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i would like to ask you a question i can not answer myself, i hope this is not too trivial and i'm not missing something too basic.

Let's suppose we have $X$ and $Y$ Kahler manifolds and $f:X\rightarrow Y$ a bimeromorphic map such that $f^*:H^2(Y)\rightarrow H^2(X)$ is an isomorphism and also an Hodge isometry. Then, given $\omega$ the class of the Kahler form on $X$ and $\omega'$ the class of the Kahler form on $Y$, is it possible $f^*(\omega')=-\omega$? Following the article i'm reading the answer seems to be no, but i can't figure out why..

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    $\begingroup$ You probably want $X$ and $Y$ compact. And could you explain what you call a Hodge isometry? $\endgroup$
    – abx
    Commented May 13, 2014 at 14:59
  • $\begingroup$ yes, i'm sorry, so let's say $X$ and $Y$ are compact Kahler surfaces and $f$ is an isometry with respect to the intersection form which also respect the Hodge decomposition (this is what i mean by Hodge isometry) $\endgroup$
    – Scott Koster
    Commented May 13, 2014 at 15:17
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    $\begingroup$ Just look at what happens to the pullback in local coordinates. You'll find that the pullback of a positive $(1,1)$-form by a holomorphic map is always semipositive. Your morphism is only bimeromorphic, but this is enough to see that we can't get $f^*(\omega') = -\omega$ on an open dense set. $\endgroup$
    – Gunnar Þór Magnússon
    Commented May 13, 2014 at 16:23
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    $\begingroup$ @Gunnar: the question is whether $f^*(\omega')=-\omega$ in $H^2(Y)$. How does your argument apply? $\endgroup$
    – abx
    Commented May 13, 2014 at 16:37
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    $\begingroup$ The cohomology class of $f^*\omega'$ should be both pseudoeffective (even big) and anti-Kähler, so it is impossible. $\endgroup$
    – Henri
    Commented May 14, 2014 at 7:45

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It is impossible, because the birational (movable) nef cone is mapped to birational nef cone, where birational nef cone is a cone of all classes which are non-negative on all curves which move in families covering the whole manifold. Clearly, $\omega$ belongs to the movable nef cone, and $-\omega$ does not.

This is for projective (or Moishezon) manifolds; if you need to see the same in non-projective situation, you need to use the pseudoeffective cone, which is the cone of all classes which can be represented by positive, closed (1,1)-currents. Demailly-Paun has shown that this cone (which coincides with the movable nef in projective situation) is a bimeromorphic invariant.

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