Structure of linearized Cauchy-Riemann equation [04AD]

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Structure of linearized Cauchy-Riemann equation

Let $\Sigma$ be a holomorphic polygon with $k\geq 0$ input ends $p_{i}$ , and one output end at $q$ . The case $k=0$ corresponds to teardrops, and $k=1$ corresponds to strips. We consider the ordinary linearized Cauchy-Riemann operator in weighted Sobolev spaces $W^{1,2;\mu}$ , to classify the structure of the first order deformation theory. As usual, the complex structure is integrable. Since $\bar{\partial}$ is elliptic, its cokernel in $L^{2}$ is finite dimensional, represented by holomorphic 1-forms on $\Sigma$ , which must have finite order of vanishing at $q$ . For large enough $\mu$ , the dual space $L^{2;-\mu}$ for $L^{2;\mu}$ imposes an exponential decay condition $O(e^{-\mu s})$ at $q$ , so the cokernel evantually vanishes for $\mu\gg 1$ . Then the kernel dimension in $W^{1,2;\mu}$ is equal to the index, computed by (25). For convenience, we use $\mu\in\pi\mathbb{N}$ , which avoids the indicial values. Then

\dim(\ker\bar{\partial}\subset W^{1,2;\mu})=\deg q-\sum_{1}^{k}\deg p_{i}+\frac{n\mu}{\pi}.

(26)

It is convenient to view the domain $\Sigma$ of the holomorphic polygon as the upper half plane with coordinate $z$ , with corners $p_{i}$ on the real line and $q$ at infinity.

Lemma 3.12.

If $v\in\ker\bar{\partial}\subset W^{1,2;\mu}$ , then $v=fw$ for some real coefficient polynomial function $f$ on the upper half plane, such that $w$ is nonvanishing on $\mathbb{R}\setminus\{p_{1},\ldots p_{k}\}$ , and vanishes minimally at $p_{i}$ (meaning $w$ is indivisible by $z-p_{i}$ ).

Proof.

If $v$ vanishes at any boundary point $a$ on $\mathbb{R}\setminus\{p_{1},\ldots p_{k}\}$ , or if $v$ vanishes at $p_{i}$ beyond minimal order, then $\frac{v}{z-a}$ (resp. $\frac{v}{z-p_{i}}$ ) is also a first order deformation with the same $TL$ boundary condition, subject to the growth constraints at infinity. Since the kernel dimension is finite, the divisions can only happen a finite number of times, producing the polynomial $f$ . ∎

Let $p\in\partial\Sigma\simeq\partial\mathbb{H}$ , and let $K$ be the maximal number depending on $p$ , such that there exist $\mathbb{R}$ -linearly independent $v_{1},\ldots v_{K}\in\ker\bar{\partial}$ , satisfying

•

In case $p$ is not a corner point, then $v_{1}(p),\ldots,v_{K}(p)$ are $\mathbb{R}$ -linearly independent vectors,
•

In case $p=p_{i}$ is a corner point, then the nonzero elements in the $\mathbb{R}$ -span of $v_{1},\ldots v_{K}$ are vector fields vanishing minimally at $p$ .

Lemma 3.13.

Any $v\in\ker\bar{\partial}\subset W^{1,2;\mu}$ is of the form $g_{1}v_{1}+\ldots g_{K}v_{K}$ for some real coefficient rational functions $g_{1},\ldots g_{K}$ , nonsingular at $p$ .

Proof.

Without loss of generality $p=0$ . Let $w_{0}$ be any first order deformation. By the maximality of $K$ , we can choose real numbers $a_{i}$ , such that the first order deformation $w_{0}-\sum_{1}^{K}a_{i}v_{i}$ vanishes at zero, so $w_{0}-\sum_{1}^{K}a_{i1}v_{i}=z^{k_{1}}w_{1}$ for some first order deformation $w_{1}$ which is nonzero at the origin. Finite dimensionality means this process can be repeated for only a finite number of times:

\begin{cases}w_{0}=\sum a_{i1}v_{i}+z^{k_{1}}w_{1},\\ w_{1}=\sum a_{i2}v_{i}+z^{k_{2}}w_{2},\\ \ldots\\ w_{N-1}=\sum a_{iN}v_{i}+z^{k_{N}}w_{N}.\end{cases}

We choose the smallest $N$ such that $v_{1},\ldots v_{K},w_{0},\ldots w_{N}$ are $\mathbb{R}$ -linearly dependent as vector fields; notice $v_{1},\ldots v_{K}$ are linearly independent, so $N\geq 0$ . We then get a linear relation

f_{0}(z)w_{N}=\sum_{1}^{K}f_{i}(z)v_{i},

where $f_{0},\ldots f_{K}$ are polynomials, and $f_{0}(0)\neq 0$ . This implies the claim. ∎

We can also apply a Möbius transform to make the output end $q$ lie at the origin. The growth condition translates to $o(|z|^{-\mu/\pi})$ at zero. Let $K$ be maximal, such that there are $\mathbb{R}$ -linearly independent vector fields $z^{-\mu/\pi}v_{1},\ldots,z^{-\mu/\pi}v_{K}\in\ker\bar{\partial}\subset W^{1,2;\mu}$ , and any nonzero element in the $\mathbb{R}$ -span of $v_{1},\ldots v_{K}$ vanishes mimimally at $q=0$ . As a caveat, this does not assume $v_{1},\ldots v_{K}$ satisfy the growth constraints at infinity to lie in $\ker\bar{\partial}\subset W^{1,2;\mu}$ . Minor adaptions give

Lemma 3.14.

Any $v\in\ker\bar{\partial}\subset W^{1,2;\mu}$ is of the form $z^{-\mu/\pi}(g_{1}v_{1}+\ldots g_{K}v_{K})$ for some real coefficient rational functions $g_{1},\ldots g_{K}$ , nonsingular at $q$ .

Corollary 3.15.

The number $K$ is independent of the boundary and corner points on $\partial\Sigma$ .

We view the boundary $\partial\Sigma\simeq\mathbb{P}^{1}(\mathbb{R})$ . By the above lemmas, there is a real algebraic vector bundle $\mathcal{E}$ of rank $K$ over $\mathbb{P}^{1}(\mathbb{R})$ such that $v_{1},\ldots v_{K}$ provide the basis of local sections. By Grothendieck’s classification of vector bundles,

Proposition 3.16.

$\mathcal{E}\simeq\oplus_{1}^{K}\mathcal{O}(n_{i})$ for some $n_{i}\in\mathbb{Z}$ .

Since the rank $K$ is nondecreasing in $\mu$ , it evantually stabilizes for $\mu\gg 0$ . Since around any given point, the same choice of $v_{1},\ldots v_{K}$ is valid for all large $\mu$ , the algebraic vector bundle $\mathcal{E}$ is independent of $\mu\gg 0$ . The elements of $\ker\bar{\partial}\subset W^{1,2;\mu}$ can be interpreted as global sections of $\mathcal{E}\otimes\mathcal{O}(\frac{\mu}{\pi})$ . Thus for all large $\mu$ ,

\dim(\ker\bar{\partial}\subset W^{1,2;\mu})=\dim\Gamma(\mathbb{P}^{1}(\mathbb{R}),\mathcal{E}\otimes\mathcal{O}(\frac{\mu}{\pi}))=K(\frac{\mu}{\pi}+1)+\sum_{1}^{K}n_{i}.

(27)

Contrasting with the index formula (26),

Corollary 3.17.

The rank $K=n$ , and the degree $\sum_{1}^{n}n_{i}=\deg q-\sum_{1}^{k}\deg p_{i}-n$ .

The structure of $\mathcal{E}\simeq\oplus_{1}^{K}\mathcal{O}(n_{i})$ provides meromorphic sections $v_{1},\ldots v_{n}$ which are a basis of local sections on $\mathbb{R}\subset\mathbb{P}^{1}(\mathbb{R})$ , and have excess vanishing orders $n_{1},\ldots n_{n}$ at $q$ . Consider the function $\Omega(v_{1},\ldots v_{n})$ . By construction, it has no boundary zero, and its excess corner vanishing order is $\sum_{1}^{n}n_{i}$ . Comparing with the index formula (23),

\begin{split}\deg q-\sum_{1}^{k}p_{i}=2\sum(\text{interior zeros})+\sum n_{i}+n.\end{split}

Since all interior vanishing orders are nonnegative by holomorphicity,

Corollary 3.18.

We have $\Omega(v_{1},\ldots v_{n})\neq 0$ in the interior of $\Sigma$ .

The significance is that the algebraic vector bundle structure on $\mathcal{E}\to\mathbb{P}^{1}(\mathbb{R})$ now extends over the entire $\Sigma$ . The $v_{1},\ldots v_{n}$ now provide the basis of local sections for the vector bundle $u^{*}TX|_{\Sigma}$ . One upshot is that an algebraic structure arises on $u^{*}TX\to\Sigma$ from solving the Cauchy-Riemann equation with Lagrangian boundary:

(u^{*}TX,u^{*}TL)\simeq(\oplus_{1}^{n}\mathcal{O}(n_{i}),\text{natural real structure}).

(28)

In contrast, the Lagrangians are only assumed to be smooth, not necessarily real analytic.

To analyze obstructions, Serre duality motivates us to consider the dualized cokernel to the ordinary (unweighted, unextended) linearized Cauchy-Riemann operator. A dualized cokernel element $\eta$ is represented by a holomorphic 1-forms in $\Omega^{1,0}(\Sigma,u^{*}T^{*}X)$ with $L^{2}$ integrablity, and its $T^{*}X$ factor lies in the annilator of the $TL$ boundary condition. Equivalently, for all test vector fields $v\in W^{1,2}(\Sigma,TX,TL)$ ,

\int_{\Sigma}\langle\bar{\partial}v\wedge\eta\rangle=0,

where $\langle,\rangle$ is the pairing of $TX$ with $T^{*}X$ , and the wedge takes care of the forms on $\Sigma$ . In the canonical form (28), this dualized cokernel is isomorphic to $\Gamma(\mathbb{RP}^{1},\oplus_{1}^{n}\mathcal{O}(-2-n_{i})).$ In particular,

Corollary 3.19.

In the teardrop curve case $k=0$ , the strip case $k=1$ and the triangle case $k=2$ , the vanishing of cokernel is equivalent to $n_{i}\geq-1$ for all $i$ .

For $k\geq 3$ the deformation of the holomorphic polygons is governed instead by the extended Cauchy-Riemann equation, since the punctured Riemann surface structure on $\Sigma$ is allowed to vary. The dualized cokernel of the extended Cauchy-Riemann operator, is the subspace of the dualized cokernel of the ordinary Cauchy-Riemann operator, which pairs trivially with $J_{X}\circ du\circ\rho$ for any $\rho$ representing some tangent vector of the Stasheff associahedron.