Now, $\sqrt{W^{T}}W_{i}^{-T}\sqrt{W^{T}}$ is a projector because it is
Hermitan and equal to its square,

$\displaystyle\left(\sqrt{W^{T}}W_{i}^{-T}\sqrt{W^{T}}\right)^{2}$
$\displaystyle=\sqrt{W^{T}}W_{i}^{-T}W^{T}W_{i}^{-T}\sqrt{W^{T}}$ (314)
$\displaystyle=\sqrt{W^{T}}W_{i}^{-T}W_{i}^{T}W_{i}^{-T}\sqrt{W^{T}}$ (315)
$\displaystyle=\sqrt{W^{T}}W_{i}^{-T}\sqrt{W^{T}}$ (316)

where the second line uses that the support of $W_{i}^{-T}$ is $P_{i}$. Being
a projector, $\sqrt{W^{T}}W_{i}^{-T}\sqrt{W^{T}}\leq I$ and we have

$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y^{\prime\prime\prime}_{ii}$
$\displaystyle=\lambda n^{-1}n_{i}\sqrt{W^{T}}W_{i}^{-T}\sqrt{W^{T}}$ (317)
$\displaystyle\leq\lambda n^{-1}n_{i}I_{\mathcal{B}}.$ (318)

We can add to that to get equality. Let $\sigma_{i}$ be the slack in the (318)
inequality and set

$\displaystyle
Y^{\prime\prime\prime\prime}=Y^{\prime\prime\prime}+\sum_{i}\dim({\mathcal{A}_{i}})I_{\mathcal{A}_{i}}\otimes\sigma,$
(319)

giving

$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y^{\prime\prime\prime\prime}_{ii}$
$\displaystyle=\lambda n^{-1}n_{i}I_{\mathcal{B}}.$ (320)

We have $Y^{\prime\prime\prime\prime}\in
S^{\prime}\otimes\mathcal{L}(\mathcal{B})$. Indeed, since
$Y^{\prime\prime}\in(S/S_{0}+\textnormal{span}\\{P_{i}\\})\otimes\mathcal{B}$
by (312), and since we’ve only multiplied by $\sqrt{W^{T}}$ on the
$\mathcal{B}$ side and added terms in
$(S/S_{0}+\textnormal{span}\\{P_{i}\\})\otimes\mathcal{B}$, we have
$Y^{\prime\prime\prime\prime}\in(S/S_{0}+\textnormal{span}\\{P_{i}\\})\otimes\mathcal{B}$.
But in fact $Y^{\prime\prime\prime\prime}$ is perpendicular to anything in
$\textnormal{span}\\{P_{i}\\}/\mathbb{C}I$. For, suppose
$R\in\textnormal{span}\\{P_{i}\\}/\mathbb{C}I$. Then
$R=\alpha_{1}P_{i}+\dots+\alpha_{n}P_{n}$ and

$\displaystyle\textnormal{Tr}_{\mathcal{A}}(RY^{\prime\prime\prime\prime})$
$\displaystyle=\sum_{i}\alpha_{i}\lambda n^{-1}n_{i}I_{\mathcal{B}}$ (321)
$\displaystyle=\lambda n^{-1}I_{\mathcal{B}}\sum_{i}\alpha_{i}n_{i}$ (322)
$\displaystyle=\lambda n^{-1}I_{\mathcal{B}}\textnormal{Tr}(R)=0.$ (323)

Therefore
$Y^{\prime\prime\prime\prime}\in(S/S_{0}+\mathbb{C}I)\otimes\mathcal{L}(\mathcal{B})=S^{\prime}\otimes\mathcal{L}(\mathcal{B})$.
Substituting $R=I_{\mathcal{A}}$ gives

$\displaystyle\textnormal{Tr}_{\mathcal{A}}(Y^{\prime\prime\prime\prime})$
$\displaystyle=\lambda I_{\mathcal{B}}.$ (324)

And finally,

$\displaystyle Y^{\prime\prime\prime\prime}$
$\displaystyle\geq(I\otimes\sqrt{W^{T}})Y^{\prime}(I\otimes\sqrt{W^{T}})$
(325)
$\displaystyle\geq(I\otimes\sqrt{W^{T}}){|{\Phi}\rangle\langle{\Phi}|}(I\otimes\sqrt{W^{T}})$
(326) $\displaystyle={|{\sqrt{W}}\rangle\langle{\sqrt{W}}|}.$ (327)

Therefore, $Y^{\prime\prime\prime\prime}$ is feasible for definition 8 for
$\tilde{\vartheta}(S^{\prime},W)$ with value $\lambda$. ∎

We now work toward the reverse inequality,
$\tilde{\vartheta}(S^{\prime},W)\geq\tilde{\vartheta}(S,n\Psi(W))$.

######  Lemma 65.

Let $U_{\alpha\beta}\in\mathcal{L}(\mathcal{A})$ be the generalized Pauli
operators

$\displaystyle U_{\alpha\beta}=\sum_{j}\omega^{\beta
j}|{j+\alpha}\rangle\langle{j}|$ (328)

where $\omega$ is a primitive root of unity of order $\dim(\mathcal{A})$ and
the indices $\alpha$ and $\beta$ range from $0$ to $\dim(\mathcal{A})-1$. Then
for any $M\in\mathcal{L}(\mathcal{A})$ we have

$\displaystyle\sum_{\alpha\beta}U_{\alpha\beta}MU_{\alpha\beta}^{\dagger}$
$\displaystyle=\dim(\mathcal{A})\textnormal{Tr}(M)I.$ (329)

###### Proof.

The generalized Paulis form a basis of $\mathcal{L}(\mathcal{A})$, so we can
write $M$ in this basis,

$\displaystyle
M=\sum_{\alpha^{\prime}\beta^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}U_{\alpha^{\prime}\beta^{\prime}}.$
(330)

The generalized Paulis satisfy the following braiding relation:

$\displaystyle U_{\alpha\beta}U_{\alpha^{\prime}\beta^{\prime}}$
$\displaystyle=\omega^{\alpha^{\prime}\beta-\alpha\beta^{\prime}}U_{\alpha^{\prime}\beta^{\prime}}U_{\alpha\beta}.$
(331)

Using this braiding relation, we can evaluate the sum,

$\displaystyle\sum_{\alpha\beta}U_{\alpha\beta}MU_{\alpha\beta}^{\dagger}$
$\displaystyle=\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}U_{\alpha\beta}U_{\alpha^{\prime}\beta^{\prime}}U_{\alpha\beta}^{\dagger}$
(332)
$\displaystyle=\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}\omega^{\alpha^{\prime}\beta-\alpha\beta^{\prime}}U_{\alpha^{\prime}\beta^{\prime}}U_{\alpha\beta}U_{\alpha\beta}^{\dagger}$
(333)
$\displaystyle=\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}\omega^{\alpha^{\prime}\beta-\alpha\beta^{\prime}}U_{\alpha^{\prime}\beta^{\prime}}$
(334) $\displaystyle=\sum_{\alpha\beta}M_{00}U_{00}$ (335)
$\displaystyle=\dim(\mathcal{A})^{2}M_{00}I$ (336)

where (335) follows because $\sum_{\alpha}\omega^{\alpha\beta^{\prime}}=0$
unless $\beta^{\prime}=0$, and similarly for the sum over $\beta$. Since
$M_{00}=\dim(\mathcal{A})^{-1}\textnormal{Tr}M$, we have the desired result. ∎

The following corollary follows by linearity.

######  Corollary 66.

Let $U_{\alpha\beta}\in\mathcal{L}(\mathcal{A})$ be as in lemma 65. For any
$M\in\mathcal{L}(\mathcal{A})\otimes\mathcal{L}(\mathcal{B})$ we have

$\displaystyle\sum_{\alpha\beta}(U_{\alpha\beta}\otimes
I_{\mathcal{B}})M(U_{\alpha\beta}^{\dagger}\otimes I_{\mathcal{B}})$
$\displaystyle=\dim(\mathcal{A})I_{\mathcal{A}}\otimes\textnormal{Tr}_{\mathcal{A}}M.$
(337)

######  Theorem 67.

Let $S$ be an $S_{0}$-graph and take $\Psi$ from definition 23. Let
$S^{\prime}=S/S_{0}+\mathbb{C}I$. For $W\in\textnormal{Pos}(\mathcal{A})$,

$\displaystyle\tilde{\vartheta}(S^{\prime},W)\geq\tilde{\vartheta}(S,n\Psi(W)).$
(338)

###### Proof.

With $S_{0}$ decomposed as in (81), let $P_{i}$ be the projector onto
${\mathcal{A}_{i}}\otimes{\mathcal{Y}_{i}}$. Set $n_{i}=\textnormal{Tr}P_{i}$,
$n=\sum_{i}n_{i}=\dim(\mathcal{A})$, and $W_{i}=P_{i}WP_{i}$.

Define $X=n\Psi(W)$. Let $Y$ be optimal for (21) for
$\tilde{\vartheta}(S^{\prime},W)$,

$\displaystyle Y$ $\displaystyle\in
S^{\prime}\otimes\mathcal{L}(\mathcal{B}),$ (339)
$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y$ $\displaystyle=\lambda W^{T},$
(340) $\displaystyle Y$ $\displaystyle\geq|{W}\rangle\langle{W}|,$ (341)

with $\lambda=\tilde{\vartheta}(S^{\prime},W)$.

We will construct a feasible solution for $\tilde{\vartheta}(S,n\Psi(W))$. For
each ${\mathcal{A}_{i}}$ let
$U_{i\alpha\beta}\in\mathcal{L}({\mathcal{A}_{i}})$ be the generalized Pauli
operators

$\displaystyle U_{i\alpha\beta}=\sum_{j}\omega^{\beta
j}|{j+\alpha}\rangle\langle{j}|,$ (342)

where $\omega$ is a primitive root of unity of order
$\dim({\mathcal{A}_{i}})$. The indices $\alpha$ and $\beta$ range from $0$ to
$\dim({\mathcal{A}_{i}})-1$ so there are $\dim({\mathcal{A}_{i}})^{2}$ such
operators.

Define the projection and twirling operation

$\displaystyle K=n\sum_{i\alpha\beta}n_{i}^{-1}V_{i\alpha\beta}\otimes
V_{i\alpha\beta}^{T}.$ (343)

where

$\displaystyle V_{i\alpha\beta}=(U_{i\alpha\beta}\otimes
I_{\mathcal{Y}})P_{i}.$ (344)

Define

$\displaystyle Y^{\prime}$ $\displaystyle=KYK^{\dagger}.$ (345)

Since $Y\in S^{\prime}\otimes\mathcal{L}(\mathcal{B})\subseteq
S\otimes\mathcal{L}(\mathcal{B})$ and $K\in
S_{0}\otimes\mathcal{L}(\mathcal{B})$ we have $Y^{\prime}\in
S_{0}SS_{0}\otimes\mathcal{L}(\mathcal{B})=S\otimes\mathcal{L}(\mathcal{B})$.
Consider now its partial trace,

$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y^{\prime}$
$\displaystyle=n^{2}\sum_{ii^{\prime}\alpha\alpha^{\prime}\beta\beta^{\prime}}n_{i}^{-1}n_{i^{\prime}}^{-1}\textnormal{Tr}\left((V_{i\alpha\beta}\otimes
V_{i\alpha\beta}^{T})Y(V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}\otimes
V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{{\dagger}T})\right).$ (346)
$\displaystyle=n^{2}\sum_{ii^{\prime}\alpha\alpha^{\prime}\beta\beta^{\prime}}n_{i}^{-1}n_{i^{\prime}}^{-1}V_{i\alpha\beta}^{T}\textnormal{Tr}_{\mathcal{A}}\left(Y(V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}V_{i\alpha\beta}\otimes
I_{\mathcal{B}})\right)V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{{\dagger}T}.$
(347)

Consider the partial trace
$\textnormal{Tr}_{\mathcal{A}}(Y(V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}V_{i\alpha\beta}\otimes
I_{\mathcal{B}}))$. We have $Y\in
S^{\prime}\otimes\mathcal{L}(\mathcal{B})=(S/S_{0}+\mathbb{C}I)\otimes\mathcal{L}(\mathcal{B})$
and
$V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}V_{i\alpha\beta}\otimes
I_{\mathcal{B}}\in S_{0}\otimes\mathcal{L}(\mathcal{B})$. These spaces only
overlap on $\mathbb{C}I\otimes\mathcal{L}(\mathcal{B})$ so we only need to
consider the component of
$V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}V_{i\alpha\beta}$ that
lies in $\mathbb{C}I$. The projection of this factor onto $\mathbb{C}I$ is

$\displaystyle
n^{-1}I_{\mathcal{A}}\textnormal{Tr}(V_{i^{\prime}\alpha^{\prime}\beta^{\prime}}^{\dagger}V_{i\alpha\beta})$
$\displaystyle=n^{-1}I_{\mathcal{A}}\textnormal{Tr}(P_{i})\delta_{ii^{\prime}}\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}$
(348)

where $\delta$ is the Kronecker delta. Continuing,

$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y^{\prime}$
$\displaystyle=n\sum_{i\alpha\beta}n_{i}^{-2}\textnormal{Tr}(P_{i})V_{i\alpha\beta}^{T}\textnormal{Tr}_{\mathcal{A}}(Y)V_{i\alpha\beta}^{{\dagger}T}$
(349) $\displaystyle=\lambda
n\sum_{i\alpha\beta}n_{i}^{-1}V_{i\alpha\beta}^{T}W^{T}V_{i\alpha\beta}^{{\dagger}T}$
(350) $\displaystyle=\lambda
n\sum_{i}n_{i}^{-1}\sum_{\alpha\beta}((U_{i\alpha\beta}\otimes
I_{\mathcal{Y}})W_{i}(U_{i\alpha\beta}\otimes
I_{\mathcal{Y}})^{\dagger})^{T}.$ (351)

Applying corollary 66,

$\displaystyle\textnormal{Tr}_{\mathcal{A}}Y^{\prime}$ $\displaystyle=\lambda
n\sum_{i}n^{-1}\dim({\mathcal{A}_{i}})I_{\mathcal{A}_{i}}\otimes\textnormal{Tr}_{\mathcal{A}_{i}}W_{i}$
(352) $\displaystyle=\lambda
n\sum_{i}\dim({\mathcal{Y}_{i}})^{-1}I_{\mathcal{A}_{i}}\otimes\textnormal{Tr}_{\mathcal{A}_{i}}W_{i}$
(353) $\displaystyle=\lambda X,$ (354)

as desired.

Conjugating both sides of (341) with $K$ yields

$\displaystyle Y^{\prime}$ $\displaystyle\geq
K|{W}\rangle\langle{W}|K^{\dagger}.$ (355)

We have

$\displaystyle K|{W}\rangle$
$\displaystyle=n\sum_{i\alpha\beta}n_{i}^{-1}(V_{i\alpha\beta}\otimes
V_{i\alpha\beta}^{T})|{W}\rangle.$ (356)
$\displaystyle=n\sum_{i\alpha\beta}n_{i}^{-1}(V_{i\alpha\beta}W\otimes
V_{i\alpha\beta}^{T})|{\Phi}\rangle.$ (357)
$\displaystyle=n\sum_{i\alpha\beta}n_{i}^{-1}(V_{i\alpha\beta}WV_{i\alpha\beta}^{\dagger}\otimes
I_{\mathcal{B}})|{\Phi}\rangle.$ (358)
$\displaystyle=n\sum_{i}n_{i}^{-1}\dim({\mathcal{A}_{i}})(I_{\mathcal{A}_{i}}\otimes\textnormal{Tr}_{\mathcal{A}_{i}}W_{i}\otimes
I_{\mathcal{B}})|{\Phi}\rangle.$ (359)
$\displaystyle=n\sum_{i}\dim({\mathcal{Y}_{i}})^{-1}(I_{\mathcal{A}_{i}}\otimes\textnormal{Tr}_{\mathcal{A}_{i}}W_{i}\otimes
I_{\mathcal{B}})|{\Phi}\rangle.$ (360) $\displaystyle=(X\otimes
I_{\mathcal{B}})|{\Phi}\rangle.$ (361) $\displaystyle=|{X}\rangle$ (362)

Therefore $Y^{\prime}\geq|{X}\rangle\langle{X}|$ and $Y^{\prime}$ satisfies
(21) for $\tilde{\vartheta}(S,X)$ with value
$\lambda=\tilde{\vartheta}(S^{\prime},W)$. ∎

##  Appendix B Proofs for convex corner geometry

###### Proof of (iii).

Suppose $\dim(\mathcal{A})>1$ and that $\mathcal{C}$ has a facet. By lemma 27
there is some $X_{0}$ in the relative interior of $\mathcal{C}$, and
$P\mathcal{C}P=\mathcal{C}$ where $P$ is the projector onto the support of
$X_{0}$. It must be the case that $X_{0}>0$, i.e., $X_{0}$ is not just in the
relative interior but in fact in the interior. Otherwise the projector $P$ is
not full rank, and $\mathcal{C}$ lies in a subspace
$P\mathcal{L}(\mathcal{A})P$ of dimension $\textnormal{rank}(X_{0})^{2}$,
strictly smaller than $\dim(\mathcal{L}(\mathcal{A}))-1$, the required
dimension of a facet. ∎

###### Proof of (iii).

Suppose the supporting hyperplane defining a facet passes through the origin.
That is, suppose the facet is of the form

$\displaystyle\mathcal{F}=\\{X\in\mathcal{C}:\textnormal{Tr}(XY)=0\\}.$ (363)

for some $Y$, with $\textnormal{Tr}(XY)\leq 0$ for all $X\in\mathcal{C}$.

By (iii) there is some $X_{0}\in\mathcal{C}$, $X_{0}>0$. Then for any $Z\geq
0$ there is some $\epsilon>0$ such that $0\leq\epsilon Z\leq X_{0}$, and hence
by hereditarity $\epsilon Z\in\mathcal{C}$. It must be the case that $Y\leq
0$. Otherwise there is some $Z\geq 0$ such that $\textnormal{Tr}(ZY)>0$ (e.g.
the projector onto the positive eigenvalues of $Y$). Then there is some
$\epsilon>0$ such that $\epsilon Z\in\mathcal{C}$, and
$\textnormal{Tr}(\epsilon ZY)>0$, contradicting that $\textnormal{Tr}(XY)\leq
0$ for all $X\in\mathcal{C}$.

Let $P$ be the projector onto the support of $Y$. The condition
$\textnormal{Tr}(XY)=0$ for all $X\in\mathcal{F}$, combined with $Y\leq 0$,
requires $PXP=0$ for all $X\in\mathcal{F}$. But this is inconsistent with
$\mathcal{F}$ having affine dimension $\dim(\mathcal{L}(\mathcal{A}))-1$. If
$P\neq 0$,
$\dim(\mathcal{F})\leq(\dim(\mathcal{A})-\textnormal{rank}(P))^{2}<\dim(\mathcal{L}(\mathcal{A}))-1$
(assuming $\dim(\mathcal{A})>1$). If $P=0$ then $Y=0$ and
$\dim(\mathcal{F})=\dim(\mathcal{C})=\dim(\mathcal{L}(\mathcal{A}))>\dim(\mathcal{L}(\mathcal{A}))-1$.
In either case $\dim(\mathcal{F})=\dim(\mathcal{L}(\mathcal{A}))-1$ is not
possible. So it is not possible for the hyperplane to pass through the origin.

A hyperplane not passing through the origin is of the form
$\\{X\in\textnormal{Herm}(\mathcal{A}):\textnormal{Tr}(XY)=\alpha\\}$ for some
$\alpha\neq 0$. Since supporting hyperplanes satisfy
$\textnormal{Tr}(XY)\leq\alpha$ for all $X\in\mathcal{C}$, and since
$0\in\mathcal{C}$, it is not possible that $\alpha<0$. Therefore $\alpha>0$.
We can then rescale, defining $Y^{\prime}=Y/\alpha$ to get
$\mathcal{F}=\\{X\in\textnormal{Herm}(\mathcal{A}):\textnormal{Tr}(XY^{\prime})=1\\}$
∎

###### Proof of (iii).

Let $X_{0}\in\textnormal{relint}(\mathcal{F})$. Note first that

$\displaystyle X_{0}+\epsilon Y\not\in\mathcal{C}$ (364)

for all $\epsilon>0$ because $Y$ forms a supporting hyperplane, and adding
$\epsilon Y$ puts the point on the wrong side of that hyperplane:
$\textnormal{Tr}((X_{0}+\epsilon Y)Y)=1+\epsilon\textnormal{Tr}(Y^{2})>1$
unless $Y=0$, which in turn is forbidden by $\textnormal{Tr}(XY)=1$ for
$X\in\mathcal{F}$.

We first show $Y\geq 0$. The hypothesis $\textnormal{Tr}(X_{0}Y)=1$ forbids
$Y\leq 0$, so $Y$ has at least one positive eigenvalue. If $Y$ also has at
least one negative eigenvalue then there is some $Z>0$ such that
$\textnormal{Tr}(ZY)=0$. Since $X_{0}\in\textnormal{relint}(\mathcal{F})$ and
$\textnormal{Tr}(ZY)=0$ (so $Z$ runs parallel to the hyperplane), $Z$ can be
chosen small enough that $X_{0}\pm Z\in\mathcal{F}$. Since $Z>0$ there is some
$\epsilon>0$ such that $-Z\leq\epsilon Y\leq Z$. Now,

$\displaystyle X_{0}-Z\in\mathcal{F}$ $\displaystyle\implies X_{0}-Z\geq 0$
$\displaystyle\implies X_{0}+\epsilon Y\geq 0$ (365)

Also, since $\epsilon Y\leq Z$,

$\displaystyle X_{0}+Z\in\mathcal{F}$ $\displaystyle\implies
X_{0}+Z\in\mathcal{C}=\mathcal{C}^{\sharp\sharp}$
$\displaystyle\implies\textnormal{Tr}((X_{0}+Z)X)\leq 1\textrm{ for all
}X\in\mathcal{C}^{\sharp}$
$\displaystyle\implies\textnormal{Tr}((X_{0}+\epsilon Y)X)\leq 1\textrm{ for
all }X\in\mathcal{C}^{\sharp}$ (366)

By the second anti-blocker theorem,
$\mathcal{C}=\mathcal{C}^{\sharp\sharp}=\\{W\geq 0:\textnormal{Tr}(WX)\leq
1\textrm{ for all }X\in\mathcal{C}^{\sharp}\\}$ so (365)-(366) imply
$X_{0}+\epsilon Y\in\mathcal{C}$. This violates (364). Therefore $Y\geq 0$.

Since $Y\geq 0$ and $\textnormal{Tr}(XY)\leq 1$ for all $X\in\mathcal{C}$
(because $Y$ is a supporting hyperplane of $\mathcal{C}$), we have
$Y\in\mathcal{C}^{\sharp}$.

Every element $X\in\mathcal{F}$ defines a supporting hyperplane of
$\mathcal{C}^{\sharp}$ at $Y$, because we have $\textnormal{Tr}(XY)=1$ from
(208) and $\textnormal{Tr}(XY^{\prime})\leq 1$ for all
$Y^{\prime}\in\mathcal{C}^{\sharp}$ due to
$X\in\mathcal{C}=\mathcal{C}^{\sharp\sharp}$.

Since $\mathcal{F}$ has affine dimension $\dim(\mathcal{L}(\mathcal{A}))-1$
and that affine space doesn’t pass through the origin (by (iii)), we have
linearly independent
$X_{1},\dots,X_{\dim(\mathcal{L}(\mathcal{A}))}\in\mathcal{F}$. These are all
supporting hyperplanes of $\mathcal{C}^{\sharp}$ at $Y$, and their
intersection is the single point $Y$, so $Y$ is a vertex. ∎

###### Proof of (iii).

Let $X\neq 0$ be a vertex of a convex corner $\mathcal{C}$. Suppose $X$ is not
maximal. Then there is $Y\geq 0,Y\neq 0$ such that $X+Y\in\mathcal{C}$. We
will show this leads to the local shape of $\mathcal{C}$ at $X$ being flat or
curved, not a sharp point as would be required for $X$ to be a vertex.

Let $R_{X},R_{Y}$ be such that $R_{X}R_{X}^{\dagger}=X$,
$R_{Y}R_{Y}^{\dagger}=Y$, and $R_{X}R_{Y}^{\dagger}+R_{Y}R_{X}^{\dagger}\neq
0$. It suffices to take $R_{X}=\sqrt{X}$ and $R_{Y}=\sqrt{Y}$ unless
$\textnormal{Tr}(XY)=0$, in which case it suffices to take $R_{Y}=\sqrt{Y}U$
where $U$ is a unitary causing the support of $R_{Y}$ to overlap with the
support of $R_{X}$.

For $0<\epsilon<1$ let

$\displaystyle Z$ $\displaystyle=X+\epsilon^{2}(Y-X)$ (367) $\displaystyle W$
$\displaystyle=\epsilon\sqrt{1-\epsilon^{2}}\left(R_{X}R_{Y}^{\dagger}+R_{Y}R_{X}^{\dagger}\right).$
(368)

Then

$\displaystyle Z\pm W$
$\displaystyle=(1-\epsilon^{2})X+\epsilon^{2}Y\pm\epsilon\sqrt{1-\epsilon^{2}}\left(R_{X}R_{Y}^{\dagger}+R_{Y}R_{X}^{\dagger}\right)$
(369) $\displaystyle=\left(\sqrt{1-\epsilon^{2}}R_{X}\pm\epsilon
R_{Y}\right)\left(\sqrt{1-\epsilon^{2}}R_{X}\pm\epsilon
R_{Y}\right)^{\dagger}$ (370) $\displaystyle\geq 0$ (371)
$\displaystyle(X+Y)-(Z\pm W)$ $\displaystyle=Y+\epsilon^{2}(X-Y)\mp W$ (372)
$\displaystyle=(1-\epsilon^{2})Y+\epsilon^{2}X\mp\epsilon\sqrt{1-\epsilon^{2}}\left(R_{X}R_{Y}^{\dagger}+R_{Y}R_{X}^{\dagger}\right)$
(373) $\displaystyle=\left(\sqrt{1-\epsilon^{2}}R_{Y}\mp\epsilon
R_{X}\right)\left(\sqrt{1-\epsilon^{2}}R_{Y}\mp\epsilon
R_{X}\right)^{\dagger}$ (374) $\displaystyle\geq 0$ (375)

so $0\leq Z\pm W\leq X+Y$. Since $X+Y\in\mathcal{C}$, by hereditarity we have
$Z\pm W\in\mathcal{C}$.

Let $V,\alpha$ define a supporting hyperplane of $\mathcal{C}$ at $X$ that is
not parallel to $W$ (note that the direction of $W$ does not depend on
$\epsilon$),

$\displaystyle\textnormal{Tr}(VX)$ $\displaystyle=\alpha$ (376)
$\displaystyle\textnormal{Tr}(VT)$ $\displaystyle\leq\alpha\textnormal{ for
all }T\in\mathcal{C}$ (377) $\displaystyle\textnormal{Tr}(VW)$
$\displaystyle\neq 0.$ (378)

If $X$ is a vertex then such a hyperplane must exist: if all hyperplanes are
parallel to $W$ then their intersection cannot be dimension 0. Now consider
the points $Z\pm W\in\mathcal{C}$,

$\displaystyle Z\pm W\in\mathcal{C}$
$\displaystyle\implies\textnormal{Tr}(V(Z\pm W))\leq\alpha$ (379)
$\displaystyle\implies\textnormal{Tr}(V(Z-X\pm W))\leq 0$ (380)

As $\epsilon\to 0$, $Z-X$ scales as $O(\epsilon^{2})$ whereas $W$ scales as
$O(\epsilon)$. So for sufficiently small $\epsilon$ the $\textnormal{Tr}(VW)$
term dominates, requiring $\pm\textnormal{Tr}(VW)\leq 0$ in contradiction of
(378). So all supporting hyperplanes of $\mathcal{C}$ at $X$ must be parallel
to $W$ and $X$ cannot be a vertex. ∎

###### Proof of (iii).

By the symmetry of the problem we need only prove that $X_{0}$ is a vertex and
forms a facet of $\mathcal{C}^{\sharp}$. Consider the semidefinite program

$\displaystyle p^{*}=\min\\{\lambda:$
$\displaystyle\textnormal{Tr}(X_{0}Y)=1,$
$\displaystyle\textnormal{Tr}(X_{i}Y)\leq\lambda\textrm{ for
}i\in\\{1,\ldots,m\\},$ $\displaystyle Y\geq 0\\}.$ (381)

After showing this is feasible with $\lambda<1$ we will show that (209) is a
facet and contains $Y$. Intuitively, the conditions of (381) mandate that $Y$
is a supporting hyperplane of $\mathcal{C}^{\sharp}$ at $X_{0}$, and remains a
supporting hyperplane if “wiggled” in a direction perpendicular to $X_{0}$. We
will return to this at the end.

The Lagrangian of this SDP is

$\displaystyle L(\lambda,Y;\mu,b_{i})$
$\displaystyle=\lambda+\mu(1-\textnormal{Tr}(X_{0}Y))+\sum_{i=1}^{m}b_{i}(\textnormal{Tr}(X_{i}Y)-\lambda)$
(382)
$\displaystyle=\mu+\textnormal{Tr}\left(Y\left(\sum_{i=1}^{m}b_{i}X_{i}-\mu
X_{0}\right)\right)+\lambda\left(1-\sum_{i=1}^{m}b_{i}\right).$ (383)

The dual program is

$\displaystyle d^{*}=\max\Big{\\{}\mu:$
$\displaystyle\sum_{i=1}^{m}b_{i}X_{i}\geq\mu X_{0}$ (384)
$\displaystyle\sum_{i=1}^{m}b_{i}=1,b_{i}\geq 0\Big{\\}}.$ (385)

This cannot be feasible for $\mu\geq 1$, since that would imply $X_{0}\leq\sum
b_{i}X_{i}$, meaning $X_{0}$ is dominated by a convex combination of the other
$X_{i}$, in contradiction to the generators being a minimal set. Therefore
$d^{*}<1$. Slater’s condition is satisfied, with $Y=I/\textnormal{Tr}(X_{0})$
and $\lambda$ large being a feasible point in the relative interior of (381).
Therefore $p^{*}=d^{*}$, giving $p^{*}<1$.

Take $Y$ to be feasible for (381) with $\lambda<1$. Let $K_{0}=0$ and let
$\\{K_{j}:j\in\\{1,\dots,\dim(\mathcal{L}(\mathcal{A}))-1\\}\\}$ be a
Hermitian basis of $\mathcal{L}(\mathcal{A})/X_{0}$ (the space perpendicular
to $X_{0}$ under the Hilbert-Schmidt inner product), with $\left\lVert
K_{j}\right\rVert\leq 1$. With $\epsilon>0$ small (we shall later see how
small), define
$Y_{j}=(1+\epsilon\textnormal{Tr}(X_{0}))^{-1}(Y+\epsilon(I+K_{j}))$ for
$j\in\\{0,\dots,\dim(\mathcal{L}(\mathcal{A}))-1\\}$. Since $\left\lVert
K_{j}\right\rVert\leq 1$, we have

$\displaystyle Y_{j}\geq 0.$ (386)

Since each $K_{j}$ is orthogonal to $X_{0}$ and $\textnormal{Tr}(X_{0}Y)=1$,
we have

$\displaystyle\textnormal{Tr}(X_{0}Y_{j})$
$\displaystyle=(1+\epsilon\textnormal{Tr}(X_{0}))^{-1}\textnormal{Tr}(X_{0}(Y+\epsilon(I+K_{j})))$
(387) $\displaystyle=1.$ (388)

As for the rest of the $X_{i}$,

$\displaystyle\textnormal{Tr}(X_{i}Y_{j})$
$\displaystyle=(1+\epsilon\textnormal{Tr}(X_{0}))^{-1}\textnormal{Tr}(X_{i}(Y+\epsilon(I+K_{j})))$
(389)
$\displaystyle\leq(1+\epsilon\textnormal{Tr}(X_{0}))^{-1}(\lambda+\epsilon\textnormal{Tr}(X_{i}(I+K_{j})))$
(390) $\displaystyle\leq\lambda+\epsilon\textnormal{Tr}(X_{i}(I+K_{j}))$ (391)
$\displaystyle<1$ (392)

where the last line uses $\lambda<1$ and $\epsilon$ sufficiently small.

Since $\\{X_{i}:i\in\\{0,\ldots,m\\}\\}$ generate $\mathcal{C}$, each element
of $\mathcal{C}$ is dominated by a convex combination of these generators. We
have then

$\displaystyle\textnormal{Tr}(XY_{j})$ $\displaystyle\leq 1\textrm{ for all
}X\in\mathcal{C}$ (393)

By (386) and (393), each $Y_{j}\in\mathcal{C}^{\sharp}$. By (388), the $Y_{j}$
are on the surface of $\mathcal{C}^{\sharp}$. Since the $K_{j}$ (and hence the
$Y_{j}$) span an affine space of dimension $\dim(\mathcal{L}(\mathcal{A}))-1$,
they define a facet of $\mathcal{C}^{\sharp}$. And by (388),
$\mathcal{F}_{0}=\\{Z\in\textnormal{Herm}(\mathcal{A}):\textnormal{Tr}(X_{0}Z)=1\\}$
is the supporting hyperplane defining this facet. ∎

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# Some Remarks on $H$-stability of Syzygy Bundles on Algebraic Surfaces

H. Torres-López  CONACyT - U. A. Matemáticas, U. Autónoma de Zacatecas
Calzada Solidaridad entronque Paseo a la Bufa,
C.P. 98000, Zacatecas, Zac. México<EMAIL_ADDRESS>and  A. G. Zamora  U. A.
Matemáticas, U. Autónoma de Zacatecas
Calzada Solidaridad entronque Paseo a la Bufa,
C.P. 98000, Zacatecas, Zac. México<EMAIL_ADDRESS>
###### Abstract.

Let $L$ be a globally generated line bundle over a smooth irreducible
projective surface$X$. The syzygy bundle $M_{L}$ is the kernel of the
evaluation map $H^{0}(L)\otimes\mathcal{O}_{X}\to L$. The main theorem proves
that the syzygy bundle defined by $nL+D$ is stable for any polarization $H$,
where $L$ is any ample , $D$ is an arbitrary divisor and $n$ is a sufficiently
large natural number. Taking $n=1$, we obtain the $L$-stability of $M_{L}$ for
Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. Finally, the
$-K$-stability of syzygy bundles $M_{L}$ over del Pezzo surfaces is proved.

###### Key words and phrases:

Syzygy bundles, $H-$stability, Algebraic Surfaces

###### 2000 Mathematics Subject Classification:

14J60

The first author was partially supported by project FORDECYT 265667. The
second author was partially supported by Conacyt Grant CB 2015-257079

##  1\. Introduction

Let $X$ be a smooth irreducible projective variety and let $L$ be a globally
generated line bundle over $X$ (from now on simply a generated bundle). The
kernel $M_{L}$ of the evaluation map
$H^{0}(L)\otimes\mathcal{O}_{X}\rightarrow L$ fits into the following exact
sequence

(1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(L)\otimes\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The bundle $M_{L}$ is called a syzygy bundle. The rank of $M_{L}^{\vee}$ is
$h^{0}(L)-1$. The vector bundles $M_{L}$ have been extensively studied from
different points of view.

When $X$ is a projective irreducible smooth curve of genus $g\geq 1$ L. Ein
and R. Lazarsfeld showed in [9] that the syzygy bundle $M_{L}$ is stable for
$d>2g$ and it is semi-stable for $d=2g$ (see also [2]). After this, the semi-
stability of $M_{L}$ was proved for line bundles with $deg(L)\geq
2g-\text{cliff}(C)$ (see [13, Corollary 5.4] and [6, Theorem 1.3]). In ([14]),
Paranjape and Ramanan proved that $M_{K_{C}}$ is semi-stable and is stable if
$C$ is non-hyperelliptic. In [15], Schneider showed that $M_{L}$ is semi-
stable for a general curve $C$ (see also [3]). The semi-stability for
incomplete linear series over general curves was proved in [5].

In [11], Flenner showed the stability of $M_{L}$ for projective spaces. The
stability of syzygy bundles for incomplete linear series in projective spaces
has been studied by many authors (see [8], [12], [1]). Recently, in [4], the
authors proved that given an Abelian variety $A$ and any ample line bundle $L$
on $A$ the syzygy bundle $M_{L^{d}}$ is $L$-stable if $d\geq 2$.

In the case of a projective surface $X$, we must start by fixing a
polarization $H$ and then ask for the $H$-stability of the syzygy bundle
$M_{L}$. Recall that $H$-stability for a vector bundle $E$ on $X$ means that
for any sub-bundle $F\subset E$

$\mu_{H}(F):=\frac{c_{1}(F).H}{\text{rank}(F)}<\mu_{H}(E):=\frac{c_{1}(E).H}{\text{rank}(E)}.$

In the pioneer work of Camere ([7]) it was proved that $M_{L}$ is $L$-stable
for any ample and generated $L$ on a $K3$ surface and for any generated $L$
with $L^{2}\geq 14$ on an abelian surface $X$. In [10] Ein, Lazarfeld and
Mustopa fixed an ample divisor $L$ and an arbitrary divisor $D$ over $X$, and
setting set $L_{d}=dL+P$ ($d\in\mathbb{N}$ showed that $M_{L_{d}}$ is
$L$-stable for $d>>0$. They raise the question (Problem 2.5) of the stability
of $M_{L_{d}}$ with respect to any polarization for $d>>0$. Our first goal is
to give an affirmative answer to this question:

Theorem 2.1 Let $X$ be a smooth projective irreducible surface. Let $L$ and
$H$ be ample line bundles on $X$. For an arbitrary divisor $D$ and
$n\in\mathbb{N}$, denote $L_{n}=nL+D$. Then for $n>>0$, the syzygy bundle
$M_{L_{n}}$ is $H$-stable.

Section 2 is devoted to the proof of this theorem. It is strongly based on the
techniques developed in [10].

Another important issue is to obtain an effective version of this result (i.e.
explicitly determining the integer $n$ for which the stability can be
achieved), it is the Problem 2.4 in [10]. Section 3 of the paper is
essentially devoted to obtaining such effective results in a variety of
particular cases, all of them concerning regular surfaces. The first general
observation is:

Theorem 3.1 Let $X$ be a smooth projective surface. Let $L$ be an ample and
generated line bundle over $X$ and $H$ be a divisor such that an irreducible
and non-singular curve $C$ exists in $|H|$. Assume that

  1. (1)

$h^{1}(L-H)=0$;

  2. (2)

$h^{0}(H)\geq h^{0}(L|_{C})$;

  3. (3)

$M_{L|_{C}}$ is semi-stable;

Then $M_{L}$ is $H$-stable.

If $L=H$ and $X$ is regular, then conditions (1) and (2) are automatically
satisfied and in order to prove the stability of $M_{L}$ it is sufficient to
prove the semi-stability of $M_{L|_{C}}$. Using this idea we obtain the
$L$-stability of the syzygy bundle $M_{L}$ in the following cases: if $|L|$
contains either a genus $\leq 1$ curve or a Brill-Noether general curve
(Corollary 3.2), for del Pezzo surfaces ( Corollary 3.4), Hirzebruch surfaces
(Corollary 3.5) and Enriques surfaces with the condition such that
$\text{cliff}(C)\geq 2$ (Corollary 3.6).

Moreover, we study the stability of syzygy bundle over del Pezzo surfaces with
respect to the anti-canonical polarization:

Theorem 3.7 Let $X$ be a del Pezzo surface and let $L$ be a generated line
bundle on $X$. If $L$ contains an irreducible curve, then the vector bundle
$M_{L}$ is $-K_{X}$-stable.

Conventions: We work over the field of complex numbers $\mathbb{C}$. Given a
coherent sheaf $\mathcal{G}$ on a variety $X$ we write $h^{i}(\mathcal{G})$ to
denote the dimension of the $i$-th cohomology group $H^{i}(X,\mathcal{G})$.
The sheaf $K_{X}$ will denote the canonical sheaf on $X$. A surface always
means a smooth (or non-singular) projective and irreducible surface.

##  2\. Stability of $M_{L_{n}}$

The aim of this section is to prove:

######  Theorem 2.1.

Let $X$ be a projective surface. Let $L$ and $H$ be ample line bundles on $X$.
For an arbitrary divisor $D$ and $n\in\mathbb{N}$ denote $L_{n}=nL+D$. Then
for $n>>0$ the syzygy bundle $M_{L_{n}}$ is $H$-stable.

###### Proof.

The proof is modeled on the argument in [10]. If necessary, replace $H$ by a
sufficiently high power $m$ in such a way that $mH$ contains an irreducible
and non-singular curve $C$ and take a point $x\in C$. For the remainder of the
argument $H$, $C\in|H|$ and $x\in C$ would be fixed. Note that $C\in|H\otimes
m_{x}|$.

Taking $n>>0$ we can assume that $h^{1}(L_{n}-H)=0$ and that $M_{L_{n}|_{C}}$
is a semi-stable bundle on $C$. Note that for the latter condition it is
sufficient to assume that $\deg L_{n}|_{C}=nL.H+D.H\geq 2g_{H}$ (see [9]).
Given any sub-bundle $F_{n}\subset M_{L_{n}}$ we have, thanks to the condition
$h^{1}(L_{n}-H)=0$, a commutative diagram:

(2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{n}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(L_{n}-H)\otimes\mathcal{O}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{L_{n}}|_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{L_{n}|_{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

(see [10] page 76 and also [7], proof of Proposition 3).

Assume that $F_{n}\subset M_{L_{n}}$ is a destabilizing sub-bundle. Note that
for any vector bundle $E$ on $X$, $\mu_{H}(E)=\mu(E|_{C})$. Thus
$\mu(F_{n}|_{C})\geq\mu(M_{L_{n}}|_{C})$.

An analogous result to the following Lemma was established in [10] (Lemma
1.1):

######  Lemma 2.2.

Let $F_{n}\subset M_{L_{n}}$ be a subbundle with
$\mu(F_{n})\geq\mu(M_{L_{n}})$. Then,

$\text{rk}(F_{n})\geq h^{0}(H)-1+\text{rk}(K_{n}).$

The proof of the Lemma is quite analogous to the one in [10]. We only need to
replace $B$ by $L_{n}-H$ and $L_{d}-B$ by $H$ and to compute the dimension of
a fiber. We include a proof for the reader convenience and to indicate how the
inequality can be bettered with respect to that obtained in [10].

###### Proof.

If $K_{n}=0$ in the exact sequence (2), then

$\displaystyle\mu(N_{n})=\mu(F_{n}|_{C})\geq\mu(M_{L_{n}}|_{C})>\mu(M_{L_{n}|_{C}})$

gives a contradiction with the semi-stability of $M_{L_{n}|_{C}}$ and thus we
have $K_{n}\neq 0$.

The multiplication map of sections

$\displaystyle\nu:|H\otimes m_{x}|\times|L_{n}-H|\rightarrow|L_{n}\otimes
m_{x}|$

is a finite morphism. After localizing at the given point $x$ we obtain a
commutative diagram:

$\textstyle{{(K_{n})}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(F_{n}|_{C})_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(L_{n}-H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{(M_{nL}|_{C})}_{x}=H^{0}(L_{n}\otimes
m_{x}).}$,

Let $Z=\nu^{-1}(|(F_{n}|_{C})_{x}|)$, since $\nu$ is finite
$\dim|(F_{n}|_{C})_{x}|\geq\dim Z$. Given $s\in|H\otimes m_{x}|$, we have that
$s$ induces the injective morphism

$\displaystyle|L_{n}-H|$ $\displaystyle\rightarrow$
$\displaystyle|L_{n}\otimes m_{x}|$ $\displaystyle\phi$ $\displaystyle\mapsto$
$\displaystyle s\otimes\phi.$

Therefore for any $\phi$ in the image of the morphism $(K_{n})_{x}\to
H^{0}(L_{n}-H)$, we have that $(s,\phi)\in Z$ and $\pi_{2}(s,\phi)=s$. It
follows that that projection

$\displaystyle\pi_{2}:Z\rightarrow\mathbb{P}(H^{0}(H\otimes m_{x}))$

is dominant and the general fiber has dimension $\text{rk}(K_{n})-1$. Hence

(3) $\displaystyle\text{rk}(F_{n})\geq h^{0}(H)+\text{rk}(K_{n})-1.$

∎

Observe that from Lemma 2.2 we have:

$\text{rk}(K_{n})+\text{rk}(N_{n})=\text{rk}(F_{n})\geq
h^{0}(H)-1+\text{rk}(K_{n}),$

i.e, $\text{rk}(N_{n})\geq h^{0}(H)-1$. Our next claim is that a stronger
inequality is valid:

$h^{0}(H)-1+h^{0}(L_{n}-H)<\text{rk}(N_{n}),$

(this serves as an alternative to Lemma 1.2 in [10] and allows us to simplify
the argument).

Indeed, assume that

$\text{rk}(N_{n})\leq h^{0}(H)-1+h^{0}(L_{n}-H).$

Since $\mu(N_{n})\leq\mu(M_{L_{n}|_{C}})$ we have:

$\deg
N_{n}\leq\frac{-nL.H-D.H}{h^{0}(L_{n}|_{C})-1}(h^{0}(H)-1+h^{0}(L_{n}-H)).$

Now,

$\deg F_{n}|_{C}=\deg N_{n}+\deg K_{n}\leq\deg N_{n},$

and

$\text{rk}(F_{n})|_{C}\geq h^{0}(H)-1,$

because of the previous Lemma 2.2. Thus,

$\mu(M_{L_{n}}|_{C})=\frac{-nL.H-D.H}{h^{0}(L_{n})-1}\leq\mu(F_{n}|_{C})\leq\frac{-nL.H-D.H}{h^{0}({L_{n}}|_{C})-1}(\frac{h^{0}(H)-1+h^{0}(L_{n}-H)}{h^{0}(H)-1}).$

From this we obtain:

$h^{0}(L_{n})-1\leq(h^{0}(L_{n}|_{C})-1)\cdot\frac{h^{0}(H)-1}{h^{0}(H)-1+h^{0}(L_{n}-H)},$

and the contradiction:

$h^{0}(L_{n})<h^{0}(L_{n}|_{C}).$

Now we can finish the proof of the theorem. So far we have deduced that:

$h^{0}(H)-1+h^{0}(L_{n}-H)<\text{rk}(N_{n})\leq\text{rk}(M_{L_{n}|_{C}})=h^{0}(L_{n}|_{C})-1.$

But taking $n>>0$ we obtain a contradiction, because the left hand term is of
order $n^{2}$ and the right hand term of order $n$. ∎

######  Remark 2.3.

The condition $H$ ample can be substituted by: $H$ is a divisor such that
$\dim|H|\geq 2$ and contains an irreducible and non-singular curve $C$.

##  3\. Effectiveness results on some regular surfaces

The method used in the proof of Theorem 2.1 does not give an answer to the
problem of effectively determining the power $n$ for which $M_{L_{n}}$ is
$H$-stable. However, under some particular conditions and a suitable choice of
$H$ the same method, combined with Camere’s kind of arguments (see [7]),
allows us to conclude that $n=1$ is sufficient.

######  Theorem 3.1.

Let $X$ be a smooth projective surface. Let $L$ be an ample and generated line
bundle over $X$ and $H$ be a divisor such that an irreducible and non-singular
curve $C$ exists in $|H|$. Assume that

  1. (1)

$h^{1}(L-H)=0$,

  2. (2)

$h^{0}(H)\geq h^{0}(L|_{C})$,

  3. (3)

$M_{L|_{C}}$ is semi-stable.

Then $M_{L}$ is $H$-stable.

###### Proof.

Assume that $M_{L}$ is unstable, let $F\subset M_{L}$ be a sub-bundle such
that $\mu(F)\geq\mu(M_{L})$. Using the conditions (1) and (3), we can repeat
the proof of Lemma 2.2 and Lemma 1.2 in [10] with $n=1$. More specifically, in
Lemma 1.2 in [10] an inequality (inequality (*) at the beginning of page 78)
is obtained that translated to our situation, just as we did with Lemma 1.1 in
the previous section, yields:

$\text{rk}(K)\geq h^{0}(L-H)\frac{\text{rk}(F)}{\text{rk}(M_{L})}.$

Therefore, we have:

$\displaystyle\text{rk}(F)$ $\displaystyle\geq$ $\displaystyle
h^{0}(H)+\text{rk}(K)-1$ $\displaystyle\geq$ $\displaystyle
h^{0}(H)+h^{0}(L-H)\cdot(\frac{\text{rk}(F)}{\text{rk}(M_{L})})-1$

That is,

$\displaystyle rk(F)\geq\frac{(h^{0}(H)-1)rk(M_{L})}{rkM_{L}-h^{0}(L-H)}.$

Since $rkM_{L}-h^{0}(L-H)=rk(M_{L_{C}})=h^{0}(L_{C})-1$, from the hypothesis
$(2)$, we obtain that $rk(F)\geq rk(M_{L})$ which is impossible because $F$ is
a sub-bundle of $M_{L}$. This proves the theorem. ∎

Theorem 3.1 is specially useful when $X$ is a regular surface and $L=H$. In
this case conditions $(1)$ and $(2)$ are automatically satisfied and only
$(3)$ remains to be verified. Also, as $H=L$ is generated, assuming an
irreducible curve $C\in|L|$ exists we can assume that it is non-singular as
well (Bertini’s Theorem).

######  Corollary 3.2.

Assume $X$ is regular and $L$ is an ample and generated line bundle on $X$. If
$C\in|L|$ exists such that $C$ is irreducible and non-singular and either:

  1. (1)

$g(C)\leq 1$ or

  2. (2)

$C$ is Brill-Noether general,

then $M_{L}$ is $L$-stable.

###### Proof.

Assume that $g(C)=0$. From the exact sequence:

$0\to M_{L|_{C}}\to H^{0}(L|_{C})\otimes\mathcal{O}_{C}\to L|_{C}\to 0,$

and taking cohomology we see that $M_{L|_{C}}\simeq\oplus\mathcal{O}_{C}(-1)$.
Therefore, $M_{L|_{C}}$ is semi-stable and the result follows at once from
Theorem 3.1.

Next, let $C$ be an elliptic non-singular curve and let $L$ be a line bundle
on $C$ of degree $d\geq 2$, observe that $M_{L}$ is stable. Indeed, the rank
of $M_{L}^{\vee}$ is $d-1$, therefore the slope of $M_{L}^{\vee}$ is given by

$\displaystyle\mu(M_{L}^{\vee})=\frac{d}{d-1}.$

Let $F$ be a quotient of $M_{L}^{\vee}$, we want to prove that
$\mu(M_{L}^{\vee})<\mu(F).$ Since $F^{\vee}$ is a sub-bundle of $M_{L}$, we
get that $H^{0}(F^{\vee})=0$ and thus $H^{1}(F)=0$, by Serre duality. By
Riemann Roch Theorem, we get

$\displaystyle
h^{0}(F)=\text{deg}(F)+\text{rk}(F)(1-g(C))+h^{1}(F)=\text{deg}(F).$

Since $F$ is globally generated over $C$, it follows that $h^{0}(F)\geq
rk(F)$. Assume that $h^{0}(F)=rk(F)$, then the evaluation map

$\displaystyle H^{0}(F)\otimes\mathcal{O}_{C}\to F$

is an isomorphism and which is impossible because $H^{1}(F)=0.$ Therefore
$\text{deg}(F)\geq\text{rk}(F)+1$ and

$\displaystyle\mu(F)\geq\frac{rk(F)+1}{rk(F)}>\frac{d}{d-1}=\mu(M_{L}^{\vee}),$

the last inequality following from $rk(F)<d-1.$

Finally, assume that $C$ is Brill-Noether general. By [15] we have that
$M_{L|_{C}}$ is semi-stable. ∎

Another general case that can be treated is when the inverse of the canonical
divisor, $-K$, is nef.

######  Proposition 3.3.

Let $X$ be a regular surface and let $L$ be an ample and generated line bundle
on $X$. Assume that in the linear system $|L|$ an irreducible $C$ exists.
Then:

  1. (1)

if $-L.K\geq 2$, then $M_{L}$ is $L$-stable.

  2. (2)

if $-L.K=1$ and $\text{cliff}(C)\geq 1$; then $M_{L}$ is $L$-stable.

  3. (3)

if $L.K=0$ and $\text{cliff}(C)\geq 2$; then $M_{L}$ is $L$-stable.

###### Proof.

We can assume $C$ is non-singular. If $g(C)\leq 1$, then apply Corollary 3.2.
If $-LK\geq 2$, then by the Adjunction Formula we have $\deg(L|_{C})=L^{2}\geq
2g(C)$. Therefore $M_{L|_{C}}$ is semi-stable and $M_{L}$ is $L$-stable by
Theorem 3.1. This proves (1). The proofs (2) and (3) are similar: using
Adjunction Formula we get $\deg(L|_{C})=L^{2}\geq 2g(C)-Cliff(C),$ therefore
$M_{L_{C}}$ is semistable.

∎

Some particular cases of this situation are:

######  Corollary 3.4.

Let $L$ be an ample and generated line bundle over a del Pezzo surface. If
$|L|$ contains an irreducible curve, then $M_{L}$ is $L$-stable.

###### Proof.

If $C\in|L|$ is irreducible and non-singular and $g(C)\leq 1$, then the result
follows from Corollary 3.2. Otherwise, from $h^{0}(-K)\geq 2$ we clearly have
$-L.K\geq 2$ and the result follows from Proposition 3.3∎

######  Corollary 3.5.

Let $\mathbb{F}_{n}$ be a non-singular Hirzebruch surface and let $L$ be an
ample and generated line bundle. Then $M_{L}$ is $L$-stable.

###### Proof.

The canonical line bundle is given by $K_{\mathbb{F}_{n}}=-2C_{n}-(n+2)F,$
where $C_{n}$ is the section and $F$ is the fiber. If $L$ is ample, then
$c_{1}(L)=aC_{n}+bF$ with $a>0$ and $b>na.$ Since
$-LK_{\mathbb{F}_{n}}=2b-an+2a>3$, it follows that $M_{L}$ is $L$-stable by
Proposition 3.3.

∎

The $L-$stability of $M_{L}$ for $X$ a $K3$ surface was studied in [7].
Similar results can be obtained for regular surfaces with numerically trivial
canonical divisor:

######  Corollary 3.6.

Let $X$ be an Enriques surface. Let $L$ be an ample and generated line bundle
on $X$. Assume that an irreducible and non-singular curve $C$ in the linear
system $|L|$ exists such that $\text{cliff}(C)\geq 2$. Then $M_{L}$ is
$L$-stable.

###### Proof.

Note that $\text{deg}(L|_{C})=L^{2}=2g-2\geq 2g-\text{cliff}(C)$, thus
$M_{L|_{C}}$ is semi-stable. ∎

Finally we address the case of $-K-$stability for del Pezzo surfaces.

######  Theorem 3.7.

Let $X$ be a del Pezzo surface and let $L$ be a globally generated line bundle
on $X$. If $|L|$ contains an irreducible curve, then the vector bundle $M_{L}$
$-K_{X}$-stable.

###### Proof.

Note that $h^{1}(L)=h^{2}(L)=0.$ From Riemann-Roch Theorem we get

$\displaystyle h^{0}(L)=1+\frac{L^{2}-LK}{2}.$

Therefore the rank of $M_{L}$ is equal to $\frac{L^{2}-LK}{2}$. We want to
prove that $M_{L}^{\vee}$ is stable with respect to the polarization $-K_{X}$.
Let $C$ be a non-singular projective curve in the linear system $|-K_{X}|.$ By
adjunction formula, we have $C$ is an elliptic curve. The slope of
$M_{L}^{\vee}$ with respect to $-K_{X}$ is given by

$\displaystyle\mu(M_{L}^{\vee})=-\frac{2LK}{L^{2}-LK}.$

Let $F$ be a torsion-free quotient sheaf of $M_{L}^{\vee}$ of rank
$0<\text{rk}F<r;$ then $F|_{C}$ is a quotient of $(M_{L}^{\vee})|_{C}$ and we
can assume that it is a vector bundle on $C$. We want to prove that

$\displaystyle\mu(M_{L}^{\vee})<\mu(F).$

We have the following properties

  1. (i)

$H^{1}(C,F|_{C})=0.$

  2. (ii)

$\text{deg}(F|_{C})\geq\text{rk}(F)+1.$

Indeed, since $F$ is globally generated, we have $F|_{C}$ is globally
generated, and therefore $h^{0}(F^{\vee}|_{C})=0$ and $h^{1}(F|_{C})=0$ by
Serre duality, this proves (i). To prove (ii), by Riemann Roch Theorem for
curves, we get

$\displaystyle
h^{0}(F|_{C})=\text{deg}(F|_{C})+\text{rk}(F|_{C})(1-g(C))+h^{1}(F|_{C})=\text{deg}(F|_{C}).$

Since $F|_{C}$ is globally generated over $C$, it follows that
$h^{0}(F|_{C})\geq rk(F|_{C})$. Assume that $h^{0}(F|_{C})=rk(F|_{C})$, then
the evaluation map

$\displaystyle H^{0}(F|_{C})\otimes\mathcal{O}_{C}\to F|_{C}$

is an isomorphism but this is impossible because $H^{1}(F|_{C})=0.$ Therefore
$\text{deg}(F|_{C})\geq\text{rk}(F)+1$ and we obtain$(ii).$

Hence, we have

$\displaystyle\mu(F)=\mu(F|_{C})\geq
1+\frac{1}{rk(F|_{C})}>1+\frac{2}{L^{2}-LK}$

therefore

$\displaystyle\mu(M_{L}^{\vee})=-\frac{2LK}{L^{2}-LK}\leq
1+\frac{2}{L^{2}-LK}<\mu(F),$

the first inequality follows by Adjunction Formula and the fact that $|L|$
contains an irreducible curve. Hence $M_{L}^{\vee}$ is stable with respect to
polarization $-K_{X}$. ∎

## References

  * [1] Brenner, H.; Looking out for stable syzygy bundles. With an appendix by Georg Hein. Adv. Math. 219 (2008), no. 2, 401–427
  * [2] Butler, D. C. Normal generation of vector bundles over a curve, J. Differential Geom. 39, no. 1,(1994), 1–34.
  * [3] D. C. Butler, Birational maps of moduli of Brill-Noether pairs, arXiv:alg-geom/9705009.
  * [4] Caucci, F. and Lahoz, M.Stability of syzygy bundles on an abelian variety, arXiv:2007.08846v1.
  * [5] Bhosle, U. N., Brambila-Paz, L. and Newstead, P. E. On linear series and a conjecture of D. C. Butler. Internat. J. Math. 26 (2015), no. 2, 1550007, 18 pp.
  * [6] Camere, C. About the stability of the tangent bundle restricted to a curve. C. R. Math. Acad. Sci. Paris 346 (2008), no. 7-8, 421–426.
  * [7] Camere, C. About the stability of the tangent bundle of $P^{n}$ restricted to a surface. Math. Z. 271 (2012), no. 1-2, 499–507.
  * [8] Coandă, I. On the stability of syzygy bundles. Internat. J. Math. 22 (2011), no. 4, 515–534.
  * [9] Ein, L. and Lazarsfeld, R. Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, In: Ellingsrud, G., Peskine, C., Sacchiero, G., Stromme, S.A. (eds.) Complex Projective Geometry (Trieste 1989/Bergen 1989). LMS Lecture Note Series, vol. 179, pp. 149–156. CUP, Cambridge (1992).
  * [10] Ein, L., Lazarsfeld, R. and Mustopa, Y. Stability of syzygy bundles on an algebraic surface. Math. Res. Lett. 20 (2013), no. 1, 73–80.
  * [11] Flenner, H. Restrictions of semistable bundles on projective varieties. Comment. Math. Helv. 59 (1984), no. 4, 635–650.
  * [12] Macias Marques, P. and Miró-Roig, R. M.; Stability of syzygy bundles. Proc. Amer. Math. Soc. 139 (2011), no. 9, 3155–3170.
  * [13] Mistretta, E. and Stoppino, L.: Linear series on curves: stability and Clifford index. Internat. J. Math. 23, no. 12 (2012).
  * [14] Paranjape, K. and Ramanan, S. On the canonical ring of a curve, In: Algebraic geometry and Commutative Algebra, in Honor of Masayoshi Nagata, vol. 2, Kinokuniya (1987), 503–516.
  * [15] Schneider, O. Stabilité des fibrés $\wedge^{p}E_{L}$ et condition de Raynaud, Fac. Sci. Toulouse Math. (6) 14(3), (2005), 515-525.


# Analytic regimes of tokamak plasma

Alexander M. Balk  Department of Mathematics, University of Utah, 155 South
1400 East, Salt Lake City, Utah 84112

###### Abstract

The paper suggests certain regimes of fusion plasma with magnetic confinement,
when the Larmor radius $\lambda$ and the angle $\alpha$ (determining the
orientation of the local coordinates) form an analytic function $w=\lambda
e^{i\alpha}$ of complex variable $z=x_{1}+ix_{2}$ ($x_{1}$ and $x_{2}$ are
coordinates of tokamak cross-section). It is known that an efficient nuclear
fusion requires transport barriers, which might be due to zonal/poloidal flow;
and the emergence of such flow is implied by the adiabatic extra conservation
(established in a local slub). The suggested analytic regimes are based on the
global extension of this extra conservation; transport barriers could spread
to the entire tokamak plasma. It is interesting that in these regimes, plasma
temperature reaches maximum often not at a single location (of tokamak cross-
section), but at several points separated by regions of lower temperature.

In nuclear fusion devices with magnetic confinement — such as tokamaks — it is
imortant to have transport barriers, which might be due to zonal/poloidal flow
Horton (2012); Diamond  _et al._ (2010). It is interesting that the emergence
of such flow in a plasma slub is implied by the conservation of the adiabatic
extra invariant, along with the energy and enstrophy. In the present paper, we
study whether this invariant can be extended from the conservation in a local
slub to the global conservation in the entire tokamak plasma. We find that
such extension leads to certain analiticity regimes.

1\. Extra invariant. Suppose a significant mode of plasma dynamics is
represented by the drift waves with dispersion relation $\omega_{\bf
k}=q/(1+p^{2}+q^{2})$. The interaction of these waves can take different
forms. The dispersion is written in dimensionless form for a local slub: The
local wave number components $p,q$ of the wave vector ${\bf k}$ are non-
dimensionalized by the Larmor radius $\lambda$.

The radial transport in a tokamak is reduced when the wave system organizes
itself to produce strong alternating zonal/poloidal flow Biglari  _et al._
(1990). This is because the ballistic transfer of heat and particles in the
radial direction — by any plasma mode — is interrupted (radial velocity field
is de-correlated) when there is strong randomly alternating poloidal current.
At the same time, the poloidal flow does not contribute to radial transport.

It is interesting that the emergence of zonal/poloidal flow follows from the
extra conservation. In addition to the energy and enstrophy, a system of drift
waves is known Balk  _et al._ (1991); Balk (1991) to conserve one more
quadratic integral $I=\int\eta_{\bf k}{\mathcal{N}}_{\bf k}\,d{\bf k}$. This
is the extra invariant with spectral density

$\displaystyle\eta_{\bf
k}=\arctan{\frac{p+q\sqrt{3}}{p^{2}+q^{2}}}-\arctan{\frac{p-q\sqrt{3}}{p^{2}+q^{2}}}\,.$
(1)

The function ${\mathcal{N}}_{\bf k}=\varepsilon_{\bf k}/\omega_{\bf k}$ is the
wave action spectrum, $\varepsilon_{\bf k}$ is the energy spectrum. The extra
conservation is adiabatic: It holds approximately over long time. The function
$\eta(p,q)$ is shown in Figure 1.

Figure 1: Contour plot of the function (1). Color represents the values of
$\eta(p,q)$. The graph is shown only for positive $p,q$, due to the symmetries
$\eta(p,q)=\eta(-p,q)=-\eta(p,-q)$. All contour lines pass through the origin
with $30^{\circ}$ slope.

As well known, the conservation of the energy and enstrophy leads to the
inverse energy cascade (local or non-local). The conservation of the extra
invariant ensures Balk (2005) that the energy is not just transferred towards
small wave numbers, but concentrates around $p$-axis ($|q|\ll|p|$), which
corresponds to zonal/poloidal flow. (In the last three references, $p$ and $q$
are interchanged for geophysical context.)

The waves similar to the plasma drift waves occur in different problems. As
well known, the waves with the same dispersion relation are the Rossby waves
in atmospheres and oceans. Much slower waves (slower than the relatively slow
Rossby waves) also appear in astrophysical magnethydrodynamics Hide (1966);
Zaqarashvili  _et al._ (2007); Balk (2022). In all these situations, the extra
invariant has important physical implications.

2\. Global conservation. Let $(x_{1},x_{2},x_{3})$ be global dimensional
coordinates; $x_{3}$ is the toroidal coordinate. In all cross-sections
$x_{3}=const$, plasma properties are assumed to be the same. In the present
paper, we assume that the confining magnetic field ${\bf B}$ is purely
toroidal, ${\bf B}=B\hat{x}_{3}$; and so, it is orthogonal to the cross-
section.

The drift waves have the dispersion relation
$\Omega=\nu(x_{1},x_{2})\,\omega(p,q)$, where $p$ and $q$ are local
dimensionless wave numbers; they are related to the global dimensional wave
vector ${\bf k}=(k^{1},k^{2})$ by a rotation through angle
$\alpha(x_{1},x_{2})$ $\quad[c\equiv\cos\alpha,\;s\equiv\sin\alpha]$

$\displaystyle p=\lambda(c\,k^{1}+s\,k^{2}),\quad
q=\lambda(-s\,k^{1}+c\,k^{2}).$ (2)

In the plane ${\bf x}=(x_{1},x_{2})$, the drift wave field is described by the
inhomogeneous wave kinetic equation

$\displaystyle\frac{\partial{\mathcal{N}}}{\partial
t}+\frac{\partial\omega}{\partial\bf
k}\cdot\frac{\partial\mathcal{N}}{\partial\bf
x}-\frac{\partial\omega}{\partial\bf
x}\cdot\frac{\partial\mathcal{N}}{\partial\bf k}=St[{\mathcal{N}}]$ (3)

for the dynamics of the wave action spectrum ${\mathcal{N}}({\bf k},{\bf
x},t)$; $t$ is dimensionless time (nondimensionalized by the frequency $\nu$).
(The wave kinetic equation has been utilized for a long time in physics and
applied mathematics; recently, kinetic equation approach was used Parker
(2016); Marston and Tobias (2022) to study a related problem: interaction
between zonal flow and turbulence.) The l.h.s. in equation (3) describes the
slow refraction of waves, while the r.h.s. (stoss-term or collision integral)
describes changes in $\mathcal{N}$ due to 3-wave resonance interactions

$\displaystyle{\bf k}_{3}={\bf k}_{1}+{\bf k}_{2},\quad\omega({\bf
k}_{3})=\omega({\bf k}_{1})+\omega({\bf k}_{2}).$ (4)

We assume that the interactions occur locally in physical space, and so, the
three interacting waves have the same plasma parameters.

Note: We can consider only linear equations but study nonlinear interactions.
This is because — in general — the conservation of a quadratic integral is
determined by the dispersion relation. This fact was noted by Zakharov and
Schulman: They called degenerative Zakharov and Schulman (1980) a dispersion
relation that allows for an additional conservation law (besides the energy,
momentum, and wave action). They showed Zakharov and Schulman (1988) that
conservation in the homogeneous wave kinetic equation is equivalent to
cancellation of the “small divisors”, and therefore, to approximate extra
conservation in the dynamic equation. The extra conservation holds for any
nonlinearity, provided the dynamic equation is Hamiltonian (which is usually
the case for a physical system). I believe the conservation by the
inhomogenous part, on the left of the wave kinetic equation (3), can also be
studied in terms of the dynamic equation.

Let us check the global conservation of an arbitrary integral

$\displaystyle I({\bf x},t)=\int F({\bf k},{\bf x})\,{\mathcal{N}}({\bf
k},{\bf x},t)\,d{\bf k},$ (5)

determined by an arbitrary spectral density $F$. According to the equation
(3),

$\displaystyle\frac{\partial I}{\partial t}$ $\displaystyle+$
$\displaystyle\frac{\partial}{\partial{\bf
x}}\cdot\int\frac{\partial\omega}{\partial{\bf k}}\,F\,{\mathcal{N}}\,d{\bf
k}+\int\frac{\partial(F,\omega)}{\partial({\bf k},{\bf
x})}{\mathcal{N}}\,d{\bf k}$ (6) $\displaystyle=$ $\displaystyle\int
St[{\mathcal{N}}]\,F\,d{\bf k},$

where we use notation

$\displaystyle\frac{\partial({\mathcal{F}},{\mathcal{G}})}{\partial(u,v)}=\frac{\partial{\mathcal{F}}}{\partial
u}\cdot\frac{\partial{\mathcal{G}}}{\partial
v}\;-\;\frac{\partial{\mathcal{F}}}{\partial
v}\cdot\frac{\partial{\mathcal{G}}}{\partial u}$

for arbitrary functions ${\mathcal{F}}$, ${\mathcal{G}}$ of arbitrary
variables $u$, $v$ (vector or scalar). The quantity $I$ is conserved if in the
equation (6), the last two integrals vanish.

The wave kinetic equation (3) always conserves the energy, when $F=\omega$.

3\. Anlyticity. Now let $F$ be the function (1), $F=\eta$, with $p,q$
determined in (2). Then the r.h.s. in (6) vanishes. Indeed by (2), $p$ and $q$
are linear combinations of the global components of the wave vector ${\bf k}$,
and so, the equations (4) lead to
$p_{3}=p_{1}+p_{2},\;q_{3}=q_{1}+q_{2},\;\omega(p_{3},q_{3})=\omega(p_{1},q_{1})+\omega(p_{2},q_{2}).$
These equations imply that

$\displaystyle\eta(p_{3},q_{3})=\eta(p_{1},q_{1})+\eta(p_{2},q_{2}),$ (7)

i.e. the function $\eta$ is conserved in 3-wave resonance interactions.
Actually, $\eta$ is the only independent function conserved in the 3-wave
interactions. This fact is valid even if the equation (7) holds only in an
arbitrarily small neighborhood anywhere on the resonance manifold Ferapontov
(1992); Balk and Ferapontov (1998). All other conserved functions are linear
combinations of $p$, $q$, $\omega(p,q)$, and $\eta(p,q)$.

In the second integral of equation (6),

$\displaystyle\frac{\partial(F,\omega)}{\partial({\bf k},{\bf x})}$
$\displaystyle=$
$\displaystyle\frac{\partial(\eta,\omega)}{\partial(p,q)}\frac{\partial(p,q)}{\partial({\bf
k},{\bf x})}.$ (8)

According to the equations (2),

$\displaystyle\frac{\partial(p,q)}{\partial({\bf k},{\bf x})}=\lambda{\bf
k}\cdot\left(-\frac{\partial\lambda}{\partial
x_{2}}-\lambda\frac{\partial\alpha}{\partial
x_{1}},\;\frac{\partial\lambda}{\partial
x_{1}}-\lambda\frac{\partial\alpha}{\partial x_{2}}\right).$ (9)

So, the expressions (9) and (8) vanish for all wave vectors ${\bf k}$ if
functions $\mu=\ln\lambda$ and $\alpha$ satisfy the Cauchy-Riemann conditions:

$\displaystyle\frac{1}{\lambda}\frac{\partial\lambda}{\partial
x_{1}}=\frac{\partial\alpha}{\partial
x_{2}},\qquad\frac{1}{\lambda}\frac{\partial\lambda}{\partial
x_{2}}=-\frac{\partial\alpha}{\partial x_{1}},$ (10)

saying that $\mu+i\alpha$ is an analytic function of $z=x_{1}+ix_{2}$. In
other words, $w=\lambda e^{i\alpha}$ should be an analytic function of $z$.

The drift waves are determined by the gradient of some macroscopic plasma
quantity. When this quantity reaches maximum, the frequency $\nu$ vanishes.
There are no drift waves at such locations, the entire reasoning based on the
wave kinetic equation (3) fails, and the analyticity is not required. The
analytic function $w(z)$ could have singularities at such points.

4\. Examples. Since $\lambda$ is related to the plasma temperature (e.g.
$\lambda\propto\sqrt{T}$, with $T$ being the ion temperature), we would like
to have smaller $\lambda$ at the tokamak boundary and larger $\lambda$ in the
central part of the tokamak. The figures below show the Larmor radius
$\lambda=|w(z)|$ in the logarithmic scale: The color bar numbers are the
values of $\ln\lambda(x_{1},x_{2})$; the white area corresponds to the values
exceeding the top color.

Example 1.

$\displaystyle w=\frac{A_{1}}{z-a_{1}}+\frac{A_{2}}{z-a_{2}}$ (11)

with four complex parameters $a_{1},a_{2},A_{1},A_{2}$. To ensure fastest
decrease of $\lambda=|w|$ as $z\rightarrow\infty$, we set $A_{1}+A_{2}=0$ so
that $w=O(z^{-2})$. Using rotation and stretching in the $w$-plane, we can
assume $A_{1}=-A_{2}=1$. Choosing the axis $x_{1},\,x_{2}$, we can have
$a_{1}=-a_{2}=ib$ ($b$ is a real number). Re-scaling length, we set $b=1$. See
figure 2.

Figure 2:  $w=\frac{1}{z-i}-\frac{1}{z+i}$.

The boundary of the tokamak cross-section should pass through the region with
smaller values of $\lambda$.

Example 2.

$\displaystyle
w=\frac{A_{1}}{z-a_{1}}+\frac{A_{2}}{z-a_{2}}+\frac{A_{3}}{z-a_{3}}$ (12)

with six complex parameters $A_{j},a_{j}\;(j=1,2,3)$. Similar to example 1, we
require $w=O(z^{-3})$, which gives us the two linear equations on
$A_{1},A_{2},A_{3}$. Using rotation and stretching in the $w$-plane, we can
set

$\displaystyle A_{1}=a_{3}-a_{2},\;A_{2}=a_{1}-a_{3},\;A_{3}=a_{2}-a_{1}.$
(13)

Finally, using rotation, shifting, and stretching in the $z$-plane, we can
exclude 4 real parameters, leaving us with two parameters. See figure 3.

Figure 3: The analytic function $w(z)$ is defined in (12)-(13) with
$a_{1}=i,\,a_{2}=-i,\,a_{3}=1/2$.

## References

  * Horton (2012) W. Horton,  _Turbulent Transport in Magnetized Plasmas_ (World Scientific, 2012).
  * Diamond  _et al._ (2010) P. H. Diamond, S.-I. Itoh, and K. Itoh,  _Modern Plasma Physics. Vol. 1: Physical Kinetics of Turbulent Plasmas_ (Cambridge University Press, 2010).
  * Biglari  _et al._ (1990) H. Biglari, P. Diamond, and P. Terry, Physics of Fluids B: Plasma Physics 2, 1 (1990).
  * Balk  _et al._ (1991) A. M. Balk, S. V. Nazarenko, and V. E. Zakharov, Phys. Lett. A 152, 276 (1991).
  * Balk (1991) A. M. Balk, Phys. Lett. A 155, 20 (1991).
  * Balk (2005) A. M. Balk, Phys. Lett. A 345, 154 (2005).
  * Hide (1966) R. Hide, Proc. Roy. Soc. London 259, 615 (1966).
  * Zaqarashvili  _et al._ (2007) T. V. Zaqarashvili, R. Oliver, J. L. Ballester, and B. M. Shergelashvili, A&A 470, 815 (2007).
  * Balk (2022) A. M. Balk, The Astrophysical Journal 926, 2 (2022).
  * Parker (2016) J. B. Parker, Journal of Plasma Physics 82, 595820602 (2016).
  * Marston and Tobias (2022) J. Marston and S. Tobias, arXiv preprint arXiv:2205.05513 (2022).
  * Zakharov and Schulman (1980) V. E. Zakharov and E. I. Schulman, Physica D 1, 192 (1980).
  * Zakharov and Schulman (1988) V. E. Zakharov and E. I. Schulman, Physica D 29, 283 (1988).
  * Ferapontov (1992) E. V. Ferapontov, “Web geometry and mathematical physics,” (Kluwer Academic Publishers, Norwell, MA, 1992) pp. 310–323, appendix in the book “Geometry and Algebra of Multidimensional Three-Webs” by M. A. Akivis and A. M. Shelekhov.
  * Balk and Ferapontov (1998) A. M. Balk and E. V. Ferapontov, in  _Nonlinear waves and weak turbulence_ , edited by V. E. Zakharov (Amer. Math. Soc. Trans. Ser. 2, vol. 182, 1998) pp. 1–30.


previously executed a clean query in Line 5. Finally, we additionally need 2
clean oracle calls, as Line 5 is executed for both matroids even if there is
no error at all. Therefore, we use at most
$2+2\eta_{r}\cdot(1+\lceil\log_{2}(n)\rceil)$ many clean oracle calls. ∎