The Hodge decomposition, diamond, and Euler characteristics

 Seminar topic

Recall the sheaf of $r$-differential forms $\Omega^r_X$ on $X$ (with $\Omega^r_X(U) = \{fdx_{i_1}\wedge \cdots \wedge dx_{i_r}\ :\ f\ $is well-defined on $U\}$ and such sums) and the structure sheaf $\mathcal O_X$ on $X$ (with $\mathcal O_X(U) = \{f/g\ :\ f,g\in k[U],\ g\neq 0\ $on$\ U\}$). Then we may consider the sheaf cohomology of $X$, with values in $\Omega^r_X$ or $\mathcal O_X$.

Definition: Let $X$ be a smooth manifold of dimension $n$. The $(p,q)$th Hodge number is $h^{p,q}=\dim(H^{p,q})$, where $H^{p,q} = H^q(X,\Omega^p_X)$. These numbers are arranged in a Hodge diamond as below.

The Hodge diamond has a lot of repetition - by complex conjugation, we get that $h^{p,q}=h^{q,p}$, so it is symmetric about its vertical axis. By the Hard Lefschetz theorem (or the Hodge star operator, or Poincare duality), we get that $h^{p,q}=h^{n-q,n-p}$, so it is symmetric about its horizontal axis.

Proposition: Let $X$ be a Kähler manifold (note that all smooth projective varieties are Kähler) of dimension $n$. Then the cohomology groups of $X$ decompose as
$$H^k(X,\C) = \bigoplus_{p+q=k}H^{p,q}(X),$$
for all $0\leqslant k\leqslant 2n$. This is called the Hodge decomposition of $X$.

This decomposition immediately gives all the Hodge numbers for $\P^n$, knowing its cohomology. For a manifold of complex dimension $n$, there are several numbers and polynomials that may be defined. These are:
$$\begin{align*} \chi_{top}(X) & = \sum_{i=1}^{2n}(-1)^i \dim(H^i(X,\C)) & \text{the (topological) Euler characteristic} \\ \chi^p(X) & = \sum_{q=0}^{n-1}(-1)^qh^{p,q} & \text{the chi-$p$ characteristic} \\ \chi_y(X) & = \sum_{p=0}^{n-1}\chi^py^p & \text{the chi-$y$ characteristic} \end{align*}$$
Note the Euler characteristic is the alternating sum of the rows of the Hodge diamond, and the chi-$p$ characteristic is the alternating sum of the left-right diagonals of the diamond.

Example: In the case $X$ is a hypersurface in projective $n$-space $\P^n$ defined by a degree $d$ polynomial,
$$\chi_y = [z^n]\frac{1}{(1+zy)(1-z)^2}\cdot\frac{(1+zy)^d-(1-z)^d}{(1+zy)^d+y(1-z)^d}.$$
Since every row except the middle row of the Hodge diamond of a hypersurface is known (as it comes from the Hodge diamond of $\P^n$ by the Lefschetz hyperplane theorem), this expression gives all the unknown numbers. This particular formula is a simplification  of Theorem 22.1.1 in Hirzebruch, which itself comes from the Riemann--Roch theorem.

References: Huybrechts (Complex Geometry: An Introduction, Chapters 3.2, 3.3), Hirzebruch (Topological Methods in Algebraic Geometry, Appendix 1, Section 22)

Exactness and derived functors

 Lecture topic

Let $0\to X\to Y\to Z\to 0$ be a short exact sequence of objects in a category $A$. Let $\mathcal F:A\to B$ be a covariant functor.

Definition: The functor $\mathcal F$ is right-exact if $\mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)\to 0$ is an exact sequence. It is left-exact if $0\to \mathcal F(X)\to\mathcal F(Y)\to \mathcal F(Z)$ is an exact sequence. It is exact if it is both left- and right-exact.

Example: These are some examples of left- and right-exact functors:
    $\Hom_A(X,-)$ is covariant left-exact
    $\Hom_A(-,X)$ is contravariant left-exact
    $-\otimes_R X$ is covariant right-exact, for $X$ a left $R$-module

Recall that $X\otimes_R Y$ is naturally isomorphic to $Y\otimes_RX$.

Definition: An object $X\in \Obj(A)$ is projective if $\Hom_A(X,-)$ is an exact functor. Similarly, $X$ is injective if $\Hom_A(-,X)$ is an exact functor.

Recall that a projective resolution of an object $X$ is a sequence of projective objects $\cdots\to P_2\to P_1\to P_0$ that may or may not terminate on the left. The homology of the sequence in degree 0 is $X$, and trivial in other degrees. Similarly, an injective resolution of $X$ is a sequence of injective objects $I_0\to I_1\to I_2\to\cdots$ that may or may not terminate on the right. The cohomology is also concentrated in degree 0, and is $X$ there. A free resolution is a projective resolution where all the objects are free (whatever that means in the context).

These types of resolutions may not exist. A category "has enough injectives (projectives)" means we can always construct injective (projective) resolutions.

Definition: Let $\mathcal F:A\to B$ be a covariant right-exact functor and $\mathcal G:A\to B$ a covariant left-exact functor. Let $X\in \Obj(A)$ with $P_\bullet$ a projective resolution of $X$ and $I_\bullet$ an injective resolution of $X$. The $i$th left-derived functor of $\mathcal F$ is $L_i\mathcal F(X) = H_i(\mathcal F(P_\bullet))$. The $i$th right-derived functor of $\mathcal G$ is $R^i\mathcal G(X) = H^i(\mathcal G(I_\bullet))$.

These objects of $B$ are well-defined up to natural isomorphism. Note that $\mathcal F^{op}:A^{op}\to B^{op}$ is a contravariant right-exact functor. Moreover, if $\mathcal F$ was contravariant right-exact and $\mathcal G$ was contravariant left-exact, then $L_i\mathcal F(X)=H_i(\mathcal F(I_\bullet))$ and $R^i\mathcal G(X)=H^i(\mathcal G(P_\bullet))$.

Example: Let $R$ be a ring with $X$ and $Y$ both $R$-bimodules. Then
$$\begin{align*} \Tor_i^R(Y,X) & =  L_i(-\otimes_RX)(Y) & \Ext^i_R(X,Y) & = R^i(\Hom_R(X,-))(Y) \\ & = L_i(Y\otimes_R - )(X), && = R^i(\Hom_R(-,Y))(X). \end{align*}$$
Recall that $\Tor_i^R(Y,X)$ is canonically isomorphic to $\Tor_i^R(X,Y)$, but it is not true for $\Ext$. Also note that $\Hom_R(X,-)$ is covariant and $\Hom_R(-,Y)$ is contravariant, while $-\otimes_R X$ and $Y\otimes_R -$ are both covariant functors.
References: Weibel (An introduction to homological algebra, Chapter 2)

Examples of limits and colimits

 Lecture topic

Let $C$ be a category and $X,Y,Z\in \Obj(C)$. Choose $I$ to be a category with $\mathcal F:I\to C$ a functor as described below. Then we may consider the limit and colimit of $\mathcal F$, noting that they may not always exist, as there may be no suitable natural transformation $i$ or $\pi$.

The limit and colimit of the category $I$ with two points and two arrows going between the points in opposite directions, namely

are not interesting to consider. That is because as a category, it must satisfy compositions, so $f\circ g=\id$, which is a restrictive condition on $f$ and $g$. We may define a new map $h:X\to X$ with $h=f\circ g$, but then more maps, such as $h\circ f$ and so on need to be defined, which complicate the situation.

References: Borceux (Handbook of Categorical Algebra I, Chapter 2)

Limits and colimits

 Lecture topic

Definition: Given categories $A,B$ and functors $\mathcal F,\mathcal G:A\to B$, a natural transformation $\eta:\mathcal F\to \mathcal G$ is a collection of elements $\eta_X\in \Hom_B(\mathcal F(X),\mathcal G(X))$ for all $X\in \Obj(A)$ such that the diagram

commutes, whenever $f\in \Hom_A(X,Y)$.

Definition: For $X\in \Obj(A)$, define the constant category $\underline X$ to be the category with $\Obj(\underline X)=\{X\}$ and $\Hom_{\underline X}(X,X)=\{\id_X\}$. For any other category $B$, this may also be viewed as a natural transformation $\underline X:B\to A$ with $\underline X(Y)=X$ and $\underline X(f)=\id_X$ for any object $Y$ and any morphism $f$ of $B$.

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The colimit $\text{colim}(\mathcal F$) of $\mathcal F$ is an object $\text{colim}(\mathcal F)\in \Obj(B)$ and a natural transformation $\iota:\mathcal F\to \underline{\text{colim}(\mathcal F)}$ that is initial among all such natural transformations. We write $\iota_X:\mathcal F(X)\to \text{colim}(\mathcal F)$ and have $\iota(f)=\id_{\text{colim}(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\eta:\mathcal F\to \underline{Z}$ is a natural transformation, there is a unique map $\zeta:\text{colim}(\mathcal F)\to Z$ such that the following diagram commutes:

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The limit $\lim(\mathcal F$) of $\mathcal F$ is an object $\lim(\mathcal F)\in \Obj(B)$ and a natural transformation $\pi:\underline{\lim(\mathcal F)}\to \mathcal F$ that is final among all such natural transformations. We write $\pi_X:\lim(\mathcal F) \to \mathcal F(X)$ and have $\pi(f)=\id_{\lim(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\epsilon:\underline{Z}\to \mathcal F$ is a natural transformation, there is a unique map $\theta:Z\to \lim(\mathcal F)$ such that the following diagram commutes:

Examples of colimits are initial objects, coproducts, cokernels, pushouts, direct limits. Examples of limits are final objects, products, kernels, pullbacks, inverse limits.

 Remark: $\Hom$ commutes with limits and tensor commutes with colimits. That is:
$$\Hom(A,\lim(B_i)) = \lim\left(\Hom(A,B_i)\right) \hspace{1cm} (\text{colim}(A_i))\otimes B = \text{colim}(A_i\otimes B)$$
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)

The canonical bundle of projective space and hypersurfaces

Let $\P^n$ be projective $n$-space with coordinates $[x_0:\cdots:x_n]$. Cover $\P^n$ with affine pieces $U_i = \{x_i\neq 0\}$, each of which are $\A^n$, in coordinates $(y_1,\dots,y_n)$, where $y_j = x_j/x_i$. Recall that the canonical bundle of $\P^n$ is the $n$-fold wedge of the cotangent bundle of $\P^n$, or $\omega_{\P^n} = \bigwedge^nT^*_{\P^n}$. The canonical bundle for an arbitrary variety is defined analogously.

Definition: Let $X$ be a projective $n$-dimensional variety. The sheaf of regular functions on $X$ is $\mathcal O_X$, with $\mathcal O_X(U)=\{f/g\ :\ f,g\in k[x_1,\dots,x_n]/I(X), g\neq 0\}$, and the restriction maps are function restriction.

There is a natural grading on $\mathcal O_X$, given by $\deg(f)-\deg(g)$. A shift in the grading may be applied, called a {\it Serre twist}, to get a differently graded (but isomorphic) module: for $\varphi\in \mathcal O_X$ with $\deg(\varphi)=k$, set $\varphi\in\mathcal O_X(\ell)$ to have $\deg(\varphi) = k-\ell$.

Let $\alpha = dy_1\wedge\cdots\wedge dy_n\in \omega_{\P^n}$, which is well-defined on all of $U_i$. We claim this is well-defined on all of $\P^n$. We check this on the overlap $U_0\cap U_n$ (for nicer notation), but the approach is analogous for $U_i\cap U_j$.
$$\begin{align*} U_0 & = \{(y_1,\dots,y_n)\ :\ y_i = x_i/x_0\} & y_i & = \frac{z_{i+1}}{z_i} & dy_i & = \frac{z_1dz_{i+1}-z_{i+1}dz_1}{z_1^2} \\ U_n & = \{(z_1,\dots,z_n)\ :\ z_i = x_{i-1}/x_n\} & y_n & = \frac1{z_1} & dy_n & = \frac{-dz_1}{z_1^2} \end{align*}$$
Therefore
$$\begin{align*} \alpha & = dy_1\wedge\cdots\wedge dy_n \\ & = \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge\cdots\wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\ & = \frac{dz_2}{z_1}\wedge\cdots\wedge \frac{dz_n}{z_1}\wedge \frac{-dz_1}{z_1^2} \\ & = \frac{(-1)^n}{z_1^{n+1}}dz_1\wedge\cdots \wedge dz_n. \end{align*}$$
Since the transition function has a pole of order $n+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\alpha$ has a pole of order $n+1$ at $\infty$. Therefore $\omega_{\P^n} \cong \mathcal O_{\P^n}(-n-1)$.

Let $X\subset \P^n$ be a smooth hypersurface defined by a degree $d$ equation $F(x_0,\dots,x_n)=0$. On the affine piece $U_0$ this becomes $f(y_1,\dots,y_n)=F(1,\frac{x_1}{x_0},\dots,\frac{x_n}{x_0})$ with $y_i = x_i/x_0$. The total derivative is
$$\frac{\dy f}{\dy y_1} dy_1 + \cdots + \frac{\dy f}{\dy y_n} dy_n = \sum_{i=1}^n\frac{\dy f}{\dy y_i}dy_i = 0,$$
and since $X$ is smooth, the terms never all vanish at the same time. Let $V_i=\{\frac{\dy f}{\dy y_i} \neq 0\}$, and set
$$\beta_i = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \in \omega_X,$$
which is well-defined on all of $V_i\subset U_0$. We claim that the choice of $V_i$ does not matter, and indeed, assuming $i<j$,
$$\begin{align*} \beta_j & = \frac{(-1)^{j-1}}{\dy f/\dy y_j} dy_1\wedge\cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{j-1+i-1}dy_i}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{j-1+i-1}\frac{-1}{\dy f/\dy y_i}\left(\frac{\dy f}{\dy y_1}dy_1+\cdots + \widehat{\frac{\dy f}{\dy y_i}dy_i} + \cdots + \frac{\dy f}{\dy y_n}dy_n\right)}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{j-1+i-1+1}\frac{1}{\dy f/\dy y_i}\cdot \frac{\dy f}{\dy y_j}dy_j}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{j-1+i-1+1+j-2}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\ & = \beta_i. \end{align*}$$
Hence $\beta_i$ is well-defined on all of $U_0$, and we call it simply $\beta$. Next we claim it is well-defined on all of $X$. Again we only check on the overlap of $U_0\cap U_n$. On the affine piece $U_n$ this becomes $g(z_1,\dots,z_n)=F(\frac{x_0}{x_n},\dots,\frac{x_{n-1}}{x_n},1)=f(\frac{z_2}{z_1},\dots,\frac{z_n}{z_1},\frac1{z_1})$ with $z_i = x_{i-1}/x_n$. We employ the chain rule $\frac{\dy f}{\dy y_i}=\frac{\dy f}{\dy z_j}\frac{\dy z_j}{\dy y_i}$ and the results above to find that
$$\begin{align*} \beta & = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\ & = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge \cdots \wedge \widehat{dy_i}\wedge \cdots \wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\ & = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{(-1)^{n-1}}{z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\ & = \frac{(-1)^{i+n}}{\left(\frac{1}{z_1}\right)^{d-1}\left(c+\cdots\right) z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\ & = \frac{(-1)^{i+n}}{z_1^{n-d+1} \left(c+\cdots \right)}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n, \end{align*}$$
for some constant $c$. This comes from expressing $f$ in terms of the $z_i$s and factoring. Since the transition function has a pole of order $n-d+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\beta$ has a pole of order $n-d+1$ at $\infty$. Therefore $\omega_{X} \cong \mathcal O_{X}(-n+d-1)$.

References: Griffiths and Harris (Principles of Algebraic Geometry, Chapter 1.2)