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The Dolbeault Laplacian is ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. On a Kähler manifold, the Kähler identities imply

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,Its formal adjoint is denoted by $\Lambda =L^{\ast }$ and lowers the bidegree of a differential form by $(1,1)$.

A Riemannian metric $g$ on a complex manifold is a Kähler metric when its complex-linear extension is a Hermitian form on each tangent space and its fundamental two-formis closed. Equivalently, $\omega$ is a positive real closed $(1,1)$-form, making $X$ a Kähler manifold.

The Lefschetz operator of a Kähler manifold and its adjoint areBecause $d\omega =0$ and $\omega$ has type $(1,1)$, both $\partial \omega$ and $\bar{\partial }\omega$ vanish. The graded Leibniz rule therefore gives $[L,\partial ]=[L,\bar{\partial }]=0$.

Writing formal adjoints with stars, define the three Laplacians byThe supplied Kähler identities identity $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$ gives, by complex conjugation and taking adjoints,Expanding these commutators and using $\partial \bar{\partial }+\bar{\partial }\partial =0$ shows that the mixed terms in $\Delta$ vanish and that ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Since $d=\partial +\bar{\partial }$, it follows that

The adjoint of $[L,\partial ]=0$ gives $[\Lambda ,{\partial }^{\ast }]=0$. Hencewhere the last equality is the adjoint of $\partial \bar{\partial }+\bar{\partial }\partial =0$. Thus $\Lambda$, and therefore every power ${\Lambda }^k$, commutes with ${\Delta }_{\partial }$. It follows that ${\Lambda }^k$ sends every $\partial$-harmonic $(p,q)$-form to a $\partial$-harmonic $(p-k,q-k)$-form whenever $0\le k\le \min (p,q)$.

Solved by gpt-5.6-sol high.

A Kähler manifold has a real closed $(1,1)$-form $\omega$ for which $g(u,v)=\omega (u,Jv)$ is positive definite. The Fubini-Study form ${\omega }_{FS}$ has these properties on ${\mathbb{CP}}^2$. Pullback by the holomorphic inclusion $i:X\hookrightarrow {\mathbb{CP}}^2$ preserves reality, type, and closedness, while positivity restricts to every nonzero vector in $TX$. Hence ${\omega }_X=i^{\ast }{\omega }_{FS}$ is a Kähler form on $X$.

Solved by gpt-5.6-sol high.

Choose a small ball $B$ whose translates by distinct lattice elements are disjoint. The restrictions of the quotient map $\pi :{\mathbb{C}}^n\to {\mathbb{C}}^n/\Lambda$ to translates of $B$ supply holomorphic charts, because every transition map is a complex translation. A closed fundamental parallelepiped is compact and surjects onto the quotient, so the resulting complex torus $X$ is compact.

The standard formis translation invariant and therefore descends uniquely to a form ${\omega }_X$ with ${\pi }^{\ast }{\omega }_X={\omega }_0$. It remains closed, of type $(1,1)$, and positive, so it is a Kähler form.

Solved by gpt-5.6-sol high.