12.27 Flat anticanonical curves0E9176045572

Proof. Put $P=\iota _ v\partial _ L$. Since the metric and $L$ are flat, $P=\nabla _ v$ and $P^*=-\nabla _{\bar v}$, and the Laplacian $\Delta _ L$ satisfies $\Delta _ L=P^*P=PP^*$ on sections and on $(0,1)$-forms. In degree $q\in \{ 0,1\} $, let $G_ q$ be the inverse of $\Delta _ L$ on the orthogonal complement of its kernel, extended by zero on the kernel. The operators $P$ and $P^*$ commute with $G_ q$. Set

The form $Pu$ is orthogonal to the parallel forms, so $\bar\partial _ Lf=Pu$; also $f$ is orthogonal to the parallel sections. Uniqueness follows because the kernel of $\bar\partial _ L$ on sections consists of parallel sections. Moreover,

Covariant differentiation along a global parallel real frame commutes with all these operators, so applying the identity to every derivative proves the $H^ k$ assertion. The argument never divides by a lower bound for a nonzero eigenvalue of $\Delta _ L$: the derivative in $Pu$ compensates for small eigenvalues. $\square$

Proof. Step 1: a first identification. Start with the normal exponential map of a smooth Hermitian metric near $D$. Its differential along $D$ is complex linear and induces the identity on the normal quotient. In a parallel local frame and a local coordinate $z_0$ on $D$ with $\vartheta _ D=\mathrm dz_0$, choose holomorphic coordinates $Z,W$ on $S$ with $D=\{ W=0\} $. Their pullbacks under the first map satisfy

The residue normalization gives $\Omega _ S=(1+Wa(Z,W))\, \mathrm dW/W\wedge \mathrm dZ$ with $a$ holomorphic. Substituting, and noting that $\Theta \wedge \mathrm dw=w\Theta \wedge \Theta =0$, we obtain $\Omega _ S-\Omega _0=O(|w|)$ in the coframe $\mathrm dz_0,\mathrm d\bar z_0,\Theta ,\bar\Theta $. Every derivative by $\partial _{z_0},\partial _{\bar z_0},w\partial _ w,\bar w\partial _{\bar w}$ preserves this bound. Since $|w|=\mathrm e^{-z/2}$, this proves the cylindrical estimate for the first map.

Step 2: holomorphic fiber disks. It remains to modify the first map so that the fiber disks become holomorphic, without changing its first jet along $D$. On a fiber disk $\Delta _{p,\rho }\subset N_ p$ of radius $\rho $, perturb the map by $\Phi _ Y(q)=\exp _ q(Y_ q)$, where $Y$ is a vector field along the image of the disk. Identify the target tangent spaces with one fixed complex vector space over $p$, by Chern parallel transport along the radial paths and the short exponential paths. In these identifications, the antiholomorphic part of the differential of the perturbed disk is

Fix a Hölder exponent $0<\nu <1$, and use $C^{k,\nu }$ norms on the disks rescaled to unit size. The Cauchy–Green operator $T_\rho $ of the disk gives the normalized right inverse

of $\bar\partial $. We restrict its domain to $(0,1)$-forms vanishing at the center; its image consists of fields whose value and first jet vanish at the center, and on this image $\bar\partial $ and $R_\rho $ are inverse to each other. The differential of $\mathcal F_{p,\rho }$ at zero has the form $\bar\partial +K_{p,\rho }$, with

Indeed, the two complex structures agree along the zero section, and rescaling by $w\mapsto \rho w$ contributes an additional factor $\rho $ to a pulled-back one-form. For small $\rho $, the operator $R_\rho (I+K_{p,\rho }R_\rho )^{-1}$ is a right inverse of the differential, and the quantitative implicit function theorem gives a correction $Y=O(|w|^2)$ with vanishing first jet that makes the disk holomorphic.

Step 3: global smoothness. Compactness of $D$ makes the choice of $\rho $ uniform in $p$. The normalized Cauchy operator is equivariant under unitary changes of the fiber frame, so the intrinsic parallel transports and the uniqueness of the small solution make the corrections agree on overlaps. The parametric implicit function theorem gives smooth dependence on $p$, and interior elliptic regularity on slightly smaller fiber disks gives smoothness along the fibers. Since the first jet is unchanged, the correction changes the map by $O(|w|^2)$ together with its cylindrical derivatives. Composing the first map with this correction therefore preserves the estimate and proves the lemma. $\square$

Proof. We look for a fiber-preserving change of the identification of lemma 12.27.2 after which the coordinate $z$ becomes pluriharmonic. The logarithm of the flat norm of the defining section is then $-z$, up to a pluriharmonic correction.

Step 1: horizontal forms. Use the identification $\Upsilon $ of lemma 12.27.2, and omit the pullback notation. A form on $N^\times $ is horizontal if its contractions with $\xi $ and $\bar\xi $ vanish. For a horizontal form $\eta $ define

A form $\eta _ m$ on $D$ with values in $N^{-m}$ defines the horizontal form $w^ m\eta _ m$, homogeneous of degree $m$ in $w$, and direct differentiation shows that $\mathrm d_ H$ acts on it as the flat covariant exterior derivative. Since the fibers are holomorphic,

with $\alpha ,\beta $ horizontal. Closedness gives $\mathcal L_{\bar\xi }\alpha =\mathcal L_{\bar\xi }\beta =0$ and $\mathcal L_\xi \beta =\mathrm d_ H\alpha $.

Step 2: a horizontal inverse. Fix $0<\delta <1/2$, and let $U_ T=\{ z\geq T\} $. For tensors define the weighted norms

Let $\mathcal H^{k+1}_{\delta ,T}$ be the space of functions $f$ on $U_ T$ with $\| f\| _{k+1,\delta ,T}<\infty $ and $\mathcal L_{\bar\xi }f=0$, and let $\mathcal D^ k_{\delta ,T}$ be the space of complex two-forms $\gamma $ on $U_ T$ with $\| \gamma \| _{k,\delta ,T}<\infty $, $\mathrm d\gamma =0$ and $\iota _{\bar\xi }\gamma =0$. We construct an operator

with constant independent of $T$, such that

Write $\gamma =\Theta \wedge \alpha +\beta $, with $\alpha ,\beta $ horizontal. The conditions on $\gamma $ give $\mathcal L_{\bar\xi }\gamma =0$ and, in the expansion in homogeneous components along the fibers, $m\beta _ m=\mathrm d_ H\alpha _ m$. Only components with $m\geq 1$ occur, because the weighted norm is finite and $\delta >0$ excludes the Laurent components of degree $m\leq 0$. Decompose

Lemma 12.27.1 gives sections $e_ m$ with $\bar\partial e_ m=\iota _ v\partial v_ m$ and $\| e_ m\| _{H^ k}\leq \| v_ m\| _{H^ k}$. Put $f_ m=-(u_ m+e_ m)/m$. Then

Also $\| \bar\partial f_ m\| _{H^ k}\leq C\| \beta _ m\| _{H^ k}$. Since $D$ is flat, integration by parts gives the same bound for $\partial f_ m$, which controls the horizontal derivatives of order $k+1$. A vertical derivative of a homogeneous component multiplies it by $m$, so every derivative involving a vertical direction is controlled by the bound for $mf_ m$. The components are orthogonal on the circles $\{ |w|=\text{constant}\} $, and the squared radial factor of the $m$th component is $\mathrm e^{-mz}$ with $m-2\delta >0$; integrating the weight gives the required estimate for $f=\sum _{m\geq 1}f_ m$, where each $f_ m$ is identified with its homogeneous horizontal lift, and we put $Q_ T\gamma =f$.

For forms with Sobolev coefficients, regularize each component by the heat operator of the flat base, which commutes with the differential equations above. The uniform estimates and orthogonality make the finite partial sums converge in the target space, and passing to the limit proves the first identity in (12.27.4). If $\gamma =\mathrm df\wedge \vartheta $, then $\alpha _ m=mf_ m\vartheta $, so $v_ m$ and the normalized $e_ m$ vanish, and the construction returns $f_ m$. This proves the second identity.

Step 3: a fixed point. Put $\eta =\Omega _ S-\Omega _0$, and choose an integer $k\geq 4$. By the cylindrical estimate, the weighted $H^{k+2}$ norm of $\eta $ on $U_{T-1}$ and its first $k+2$ derivatives are bounded by a number $a_ T$ with $a_ T\to 0$. For small $f\in \mathcal H^{k+1}_{\delta ,T}$, the map $\Phi _ f(\zeta )=\mathrm e^{f(\zeta )}\zeta $, which rescales each fiber point $\zeta $, is a $C^2$ diffeomorphism onto its image and maps $U_ T$ into $U_{T-1}$. Its restriction to every fiber is holomorphic, so $\Phi _ f^*\Omega _ S-\Omega _0$ lies in the domain of $Q_ T$. We look for $f$ such that $\Phi _ f^*\Omega _ S$ has no horizontal component. Since

(12.27.4) reduces this to the equation

On a fixed sufficiently small ball, multiplication and composition estimates in weighted Sobolev spaces on the flat cylinder give

The constants are independent of $T$, because translation in the cylindrical direction is an isometry, the weight is normalized at $T$, and the shifts of $z$ are uniformly bounded. Differentiating the pullback along the segment from $f$ to $g$ involves one Lie derivative of the fixed form $\eta $ and at most one derivative of $f-g$, and the gain of one derivative in $Q_ T$ restores the norm. For $T$ sufficiently large, $-E_ T$ is therefore a contraction of the closed ball of radius $2Ca_ T$ into itself. Let $f$ be its fixed point.

Step 4: a pluriharmonic function. By the first identity in (12.27.4), the horizontal component of $\Phi _ f^*\Omega _ S$ vanishes, so $\Theta \wedge \Phi _ f^*\Omega _ S=0$. The pulled-back two-form is nowhere zero and of type $(2,0)$ for $\Phi _ f^*J_ S$, so $\Theta $ is of type $(1,0)$ for this complex structure. Since $\mathrm dz=-(\Theta +\bar\Theta )$ and $\mathrm d\Theta =0$, the function $z$ is pluriharmonic for $\Phi _ f^*J_ S$.

Step 5: the flat metric. Return to $S$ through $A=\Upsilon \circ \Phi _ f$. On each fiber, $f$ is bounded and holomorphic on a punctured disk, so it extends holomorphically across the center, and its expansion in positive powers of $w$ gives $f(0)=0$. Since $f$ is uniformly small, Rouché's theorem, comparing $w\mathrm e^{f(w)}-c$ with $w-c$ on a fixed smaller circle, shows that every sufficiently small value $c$ is attained. Hence the image of $A$ contains a punctured neighborhood of $D$. The function $r=z\circ A^{-1}$ is $C^2$ and pluriharmonic there, hence smooth by elliptic regularity. If $w_ S$ is a local holomorphic defining function for $D$, the unchanged first normal jet of $\Upsilon $ and the bounded factor $\mathrm e^ f$ give

Thus $r+\log |w_ S|^2$ is bounded and pluriharmonic off $D$. Both this function and its negative are bounded plurisubharmonic functions off $D$, so they extend plurisubharmonically across $D$; the sum of the extensions vanishes almost everywhere, hence everywhere, so the extension is pluriharmonic. It follows that $F=-r$, which has a logarithmic pole along $D$, satisfies $\mathrm{dd}^{\mathrm{c}}F=[D]$ near $D$. Locally, the metric for which the squared norm of the defining section equals $\mathrm e^{F}$ is smooth and flat, since $F-\log |w_ S|^2$ is smooth and pluriharmonic. These local metrics agree on overlaps and give the required unitary flat metric. $\square$

Proof. Let $s_ D$ be the defining section and $h_0$ a flat metric on a neighborhood $U$ of $D$. Put $F=\log |s_ D|_{h_0}^2$, which is pluriharmonic on $U\setminus D$. Choose a relatively compact tubular neighborhood $U_0\Subset U$ of $D$. Since $F$ tends uniformly to $-\infty $ along $D$ and is bounded on $\partial U_0$, after adding a constant to $F$ we may assume $F>0$ near $\partial U_0$. Choose a smooth concave function $\chi $ and a constant $c$ with $\chi (t)=t+c$ for $t\leq -1$ and $\chi (t)=0$ for $t\geq 0$. Define $\widetilde F=\chi (F)$ on $U_0$, extended by zero outside $U_0$, and define a metric $h$ on $S\setminus D$ by

Near $D$ this gives $h=\mathrm e^ c h_0$, so $h$ extends smoothly across $D$; it is also smooth near $\partial U_0$, where $\widetilde F=0$. By the Poincaré–Lelong formula its curvature is $B=[D]-\mathrm{dd}^{\mathrm{c}}\widetilde F$, and off $D$, since $F$ is pluriharmonic,

This smooth closed form has rank at most one and vanishes where asserted. On a compact Kähler surface with Kähler form $\kappa $,

so $B$ is not identically zero and its maximal rank is one. $\square$

Proof. The inequalities $(\min f)B\leq fB\leq (\max f)B$ and proposition 12.26.1 give the two-sided bounds. The polynomial follows from the closed nef formula and $B^2=0$. At a point where $B$ has rank one, choose a smooth positive $f$ whose differential has a nonzero component along the kernel of $B$. Then $\mathrm d(fB)=\mathrm df\wedge B\ne 0$ there. $\square$

Proof. A blowup at a point of the cubic replaces the class of the cubic by its pullback minus the exceptional curve, exactly as it does for the anticanonical class. The strict transform remains isomorphic to the cubic, and its self-intersection decreases by one. After nine steps, $D$ is anticanonical and $D^2=9-9=0$. Theorem 12.27.3, Proposition 12.27.5 give the metric.

For $k\geq 1$ we use the restriction sequences

A nontrivial line bundle of degree zero on an elliptic curve has no nonzero section, since such a section would have a zero divisor of degree zero and would trivialize the bundle. If $N$ is not torsion, induction on $k$ gives $h^0(S,\mathcal O_ S(kD))=h^0(S,\mathcal O_ S)=1$. This proves that every $|kD|$ is the single divisor $kD$, and in particular that the anticanonical bundle is not semiample.

Suppose now that $N$ has exact order $m$. For $1\leq k<m$ we have $H^0(D,N^ k)=0$, and $H^1(D,N^ k)=0$ by Serre duality on the elliptic curve. Since $S$ is rational, $H^1(S,\mathcal O_ S)=0$. The same sequences give $h^0(S,\mathcal O_ S(kD))=1$ and $H^1(S,\mathcal O_ S(kD))=0$ for $0\leq k<m$. For $k=m$ the restriction map to $H^0(D,N^ m)$ is therefore surjective, and the constant section of $N^ m\simeq \mathcal O_ D$ lifts to a section of $\mathcal O_ S(mD)$ that does not vanish on $D$. Together with $s_ D^ m$, this section generates $\mathcal O_ S(mD)$ everywhere. Thus $|mD|$ is basepoint-free, while no smaller positive multiple of $D$ moves.

Finally, non-torsion configurations exist, with nine distinct centers. Fix the first eight points on the cubic and vary the ninth. Under the identification of $D$ with the cubic, the normal bundle is $\mathcal O_ D(3)\otimes \mathcal O_ D(-p_1-\cdots -p_9)$, where $\mathcal O_ D(3)$ is the restriction of $\mathcal O_{\mathbb P^2}(3)$. As $p_9$ varies, these bundles run through the whole group of line bundles of degree zero, by the Abel map of the elliptic curve. The torsion subgroup is countable, whereas the curve is uncountable, so even after excluding the first eight points there are non-torsion choices. For them, corollary 12.27.6 gives nonclosed backgrounds of order one, although the anticanonical bundle is not semiample. $\square$