5.27.20 Sobolev regularity at divisorial polesC37C319849E5

Proof. Shrink $U$ so that $|h_ j|\leq 1$ and the unit $g$, its inverse and the first derivatives of $g$ and of all $h_ j$ are bounded. For each $j$, $|f|^{1/m_0}\leq C|h_ j|^{m_ j/m_0}\leq C|h_ j|$. Logarithmic differentiation of the factorization therefore gives

which proves the first bound.

For small $\zeta \in \mathbb C$, let $\mu _\zeta $ be the Euclidean volume measure of the fiber divisor $\{ f=\zeta \} $, counted with multiplicity. As $\zeta \to 0$, we have $\log |f-\zeta |\to \log |f|$ in $L^1_{\mathrm{loc}}$: by the local compactness theorem recalled at the beginning of section 5.25, it suffices to identify subsequential limits, and the convergence is locally uniform off $Z$, which identifies every such limit and excludes convergence to $-\infty $. The Poincaré–Lelong formula therefore gives weak convergence of the fiber currents, and hence of their Euclidean trace measures. Pairing with a fixed nonnegative cutoff function equal to one near $K$ shows that $\sup _{|\zeta |<r_ K}\mu _\zeta (K)<\infty $ for some $r_ K>0$.

The critical locus of $f$ has finitely many local components, and $f$ is constant on each of them. After shrinking $U$, every remaining critical component passes through $0$, where $f$ vanishes. Thus the small nonzero fibers are smooth. On $\{ |f|=r\} $, the coarea formula for the map $f$ to the circle of radius $r$ gives

The coarea formula for the function $|f|$, together with $|\nabla |f||=|\partial f|$, proves the last inequality. The same formulas hold in dimension one, where the fiber volume is the counting measure. $\square$

Proof. At a smooth point of a component of multiplicity $m_0$, outside the other components, multiplying the defining function of the component by a holomorphic $m_0$-th root of the remaining unit gives a normal coordinate $w_1$ with $f=w_1^{m_0}$. Coordinate changes with bounded derivatives and Jacobians preserve local first-order Sobolev integrability. With $r=|w_1|$ and $s=m_0\log r$, polar integration of the classical gradient gives, up to fixed positive factors,

Such smooth points occur in every neighborhood of $0$. This proves necessity, since changing the finite upper endpoint does not affect convergence.

For sufficiency, put

For small $\varepsilon >0$, these are $C^2$ plurisubharmonic functions decreasing to $v$. By convexity, $|\chi (s)|\leq C(1+|s|)$ for $s\leq A$, so the local exponential integrability of $\log |f|$, recalled in [SGPT], gives convergence in every local $L^ p$ space with $p<\infty $. By the chain rule,

Apply lemma 5.27.21 with the coarea formula, keeping two powers of $|\partial f|$ for the level estimate and bounding the other $p-2$ powers by the first inequality of the lemma. The local integral of $|\nabla v_\varepsilon |^ p$ over $K\cap \{ 0<|f|<r_ K\} $ is then bounded by

With $s=\tfrac 12\log (r^2+\varepsilon ^2)$, we have $\mathrm ds=r\, \mathrm dr/(r^2+\varepsilon ^2)$ and $r\leq (r^2+\varepsilon ^2)^{1/2}$, so this is at most

The integral over the part where $|f|\geq r_ K$ is uniformly bounded as well. Weak compactness in Sobolev spaces and the local $L^ p$ convergence therefore show that $v\in W^{1,p}_{\mathrm{loc}}$. Its weak gradient agrees with the classical one off $Z$, and $Z$ has Lebesgue measure zero, which proves the final assertion. $\square$

Proof. The limit $\ell \geq 0$ exists because $\chi '$ is nonnegative and nondecreasing. Near a smooth point of a component of multiplicity $m_0$, outside the other components, the coordinate reduction to $f=w_1^{m_0}$ makes the positive complex Hessian of $v$ off $Z$ a single factor in the normal direction. Its Euclidean traces in the two coordinate systems are uniformly comparable, so polar integration gives necessity. In general, $G=\chi ''(\log |f|)|\partial f|^2/|f|^2$ off $Z$. The coarea formula and lemma 5.27.21, now bounding the remaining $2p-2$ powers of $|\partial f|$ by the first inequality of the lemma, give

The substitution $s=\log r$ proves sufficiency. For $p=1$ the integral is always finite, since $\int _{-\infty }^ A\chi ''=\chi '(A)-\ell $.

For the distributional identity, use $s_\varepsilon $ and $v_\varepsilon $ from the preceding proof. Direct differentiation gives

Off $Z$, the first term tends locally uniformly to $G$. By the level estimate of lemma 5.27.21 and the same substitution, its integral over $K\cap \{ |f|<r_0\} $ is at most

This bound tends to zero if we first let $\varepsilon \to 0$ and then $r_0\to 0$, so the first term converges to $G$ in $L^1_{\mathrm{loc}}$.

The positive measures $\Delta s_\varepsilon $ converge weakly to $\Delta \log |f|$, with locally bounded masses, and the Poincaré–Lelong formula shows that this limit is a measure supported on $Z$. On $\{ |f|<r_0\} $,

while $\Delta s_\varepsilon \to 0$ uniformly on compact subsets of $\{ |f|\geq r_0\} $. Taking the same successive limits shows that $(\chi '(s_\varepsilon )-\ell )\Delta s_\varepsilon \to 0$ weakly. Since $v_\varepsilon \to v$ in $L^1_{\mathrm{loc}}$, passing to the limit in the formula for $\Delta v_\varepsilon $ gives the identity. If $\ell >0$, the measure $\ell \, \Delta \log |f|$ is nonzero and singular with respect to Lebesgue measure; uniqueness of the Lebesgue decomposition therefore proves the final criterion. $\square$

Proof. Here $\chi (s)=-(-s)^ a$ is of class $C^2$ on $(-\infty ,-1]$, so the preceding propositions apply with $A=-1$. Moreover, $\chi '(s)=a(-s)^{a-1}$ and $\chi ''(s)=a(1-a)(-s)^{a-2}$, so $\ell =0$. The two propositions give the assertion about the Laplacian and the first-order thresholds. At $a=1/2$, the first-order integral diverges logarithmically; for $p>2$, its exponential factor forces divergence.

For second derivatives, take a smooth point of a component of multiplicity $m$, outside the other components, with a normal coordinate $w_1$ such that $f=w_1^ m$. Write $F(r)=\chi (m\log r)$ with $r=|w_1|$. The real second derivative of $v_ a$ in the unit direction tangent to the circles $|w_1|=r$ equals $F'(r)/r=m\chi '(m\log r)/r^2$. Thus the integral of its absolute value over the normal disk is bounded below, up to a fixed positive factor, by

Hence the classical Hessian of $v_ a$ off the divisor is not locally integrable, whereas local integrability is necessary for $W^{2,1}_{\mathrm{loc}}$. Since smooth coordinate changes preserve this Sobolev class, the argument is complete. $\square$