Pome

Example 18.39.4FE69C8A3CBBF

Example 18.39.4.FE69C8A3CBBF For all integers $q,r,k\geq 1$ and every real $0<c<k$, the hypotheses above are satisfied on $X=\mathbb P^ q\times \mathbb P^ r$ with $B=\mathbb P^ q$ and $A=\mathcal O_{\mathbb P^ q}(k)$, for suitable choices made in the proof. For $q\geq 2$ the defect is

\[ \dim V_ m-\dim W_ m =\frac{r(k-c)^{q-1}}{(q-1)!}m^{q-1}+O(m^{q-2}), \]

and for $q=1$ it equals $\min \{ r,\lfloor mc\rfloor \} $ for every $m$.

Proof. Use homogeneous coordinates $[Z_0:\cdots :Z_ q]$ on the base and $[W_0:\cdots :W_ r]$ on the fiber. Let $Z=\{ Z_0=0\} $ and $Y=Z\times \{ [1:0:\cdots :0]\} $, and let $\omega _ B,\omega _ F$ be the Fubini–Study forms normalized to have top integrals one. Define

\[ R_ B=\frac{|Z_0|^2}{\sum _{i=0}^ q|Z_ i|^2},\qquad R_ F=\frac{\sum _{j=1}^ r|W_ j|^2}{\sum _{j=0}^ r|W_ j|^2},\qquad \varphi =c\log (R_ B+R_ F),\quad \psi =c\log R_ B. \]

The function $\varphi $ is smooth off $Y$ and satisfies (18.39.2). Over a common denominator, the numerator of $R_ B+R_ F$ is

\[ \sum _{j=0}^ r|Z_0W_ j|^2+\sum _{i=0}^ q\sum _{j=1}^ r|Z_ iW_ j|^2, \]

whose logarithm is plurisubharmonic in each affine chart. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -c(\omega _ B+\omega _ F)$. Choose an integer $b\geq c$ and set $\omega =k\omega _ B+b\omega _ F$. This is a Hermitian form, and $\omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq (k-c)\omega _ B+(b-c)\omega _ F\geq 0$. On the base, $\theta =k\omega _ B$, $\Theta _ Z=\omega _ B$ and $\alpha =(k-c)\omega _ B>0$.

Put $\ell _ m=\lfloor mc\rfloor $. Divisibility of homogeneous polynomials by powers of $Z_0$ gives the exact dimensions

\[ \dim W_ m=\binom {km-\ell _ m+q}{q},\qquad \dim V_ m=\binom {km-\max \{ \ell _ m-r,0\} +q}{q}. \]

For large $m$, the two residual degrees differ by $r$ and equal $(k-c)m+O(1)$. Subtracting the two binomial coefficients, which are polynomials of degree $q$ in the residual degrees, gives the asserted leading coefficient and remainder. For $q=1$ the difference is $\ell _ m-\max \{ \ell _ m-r,0\} =\min \{ r,\ell _ m\} $ for every $m$. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub