Proposition 24.4.11.E63ACDB15696 Suppose $c_1(L)\ne 0$ in real cohomology, and put
\[ \kappa =-\frac{\int _ Bc^2}{A}>0. \]
Every smooth potential $u=f(t)+\pi ^*v$ with $f$ of period $T$, $v\in C^\infty (B,\mathbb R)$ and $\omega +\mathrm{dd}^{\mathrm{c}}u>0$ satisfies
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 >3TA\bigl (1-\kappa F(\kappa ^{-1/2},T)\bigr )>0. \]
Let $\lambda \geq 0$ be the nonnegative eigenvalue function of $c$ relative to $\eta $, so $c^2=-\lambda ^2\eta ^2$, and put $M_0=1/\max _ B\lambda $. For the subclass $v=0$, the exact mass infimum is
\[ 3TA\bigl (1-\kappa F(M_0,T)\bigr )>0. \]
It is approached by smooth potentials $f(t)$ with $\omega +\mathrm{dd}^{\mathrm{c}}f(t)>0$, and it is not attained by such potentials.
Proof.
Put $p=f'$. The chain rule gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}u =\pi ^*(\eta +pc+\mathrm{dd}^{\mathrm{c}}v)+(1+p')D. \]
Positivity on the vertical line implies $p'>-1$. Over every point of $B$, the function $t$ takes every value modulo $T$, so restriction to $\ker \partial t$ shows that $\eta +pc+\mathrm{dd}^{\mathrm{c}}v$ is a Kähler form on $B$ for every value of $p$. Its square has integral
\[ \int _ B(\eta +pc+\mathrm{dd}^{\mathrm{c}}v)^2=A(1-\kappa p^2)>0, \]
since primitivity removes the mixed term and Stokes' theorem removes the exact terms. Hence $|p|<\kappa ^{-1/2}$.
Only the vertical derivative enters (24.4.9), because $-\pi ^*(c^2)$ has top degree in the base directions. For a $T$-periodic function $F_0$, integration over a fiber gives $\int _ E F_0(t)D=\int _0^ T F_0(s)\, \mathrm{d}s$, and therefore
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 =3TA\left(1-\frac\kappa T\int _0^ T p^2\, \mathrm{d}s\right). \]
Now apply lemma 24.4.10 with $M=\kappa ^{-1/2}$. The formula for $F$ gives $F(M,T)<M^2$, which proves positivity of the bound.
For $v=0$, diagonalizing $c$ with respect to $\eta $ shows that the horizontal positivity condition is exactly $|p|<M_0$; together with $p'>-1$ it also suffices for positivity of the full form. Moreover $\kappa $ is the $\eta ^2$-average of $\lambda ^2$, so $\kappa \leq M_0^{-2}$. Lemma 24.4.10 with $M=M_0$ therefore proves the claimed positive lower bound and its nonattainment. The smooth approximating functions in its proof have mean zero and hence periodic primitives $f_\varepsilon $, and the potentials $f_\varepsilon (t)$ prove that the bound is sharp.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.