9.11.27 Equations on projective cones821709E49BC3

The residual construction of section 15.22 has a counterpart on a projective cone, which is a singular space. A logarithmic pole at the vertex becomes a divisor on the incidence resolution, and removing it leaves a positive residual background. The radial equation below determines both the solution and its normalization scalar. Uniqueness among bounded residuals holds because the lifted density is smooth and its positive locus is connected and dense, even though the density vanishes on the exceptional section.

Let $B\subset \mathbb P^ N$ be a smooth connected, linearly nondegenerate, projectively normal variety of dimension $n-1$, where $N\geq n\geq 2$. Let $X\subset \mathbb P^{N+1}$ be its projective cone, with vertex $p=[0:\cdots :0:1]$, and let $\theta _0=\omega _{\mathrm{FS}}|_ X$ be the restriction of the Fubini–Study form of $\mathbb P^{N+1}$. Write the ambient homogeneous coordinates as $[Z:W]$, with $Z=(Z_0,\ldots ,Z_ N)$. The incidence resolution $\pi :Y\to X$ is a smooth $\mathbb P^1$-bundle $f:Y\to B$; its section $E=\pi ^{-1}(p)$ is contracted, and $\pi $ is an isomorphism off $E$. These cones are reduced, irreducible, compact and normal, hence unibranch, and their vertices are singular.

Indeed, the homogeneous coordinate ring of $B$ is normal by projective normality. Adjoining $W$ preserves normality, and so do the degree-zero projective charts. Algebraic normality passes to the associated analytic space by the algebraic–analytic normality theorem. Smoothness and connectedness of $B$ give irreducibility. In the chart $W=1$, the ideal of the affine cone has no linear terms, by linear nondegeneracy. Its tangent space at the origin therefore has dimension $N+1>n$, so $p$ is singular. The incidence construction is the bundle of lines in $\mathcal O_ B\oplus \mathcal O_ B(-1)$ and has the stated contraction and isomorphism properties.

Write

The same symbol $x$ denotes its pullback to $Y$. Choose $\Psi \in C^\infty (\mathbb P^ N,\mathbb R)$ with $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}\Psi >0$ and $k\in C^\infty ([0,1])$ with $k>0$. Put $\psi =\Psi |_ B$ and define on $Y$

These data prescribe the measure $\mu $ before the equation is solved. The local form and the fiber integral below describe its degeneracy along $E$ and show that its total mass is $M$.

Choose a local holomorphic lift $b(z)$ of the embedding of $B$, and write $S(z)=\sum |b_ j(z)|^2$ and $Z=t b(z)$. In these local coordinates $(z,t)$ on $Y$,

On $0<x<1$ put $\rho =\log (|t|^2S)$, $\rho _\psi =\rho +\psi $, $h(r)=\log (1+\mathrm{e}^ r)$ and $\zeta _\psi =\mathrm{d}\rho _\psi \wedge \mathrm d^{\mathrm c}\rho _\psi $. Then $x=h'(\rho )$, $s=h'(\rho _\psi )$ and $\ell =h(\rho _\psi )-h(\rho )$. Differentiation gives

Near $E$, a local potential of $B_\psi $ is $\log (1+\mathrm{e}^\psi |t|^2S)$; at $t=0$ its Hessian has rank one, in the transverse direction. At the infinity section, with $v=1/t$, a local potential is $\log (\mathrm{e}^\psi S+|v|^2)$; its tangential Hessian is $\sigma _\psi $ and its transverse coefficient is positive. These computations show that $B_\psi $ is smooth and semipositive, and positive off $E$. Since $n\geq 2$, its top power vanishes on $E$.

Along a fiber, $\mathrm d^{\mathrm c}\rho _\psi =\mathrm{d}(\arg t)/(2\pi )$ and $\mathrm{d}\rho _\psi =\mathrm{d} s/[s(1-s)]$. Base components disappear after wedging with $\sigma _\psi ^{n-1}$, so (9.11.29) gives

Since $\omega _ B+\mathrm{dd}^{\mathrm{c}}\psi $ is closed, Stokes' theorem on $B$ gives $\int _ B(\omega _ B+\mathrm{dd}^{\mathrm{c}}\psi )^{n-1}=d$, hence $\nu (Y)=M$. Smooth measures give $E$ zero mass. Therefore $\mu (X)=M$ and $\mu (\{ p\} )=0$. Taking $\psi =0$ and $k=1$ gives $\int _ Y\beta ^ n=d$.