# Why does the bisectional curvature blow up?

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?

• 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 $X_1$. – YangMills Mar 15 at 3:44
• 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. – YangMills Mar 15 at 3:45