$ is given in local trivializations by multiplication by a smooth function $\sum_{\abs I = N} g_I y^I$. ::: :::{.example title="?"} Consider $\OO \mapsvia{d} \Omega^1$. In local coordinates, this is given by $d = \qty{\del_1, \cdots, \del_n}$, i.e. coordinate-wise differentiation, since we can write a local trivialization \( \Omega^1 = \OO dz_1 \oplus \cdots \oplus \OO dz_n \). Then the symbol of $d$ is given by \[ \sigma(d): \pi^* \OO &\to \pi^* \Omega^1 \\ 1 &\mapsto (y_1, \cdots, y_n) ,\] thought of as vector bundles over $T\dual X$, and this is projection onto to cotangent factor. Locally, the image of 1 is given by $y_1 dx_1 + \cdots y_n dx_n$, which is a point in $T_p\dual X$ for all $(p, \alpha) \in T\dual X$ which is an assignment to every point $(p, \alpha) \in T_p\dual X$ a point in $(\pi^* \Omega^1)_{p, \alpha} \cong T_p\dual X$. There is a tautological section $(p, \alpha) \to \alpha\in T_p\dual X\in (\pi^* \Omega^1)_{p, \alpha}$, or really $(p, \alpha) \mapsto ( (p, \alpha), \alpha)$. ::: :::{.remark} See similarly to the canonical symplectic structure of the cotangent bundle. ::: :::{.remark} More generally, for $d: \Omega^p \to \Omega^{p+1}$, \( \sigma(d) \) acts on the frame \( dx_{i_1} \wedge \cdots dx_{i_p} \) in the following way: \[ \sigma(d)(dx_{i_1} \wedge \cdots \wedge dx_{i_p}) = \sum_y y_y dx_j \wedge dx_{i_1} \wedge \cdots dx_{i_p} \] where \[ d: fdx_{i_1} \wedge \cdots \wedge dx_{i_p} \mapsto \sum_j \dd{f}{x_j} dx_j \wedge \qty{dx_{i_1} \wedge \cdots \wedge dx_{i_p}} .\] The symbol complex is \[ \pi^* \OO \mapsvia{\sigma(d)} \pi^* \Omega^1 \mapsvia{\sigma(d)} \pi^* \Omega^2 \to \cdots \to \pi^* \Omega^n \to 0 \] for $n$ the dimension. In this case, $\sigma(d)$ has the same formula everywhere, since it's $C^ \infty \dash$linear: \[ \sigma(d) = \sum_j y_j dx_j \wedge \qty{\cdots} .\] ::: :::{.definition title="Elliptic Complex"} A differential complex $(\complex{\bundle{E}}, d)$ is **elliptic** if the symbol complex $(\pi^* \complex{\bundle{E}}, \sigma(d))$ is an exact sequence of sheaves (importantly) on $T\dual X \sm \ts{s_z}$ for $s_z$ the zero section. ::: :::{.claim} $(\complex{\Omega}, d)$ is elliptic. To check exactness of a sequence of vector bundles, it suffices to check exactness on every fiber. Fix $(p, \alpha) \in T\dual X \sm \ts{ s_z }$, then \[ 0 \to \CC \mapsvia{\wedge \alpha} T\dual_p X \mapsvia{\wedge \alpha} \Wedge^2 T_p\dual X \mapsvia{\wedge \alpha} \Wedge^3 T_p\dual X \to \cdots .\] Moreover, if \( \alpha\wedge \beta = 0 \) implies that \( \beta = \alpha\wedge \gamma \) for some \( \gamma \), which implies that this sequence is exact. :::