Equivalent definitions of Hyperkahler manifolds.

Question

I am reading the paper HYPERKAHLER METRICS ON COTANGENT BUNDLES OF ¨ HERMITIAN SYMMETRIC SPACE by OLIVIER BIQUARD AND PAUL GAUDUCHON.

Suppose $M$ is a manifold with a triple $(g,I,J)$ where g is a metric and $I,J$ are anti-commuting $g$ orthogonal almost complex structures. Then in the paper $M$ is said to be hyperkahler if and only if

$d\omega_I=d\omega_J=d\omega_K=0$ where $K=IJ$. (And this would imply $I,J,K$ are integrable almost complex structures.)

Now suppose $M$ is the cotangent bundle $T^*\Sigma$ of a complex manifold $\Sigma$. Let $I$ be the complex structure induced from the complex structure of $\Sigma$. Let $J$ a $(1,1)$ tensor such that $\omega_J=g(J(\cdot),\cdot)$ is equal to the canonical symplectic form of $M$ as a contangent bundle. Then the paper claims that $(g,I,J)$ being Hyperkahler is equivalent to the following conditions:

1. $g(I(\cdot),I(\cdot))=g(\cdot,\cdot)$

2. $J^2=-1$

3. $d\omega_I=0$ where $\omega_I=g(I(\cdot),\cdot)$.

But from these conditions, I can only obtain that

$d\omega_I=d\omega_J=0$, and $\nabla I=0$.

So my problem is, are these conditions equivalent to saying that $M$ is hyperKahler? Any help or hints or reference are appreciated. Thank you in advanced.

Answer 1

Let $\omega_c$ be the holomorphic symplectic form on $T^*\Sigma$. Then $\omega_c(IX,IY) = - \omega_c(X,Y)$, since $\omega_c$ is of type $(2,0)$ with respect to $I$. Taking the real part on both sides, we get: $$\omega_J(IX,IY) = -\omega_J(X,Y).$$ Therefore we have $$g(JIX,IY) = -g(JX,Y) = -g(IJX,IY).$$ And since this is true for any $X$ and $Y$, this implies that $IJ = -JI$.

Moreover, since $K = IJ$ and since $\omega_J$ is of type $(2,0)$ with respect to $I$, therefore $\omega_K$ is the imaginary part of $\omega_c$, and it follows that $\omega_K$ is also closed, so that $(g,I,J)$ is hyperkähler, assuming the $3$ conditions you wrote down.