Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, x_2) = e^{-t} \omega_1(x_1) + \omega_2(x_2)$, where $Ric(\omega_1) = 0$ and $Ric(\omega_2) = -\omega_2$. The claim is that the bisectional curvature of $\omega(t, x_1, x_2)$ blows up as $t \to \infty$. I do not quite see why it is true. Isn't $\omega(t)$ converging to a fixed metric smoothly?

  • 3
    $\begingroup$ No, $\omega(t)$ is not converging smoothly to a fixed metric. It converges smoothly to a semipositive definite symmetric $2$-tensor which is degenerate in the directions tangent to $X_1$. $\endgroup$ – YangMills Mar 15 at 3:44
  • 4
    $\begingroup$ About your question: your claim is false if $\omega_1$ is flat (which happens on tori and their quotients), and it is true if $\omega_1$ is not flat: just compute the curvature of $\omega$, using its explicit product structure. $\endgroup$ – YangMills Mar 15 at 3:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.