Proposition 10.19.2.24BCA82D5034 Pullback
\[ \pi ^*:H^{1,1}_{\mathrm{BC}}(B,\mathbb R) \longrightarrow H^{1,1}_{\mathrm{BC}}(X,\mathbb R) \]
is injective. The pseudoeffective and nef cones on $X$ coincide and are
\[ \operatorname {Psef}(X)=\operatorname {Nef}(X) =\{ a\{ \alpha \} _{\mathrm{BC}}+b\{ \beta \} _{\mathrm{BC}}: a,b\geq 0\} . \]
For the trace functional of the Gauduchon metric $\omega $, the slice of this cone at trace one is the segment $a+b=(8\pi )^{-1}$. Every class in the cone has a smooth semipositive representative, and none is big. The real form $\beta -\alpha $ is de Rham exact but represents a nonzero Bott–Chern class.
Proof.
Suppose a smooth closed real form $\tau $ on $B$ satisfies $\pi ^*\tau =\mathrm{dd}^{\mathrm{c}}u$ with $u$ smooth on $X$. The restriction of $u$ to each compact connected elliptic fiber is harmonic, hence constant, and local holomorphic sections show that $u=v\circ \pi $ for a smooth function $v$ on $B$. Thus $\pi ^*(\tau -\mathrm{dd}^{\mathrm{c}}v)=0$, which implies $\tau =\mathrm{dd}^{\mathrm{c}}v$ because $\pi $ is a submersion. This proves injectivity.
The usual cohomology computation for $\mathbb P^1\times \mathbb P^1$, together with the $\partial \bar\partial $-lemma on this Kähler surface, gives
\[ H^{1,1}_{\mathrm{BC}}(B,\mathbb R) =\mathbb R\{ \sigma _1\} \oplus \mathbb R\{ \sigma _2\} , \qquad \int _ B\sigma _1\wedge \sigma _2=1, \qquad \sigma _1^2=\sigma _2^2=0. \]
If $S$ is a positive closed current on $B$, the two coefficients of its class are
\[ a=\int _ BS\wedge \sigma _2\geq 0, \qquad b=\int _ BS\wedge \sigma _1\geq 0. \]
By lemma 20.4.10 and lemma 10.19.1, every pseudoeffective class on $X$ is the pullback of the class of such a current, and it therefore lies in the cone of the statement. Conversely, every class in that cone has the smooth semipositive representative $a\alpha +b\beta $ and is therefore nef. A nef class is pseudoeffective by the compactness argument following proposition 10.11.2. This proves the equality of the cones; the whole Bott–Chern group of $X$ is not needed.
Since $\omega ^2=2\alpha \wedge \beta +2(\alpha +\beta )\wedge D$ and $\int _ E D=4\pi $ on each fiber, integration along the fibers gives
\[ \int _ X(a\alpha +b\beta )\wedge \omega ^2=8\pi (a+b). \]
This proves the assertion about the slice. By proposition 12.19.22, $X$ carries no closed Kähler current, so none of these classes is big. Finally, the one-form $\mathrm d^{\mathrm c}t$ is globally defined on $X$ and satisfies
\[ \mathrm{d}(\mathrm d^{\mathrm c}t)=\beta -\alpha . \]
The class of $\sigma _2-\sigma _1$ on $B$ is nonzero, and injectivity makes its pullback nonzero in Bott–Chern cohomology. Thus de Rham exactness does not imply $\mathrm{dd}^{\mathrm{c}}$-exactness in this example.
$\square$
Pome
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