Geometric Methods in Partial Differential Equations

Abstract

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem.

Appendix 1: The Twistor Correspondence and Solutions of Differential Equations

In this appendix we present an interesting approach to the determination of the solutions of linear partial differential equations due to [12]. Let $f(z_0,z_1,\cdots ,z_n)$ be a complex homogeneous polynomial of degree $k>1$ in the indicated variables with $n>1$. In this section we will show how to use a general twistor correspondence to describe all the solutions of

$$\begin{aligned} \displaystyle D_f\phi :=f\left( \dfrac{\partial }{\partial z_0},\ldots , \dfrac{\partial }{\partial z_n}\right) \phi (z_0,\ldots ,z_n)=0. \end{aligned}$$

(5.1)

We will do so by showing that there is a twistor space Z, a vector space $H^{n-1}(Z,{\mathcal {O}}(-n-1+k))$ of Dolbeault cohomology classes and a twistor correspondence

$$\begin{aligned} T:H^{n-1}(Z,{\mathcal {O}}(-n-1+k))\rightarrow H^0({\mathbb {C}}^{n+1},{\mathcal {O}}), \end{aligned}$$

(5.2)

whose image is the space of solutions of 5.1. Here and in what follows we will denote by ${\mathcal {O}}$ the sheaf of holomorphic functions on ${\mathbb {C}}^m$ for some $m\in {\mathbb {N}}^\times$, and we will show that T is an injective map, obtained by integrating the cohomology class against “a cycle”, and onto the space of analytic functions in the kernel of $D_f$ for all $k>1$.

To begin with, we recall that the complex projective space ${\mathbb {C}}P_n$ is the set of all lines through the origin in ${\mathbb {C}}^{n+1}$, namely the set of all $\left[ z\right]$, $z\in {\mathbb {C}}^{n+1}$-$\{0\}$ with

$$\begin{aligned} \displaystyle \left[ z\right] :=\left[ z_0:z_1:\ldots :z_n\right] =\{\lambda z,\lambda \in {\mathbb {C}}^{\times }\}. \end{aligned}$$

It is a compact complex manifold with a covering family of charts given by the open sets $U_i$ where the i-th homogeneous coordinate is non-zero.

$$\begin{aligned} U_i&\rightarrow {\mathbb {C}}^n\\ \left[ z_0:z_1:\ldots :z_n\right]&\mapsto \left( \dfrac{z_0}{z_i}, \ldots ,\dfrac{z_{i-1}}{z_i}, \dfrac{z_{i+1}}{z_i},\ldots ,\dfrac{z_n}{z_i}\right) . \end{aligned}$$

As in the previous sections let H be the tautological line bundle over ${\mathbb {C}}P_n$ whose fibre over a point $\left[ z\right] \in {\mathbb {C}}P_n$ is just the line $\left[ z\right]$, i.e.

$$\begin{aligned} \displaystyle H=\{(\left[ z\right] ,w),w=\lambda z,\lambda \in {\mathbb {C}}^\times \} \subset {\mathbb {C}}P_n\times {\mathbb {C}}^{n+1}. \end{aligned}$$

(5.3)

We define local sections $\psi _i:U_i\rightarrow H$ for each $i=0,\ldots ,n$ by

$$\begin{aligned}&U_i\rightarrow H\nonumber \\&\psi _i(\left[ z\right] )=\left( \left[ z\right] , \left( \dfrac{z_0}{z_i},\ldots ,\dfrac{z_{i-1}}{z_i},1, \dfrac{z_{i+1}}{z_i},\ldots ,\dfrac{z_n}{z_i}\right) \right) . \end{aligned}$$

(5.4)

Note that $\psi _i$ is H-valued because

$$\begin{aligned} \left( \dfrac{z_0}{z_i},\ldots ,\dfrac{z_{i-1}}{z_i},1, \dfrac{z_{i+1}}{z_i},\ldots ,\dfrac{z_n}{z_i}\right) =\dfrac{1}{z_i}\left( z_0:z_1:\ldots :z_n\right) . \end{aligned}$$

From the definition of the $\psi _i$ we have that

$$\begin{aligned} \psi _i(\left[ z\right] )=\dfrac{z_j}{z_i}\psi _j(\left[ z\right] ) \end{aligned}$$

and hence the transition functions of H are

$$\begin{aligned} g_{ij}=\dfrac{z_j}{z_i}. \end{aligned}$$

We set ${\mathcal {O}}(-1):=H$. For $p>0$ we define ${\mathcal {O}}\left( p\right) =\bigotimes _1^pH^{\star }$, with $H^{\star }$ the line bundle dual to H. We also define ${\mathcal {O}}\left( p\right) =\bigotimes _1^{-p}H$, for $p<0$ and ${\mathcal {O}}(0)$ as the trivial line bundle on ${\mathbb {C}}P_n$.

For a given sheaf ${\mathcal {S}}$ on ${\mathbb {C}}P_n$, we denote by $H^p({\mathbb {C}}P_n,{\mathcal {S}})$ its p-th cohomology group $p\geqslant 0$. The dimension of the vector space $H^p({\mathbb {C}}P_n,{\mathcal {S}})$ (which is finite by Hodge theory) is denoted by $h^p({\mathbb {C}}P_n,{\mathcal {S}})$. We have the following formulas of Bott [13, p. 4]

$$\begin{aligned} h^q({\mathbb {C}}P_n,\mathscr {O}(k))={\left\{ \begin{array}{ll} \genfrac(){0.0pt}0{k+n}{k},\,k\geqslant 0,\,q=0 \\ \genfrac(){0.0pt}0{-k-1}{-k-1-n},\,q=n,\,k\leqslant -n-1\\ 0\,\,\,\text {otherwise}. \end{array}\right. } \end{aligned}$$

(5.5)

We have [9, p. 165]

$$\begin{aligned} Sym^d(({\mathbb {C}}^{n+1})^{\star })\simeq H^0({\mathbb {C}}P_n,\mathscr {O}(d)) \end{aligned}$$

here $Sym^d(({\mathbb {C}}^{n+1})^{\star })$ is the space of homogeneous polynomials of degree d in the variables $z_0, \ldots ,z_n$. Moreover from [9, p. 135] one knows that to any global section in $H^0({\mathbb {C}}P_n,\mathscr {O}(d))$, one can associate an effective divisor which is precisely where the considered section vanishes.

Definition 5.1

A divisor D is a locally finite formal linear combination

$$\begin{aligned} D=\sum a_i V_i \end{aligned}$$

of irreducible analytic subvarieties $V_i$, of codimension 1. If all the $a_i\geqslant 0$, D is called effective.

Let us now go back to the homogeneous polynomial $f(z_0,z_1,\ldots ,z_n)$ of degree k we started with. From what we have explained we can associate to it a section $f(\xi _0,\ldots ,\xi _n)$ of $\mathscr {O}(k)$ which vanishes exactly on an effective divisor denoted X.

We then have the exact sequence of sheaves

(5.6)

By taking the tensor product of the sequence (5.6) with the locally free sheaf $\mathscr {O}(p)$ and using the Bott’s formulas we get (by the long exact sequence in cohomology) the short exact sequences

(5.7)

and

(5.8)

Set $p=-n-1=k$ in (5.8). Then since $k>0$ we have $-n-1+k>-n-1$ and from formulas (5.5) we get

$$\begin{aligned} H^{n-1}({\mathbb {C}}P_n,\mathscr {O}(-n-1+k))\simeq H^{n} ({\mathbb {C}}P_n,\mathscr {O} (-n-1))\simeq {\mathbb {C}}\end{aligned}$$

where the last isomorphism also follows from (5.5).

If we assume X to be smooth, this isomorphism can be realized by integrating smooth $(0,n-1)$-forms with values in $\mathscr {O}(-n-1+k)$.

(5.9)

Recall now that $H^0({\mathbb {C}}P_n,{\mathcal {O}}(1))$ is the space of homogeneous polynomials of degree 1 in ${\mathbb {C}}^{n+1}$, and let $\xi _0,\ldots ,\xi _n$ be a basis of $H^0({\mathbb {C}}P_n,{\mathcal {O}}(1))$. We then have an isomorphism

$$\begin{aligned} {\mathbb {C}}^{n+1}&\rightarrow H^0({\mathbb {C}}P_n,{\mathcal {O}}(1))\nonumber \\ (x_0,\ldots ,x_n)&\mapsto \sum _{i=0}^{n}x_i\xi _i. \end{aligned}$$

(5.10)

Moreover the restriction map

$$\begin{aligned} H^0({\mathbb {C}}P_n,{\mathcal {O}}(1))\rightarrow H^0(X,{\mathcal {O}}(1)) \end{aligned}$$

(5.11)

is an isomorphism by Lefschetz’s hyperplane theorem, and therefore if X is the divisor defined by f we obtain

$$\begin{aligned} {\mathbb {C}}^{n+1}=H^0({\mathbb {C}}P_n,{\mathcal {O}}(1))=H^0(X,{\mathcal {O}}(1)). \end{aligned}$$

(5.12)

Remark 5.2

If E is a vector bundle over a manifold M then there is a sequence of induced jet bundles $J^pE$, $p=0,1,\ldots$. The fibre of $J^pE$ at $m\in M$ is the vector space of all p-jets at m or of equivalence classes of local sections about m where two sections are equivalent if they have the same Taylor series up to order p. This equivalence relation is independent of the charts necessary to define it and the fibers fit together to form smooth vector bundles. There is a natural projection from $J^pE$ to $J^{p-1}E$ which “forgets” the p-th term in the Taylor series. Associated with the jet bundles one has the following exact sequence

(5.13)

In this invariant language a $p^{th}$ order differential operator acting on sections of E is just a bundle map

$$\begin{aligned} D:J^pE\rightarrow E. \end{aligned}$$

(5.14)

When $E={\mathbb {C}}^{n+1}\times {\mathbb {C}}$, $J^pE$ will be denoted $J^p{\mathbb {C}}$ and (5.13) becomes

(5.15)

where ${\mathbb {C}}^{n+1}$ is identified with its dual. If $p=k$, then the natural flat connection on ${\mathbb {C}}^{n+1}$, which is just the exterior derivative, defines a splitting $J^k{\mathbb {C}}\rightarrow ({\mathbb {C}}^{n+1}\times S^k{\mathbb {C}}^{n+1})$ and composing with the projection on ${\mathbb {C}}^{n+1}$ followed by the composition with f defines the homogeneous polynomial differential operator

$$\begin{aligned} D_f:J^k{\mathbb {C}}\rightarrow {\mathbb {C}}. \end{aligned}$$

(5.16)

This is how one should interpret the differential operator given in equation (5.1).

Now let us define the twistor space Z as the total space of the line bundle ${\mathcal {O}}(1)$ on ${\mathbb {C}}P_n$ restricted (or pulled-back) to X.

The relation between Z and ${\mathbb {C}}^{n+1}$ revolves around the following double fibration where we identify ${\mathbb {C}}^{n+1}$ with $H^0(X,{\mathcal {O}}(1))$ via the isomorphism (5.12).

(5.17)

If $(\sum _0^nx_i\xi _i,z)\in {\mathbb {C}}^{n+1}\times X$, then the left hand map sends it to $(x_1,\ldots ,x_n)$ and the right hand arrow sends it to $\sum _0^n\xi _i(z)$. If we fix a point $x\in {\mathbb {C}}^{n+1}$ then the image of the section

$$\begin{aligned} \displaystyle \sum _0^nx_i\xi _i:X\rightarrow Z \end{aligned}$$

(5.18)

is a subvariety $X_x$ of Z which is identified with X by the projection map $\pi :Z\rightarrow X$. We will need various line bundles on Z. We use the mappping $\pi$ to pull-back line bundles from X. That is if L is a line bundle on X, we define a line bundle $\pi ^{\star } L$ on Z by $(\pi ^{\star } L)_z=L_{\pi (z)}$ (the fiber of $(\pi ^{\star } L)$ over z is by definition given by $L_{\pi (z)}$). Notice that a peculiar thing happens for the bundle ${\mathcal {O}}(1)$. Here if z is an element of Z then because Z is itself the line bundle ${\mathcal {O}}(1)$, $z\in {\mathcal {O}}(1)_{\pi (z)}$, that is $z\in (\pi ^{\star }{\mathcal {O}}(1))_z$. We denote this section of ${\mathcal {O}}(1)$, the one that sends z to z, by $\eta$. It is the section whose divisor is the zero section $Z_0$ of the line bundle $Z\overset{\pi }{\rightarrow }X$. Notice that the subvariety $X_x$ in this notation is the subset of Z where

$$\begin{aligned} \displaystyle \eta =\sum _0^n\xi _ix_i. \end{aligned}$$

(5.19)

Let $\omega _l,\, l=0,\ldots ,m$ be $(0,n-1)$ forms on X with values in ${\mathcal {O}}(-n-1+k-l)$. Consider now the $(0,n-1)$ form on Z with values in ${\mathcal {O}}(-n-1+k)$ defined by

$$\begin{aligned} \displaystyle \omega =\sum _0^m\pi ^{\star }(\omega _l)\eta ^l. \end{aligned}$$

(5.20)

Because each $X_x$ is a copy of X we can restrict $\omega$ to $X_x$ and integrate. We obtain

$$\begin{aligned} \int _{X_x=\pi ^{-1}(X)}\omega&=\int _{X_x}\sum _0^m \pi ^{\star }(\omega _l)\eta ^l\nonumber \\&=\int _{X_x}\sum _0^m\pi ^{\star }(\omega _l)(\sum _0^n\xi _ix_i)^l\nonumber \\&=\int _X\sum _0^m\omega _l(\sum _0^n\xi _ix_i)^l=:\phi (x). \end{aligned}$$

(5.21)

This $\phi$ is clearly in $\ker D_f$, because if we apply a monomial differential operator

$$\begin{aligned} \left( \dfrac{\partial }{\partial x_0}\right) ^{i_0} \left( \dfrac{\partial }{\partial x_1^{i_1}}\right) ^{i_1} \ldots \left( \dfrac{\partial }{\partial x_n}\right) ^{i_n} \end{aligned}$$

to $\phi (x)$ we obtain

$$\begin{aligned} \displaystyle \int _X\sum _{l=k}^m\omega _ll(l-1)\ldots (l-k+1) \left( \sum _0^n\xi _ix_i\right) ^{l-k}\xi ^{i_1}\xi ^{i_2}\ldots \xi ^{i_k}. \end{aligned}$$

Hence if we apply $D_f$ to $\phi (x)$ we obtain

$$\begin{aligned} \displaystyle \displaystyle \int _X\sum _0^m\omega _ll(l-1) \ldots (l-k+1)(\sum _0^n\xi _ix_i)^{l-k}f(\xi ^{0},\xi ^{1},\ldots ,\xi ^{k}) \end{aligned}$$

which vanishes because f vanishes on X.

In [12] it is shown that more generally one should integrate over $X_x$ elements belonging to $H^{n-1}(Z,\pi ^{\star }{\mathcal {O}}(-n-1+k))$.

Finally, as we indicated at the beginning of this section, one has in full generality the twistor transform

$$\begin{aligned} T:H^{n-1}(Z,\pi ^{\star }{\mathcal {O}}(-n-1+k))&\rightarrow H^0({\mathbb {C}}^{n+1},{\mathcal {O}})\nonumber \\ T(\omega )(x)&=\int _{X_x}\omega , \end{aligned}$$

(5.22)

which is a bijection onto the kernel of $D_f$ for $k\leqslant n$.

Remark 5.3

$H^{n-1}(Z,\pi ^{\star }{\mathcal {O}}(-n-1+k))$ is an infinite dimensional vector space because Z is non-compact and $-n-1+k<0$ as $k\leqslant n$. Also the appearance of ${\mathcal {O}}(-n-1+k)$ is due to the fact that the canonical bundle of X is exactly ${\mathcal {O}}(-n-1+k)$. Moreover the fact that one should integrate $(0,n-1)$-forms is a consequence of the Dolbeault resolution which implies that $H^{n-1}(X,\mathscr {O}(-n-1+k))\simeq H^{n-1}(\Gamma (X,{\mathcal {E}}^{0,\bullet }(\mathscr {O}(-n-1+k))))$. Finally if $f(x^0,x^1,\ldots ,x^n)$ is a homogenous polynomial then $f(\xi )=f(\xi ^0,\xi ^1,\ldots ,\xi ^n)$ defines a section of the line bundle $\mathscr {O}(k)$ which vanishes precisely on X, namely $\{f=0\}$.

Let us now see from what we have said in this section how the classical contour integral formula, for the Laplacian in three dimensions given in [17], arises. First, from [17], the general solution of the Laplacian in three dimensions is

$$\begin{aligned} \phi (x,y,z)=\displaystyle \int _{-\pi }^\pi f(z+ix\cos u+iy\sin u,u)du \end{aligned}$$

(5.23)

for an arbitrary real analytic function f. In three dimensions the manifold X is the quadric Q which is the vanishing set in ${\mathbb {C}}P_2$ of $z^2_0+z_1^2+z_2^2$. We have ${\mathbb {C}}P_2\supseteq Q\simeq {\mathbb {C}}P_1$. From equation (5.21) the general solution is

$$\begin{aligned} \phi (x,y,z)=\displaystyle \int _{Q} \sum _0^\infty f_p(\xi ) (x\xi _0+y\xi _1+z\xi _2)^pd\xi . \end{aligned}$$

(5.24)

Indeed from (5.21), and with the notations used before, $\sum \nolimits _{0}^m\omega _p(\sum _0^n\xi _ix_i)^p$ is a (0, 1) form with values in $\mathscr {O}(-1)$. A crucial remark is that for any complex projective hypersurface $M=\{g=0\}$, of degree k of ${\mathbb {C}}P^n$, its canonical bundle with sheaf of sections given by the local holomorphic one forms on M, is isomorphic to $\mathscr {O}(-n-1+k)$. Therefore in the case at hand $\mathscr {O}(-1)$ is the canonical bundle of Q, and $\displaystyle \sum \nolimits _{0}^m\omega _p(\sum \nolimits _0^n\xi _ix_i)^p$ is a (1, 1)-form on Q (where we interpret $d\xi$ as a local (1, 0)-form on Q).

But $f(\xi _0,\xi _1,\xi _2)=0$ precisely on Q. Thus we can identify $(\xi _0,\xi _1,\xi _2)$ as a variable point which lies on Q. The embedding of ${\mathbb {C}}$ in the quadric which will be used is

$$\begin{aligned} w\mapsto \left[ i(w^2+1),(w^2-1),2w\right] . \end{aligned}$$

Restricting to the circle this gives

$$\begin{aligned} u\mapsto \left[ i(e^{2iu}+1),(e^{2iu}-1),2e^{iu}\right] , \,u\in \left[ -\pi ,\pi \right] \end{aligned}$$

where square brackets denote the point of ${\mathbb {C}}P_2$ which is the line through $(i(e^{2iu}+1),(e^{2iu}-1),2e^{iu})$. So $(x\xi _0+y\xi _1+z\xi _2)$ becomes

$$\begin{aligned} 2e^{iu}(xi\cos u+yi\sin u+z). \end{aligned}$$

Therefore we have

$$\begin{aligned} \phi (x,y,z)&=\displaystyle \displaystyle \int _{-\pi }^\pi \sum _pf_p(u)2^pe^{ipu}(z+ix\cos u+iy\sin u)^ph(u)du\nonumber \\&=\int _{-\pi }^\pi f(z+ix\cos u+iy\sin u,u)du \end{aligned}$$

(5.25)

by change of variables and this recovers the result (5.23).

We now note that the formula in [17] for a solution of the wave equation is

$$\begin{aligned} \phi (x,y,z,t)=\displaystyle \int _{-\pi }^\pi \int _{-\pi }^\pi f (t+x\sin u\cos v+y\sin u\sin v+z\cos u,u,v)dudv. \end{aligned}$$

(5.26)

Again the hypersurface $X\subseteq {\mathbb {C}}P_3$ is the quadric $z^2_0+z_1^2+z_2^2-z_3^2=0$ which is simply ${\mathbb {C}}P_1\times {\mathbb {C}}P_1,$ which accounts for the double coutour integral in (5.26) and $Z=\mathscr {O}(1)|_X$. To obtain (5.26) one uses the 2-1 ramified map given by

$$\begin{aligned}&{\mathbb {C}}\times {\mathbb {C}}\rightarrow Q\subseteq {\mathbb {C}}P_3,\\&(\xi ,\xi ^\prime )\mapsto \left[ (1+\xi ^2)(-1+\xi ^{\prime 2}), -i(-1+\xi ^2)(-1+\xi ^{\prime 2}),2i(1+\xi ^{2})\xi ^\prime ,4i \xi \xi ^\prime \right] . \end{aligned}$$

Setting $\xi =e^{iu},\,\xi ^\prime =e^{iv}$ (we restrict to the product of circles) and using formula (5.21), we obtain

$$\begin{aligned} \phi (x,y,z,t)&=\displaystyle \displaystyle \int _{\xi } \int _{\xi ^\prime }\sum _pf_p(\xi ,\xi ^\prime )(2i(1+\xi ^{2}) \xi ^\prime z+x(1+\xi ^2)(-1+\xi ^{\prime 2})\nonumber \\&\quad -i(-1+\xi ^2)(-1+\xi ^{\prime 2})y+4i\xi \xi ^\prime t)^p d\xi d\xi ^\prime \nonumber \\&=\int _{-\pi }^\pi \int _{-\pi }^\pi \sum _pf_p(u,v) (t+x\sin u\cos v+y\sin u\sin v+z\cos u)^pdudv\nonumber \\&=\displaystyle \int _{-\pi }^\pi \int _{-\pi }^\pi f (t+x\sin u\cos v+y\sin u\sin v+z\cos u,u,v)dudv. \end{aligned}$$

(5.27)