Pluriclosed star split Hermitian metrics

Abstract

We introduce a class of Hermitian metrics, that we call pluriclosed star split, generalising both the astheno-Kähler metrics of Jost and Yau and the $(n-2)$-Gauduchon metrics of Fu-Wang-Wu on complex manifolds. They have links with Gauduchon and balanced metrics through the properties of a smooth function associated with any Hermitian metric. After pointing out several examples, we generalise the property to pairs of Hermitian metrics and to triples consisting of a holomorphic map between two complex manifolds and two Hermitian metrics, one on each of these manifolds. Applications include an attack on the Fino-Vezzoni conjecture predicting that any compact complex manifold admitting both SKT and balanced metrics must be Kähler, that we answer affirmatively under extra assumptions. We also introduce and study a Laplace-like differential operator of order two acting on the smooth $(1,\,1)$-forms of a Hermitian manifold. We prove it to be elliptic and we point out its links with the pluriclosed star split metrics and pairs defined in the first part of the paper.

Appendix: Commutation relations

We briefly recall here some standard formulae that were used throughout the paper.

Let $(X,\,\omega )$ be a compact complex Hermitian manifold. Recall the following standard Hermitian commutation relations ( [3], see also [ [4], VII, $\S .1$]):

$$\begin{aligned}{} & {} \mathrm{(i)}\,\,(\partial + \tau )^{\star } = i\,[\Lambda ,\,{{\bar{\partial }}}]; \hspace{3ex} \mathrm{(ii)}\,\,({{\bar{\partial }}} + {{\bar{\tau }}})^{\star } = - i\,[\Lambda ,\,\partial ]; \nonumber \\{} & {} \mathrm{(iii)}\,\, \partial + \tau = -i\,[{{\bar{\partial }}}^{\star },\,L]; \hspace{3ex} (iv)\,\, {{\bar{\partial }}} + {{\bar{\tau }}} = i\,[\partial ^{\star },\,L], \end{aligned}$$

(95)

where the upper symbol $\star$ stands for the formal adjoint w.r.t. the $L^2$ inner product induced by $\omega$, $L=L_{\omega }:=\omega \wedge \cdot$ is the Lefschetz operator of multiplication by $\omega$, $\Lambda =\Lambda _{\omega }:=L^{\star }$ and $\tau _\omega =\tau :=[\Lambda ,\,\partial \omega \wedge \cdot ]$ is the torsion operator (of order zero and type $(1,\,0)$) associated with the metric $\omega$.

Other standard formulae (see e.g. [20]) are the following:

$$\begin{aligned}{} & {} \mathrm{(i)}\,\, [\Lambda ,\,L] = (n-k)\,\text{ Id } \hspace{5ex} \text{ on }\hspace{1ex} k\text{-forms, } \text{ for } \text{ every } \text{ non-negative } \text{ integer }\hspace{1ex} k; \nonumber \\{} & {} \mathrm{(ii)}\,\, [L^r,\,\Lambda ] = r(k-n+r-1)\,L^{r-1} \hspace{5ex} \text{ on }\hspace{1ex} k\text{-forms, } \text{ for } \text{ all } \text{ integers }\hspace{1ex} k\ge 0, r\ge 2; \nonumber \\{} & {} \mathrm{(iii)}\,\,\star \,L = \Lambda \,\star \hspace{3ex}\text{ and }\hspace{3ex} \star \,\Lambda = L\,\star . \end{aligned}$$

(96)

We also used the following result involving again the torsion operator $\tau _\omega$.

Lemma 1.63

Let $(X,\,\omega )$ be a compact complex Hermitian manifold with $\text{ dim}_{\mathbb {C}}X=n$. The following identities hold:

$$\begin{aligned} -\frac{1}{2}\,{{\bar{\tau }}}_\omega ^\star \omega {\mathop {=}\limits ^{(i)}} {{\bar{\partial }}}^\star _\omega \omega {\mathop {=}\limits ^{(ii)}} i\,\Lambda _\omega (\partial \omega ). \end{aligned}$$

(97)

In particular, $\omega$ is balanced if and only if ${{\bar{\tau }}}_\omega ^\star \omega =0$.

Proof

$\bullet$ To prove identity (i) in (97), we will show that the multiplication operators by the $(1,\,0)$-forms ${{\bar{\tau }}}^\star \omega$ and $-2{{\bar{\partial }}}^\star \omega$ acting on functions, namely

$$\begin{aligned} {{\bar{\tau }}}^\star \omega \wedge \cdot , \hspace{1ex} -2{{\bar{\partial }}}^\star \omega \wedge \cdot :C^\infty _{0,\,0}(X,\,{\mathbb {C}})\longrightarrow C^\infty _{1,\,0}(X,\,{\mathbb {C}}), \end{aligned}$$

coincide by showing that their adjoints

$$\begin{aligned} ({{\bar{\tau }}}^\star \omega \wedge \cdot )^\star , \hspace{1ex} (-2{{\bar{\partial }}}^\star \omega \wedge \cdot )^\star :C^\infty _{1,\,0}(X,\,{\mathbb {C}})\longrightarrow C^\infty _{0,\,0}(X,\,{\mathbb {C}}) \end{aligned}$$

coincide.

Let $\alpha \in C^\infty _{1,\,0}(X,\,{\mathbb {C}})$ and $g\in C^\infty _{0,\,0}(X,\,{\mathbb {C}})$ be arbitrary. We have:

$$\begin{aligned} \langle \langle ({{\bar{\partial }}}^\star \omega \wedge \cdot )^\star \alpha ,\,g \rangle \rangle = \langle \langle {\bar{g}}\alpha ,\,{{\bar{\partial }}}^\star \omega \rangle \rangle = \langle \langle {{\bar{\partial }}}({\bar{g}}\alpha ),\,\omega \rangle \rangle = \int \limits _X{{\bar{\partial }}}({\bar{g}}\alpha )\wedge \star \omega = \int \limits _X{\bar{g}}\alpha \wedge {{\bar{\partial }}}\omega _{n-1},\nonumber \\ \end{aligned}$$

(98)

where we put $\omega _{n-1}:=\omega ^{n-1}/(n-1)!$ and we used the standard identity $\star \omega = \omega _{n-1}$.

Meanwhile, we have:

$$\begin{aligned} \langle \langle ({{\bar{\tau }}}^\star \omega \wedge \cdot )^\star \alpha ,\,g \rangle \rangle= & {} \langle \langle {\bar{g}}\alpha ,\,{{\bar{\tau }}}^\star \omega \rangle \rangle = \langle \langle {\bar{g}}\,{{\bar{\tau }}}(\alpha ),\,\omega \rangle \rangle = \langle \langle {\bar{g}}\,\Lambda ({{\bar{\partial }}}\omega \wedge \alpha ),\,\omega \rangle \rangle \nonumber \\= & {} \langle \langle {{\bar{\partial }}}\omega \wedge \alpha ,\,g\,\omega ^2\rangle \rangle = \int \limits _X{{\bar{\partial }}}\omega \wedge \alpha \wedge \star ({\bar{g}}\omega ^2) = -2\,\int \limits _X{\bar{g}}\alpha \wedge {{\bar{\partial }}}\omega \wedge \omega _{n-2} \nonumber \\= & {} -2\,\int \limits _X{\bar{g}}\alpha \wedge {{\bar{\partial }}}\omega _{n-1}, \end{aligned}$$

(99)

where for the third identity on the first line we used the definition ${{\bar{\tau }}} = [\Lambda ,\,{{\bar{\partial }}}\omega \wedge \cdot ]$ of ${{\bar{\tau }}}$ and the fact that $\Lambda (\alpha )=0$ for bidegree reasons, while for the third identity on the second line we used the standard identity $\star \omega _2 = \omega _{n-2}$, where $\omega _2:=\omega ^2/2!$.

Comparing (98) and (99), we get $\langle \langle ({{\bar{\tau }}}^\star \omega \wedge \cdot )^\star \alpha ,\,g \rangle \rangle = -2\,\langle \langle ({{\bar{\partial }}}^\star \omega \wedge \cdot )^\star \alpha ,\,g \rangle \rangle$ for all $\alpha$ and g. Hence $({{\bar{\tau }}}^\star \omega \wedge \cdot )^\star = -2\,({{\bar{\partial }}}^\star \omega \wedge \cdot )^\star$, which proves (i) of (97).

$\bullet$ To prove identity (ii) in (97), we start from the Hermitian commutation relation (ii) in (95):

$$\begin{aligned} {[}\Lambda ,\,\partial ] = i\,({{\bar{\partial }}}^\star +{{\bar{\tau }}}^\star ) \end{aligned}$$

that we apply to $\omega$. We get the equivalent identities:

$$\begin{aligned} {[}\Lambda ,\,\partial ]\,\omega = i{{\bar{\partial }}}^\star \omega + i{{\bar{\tau }}}^\star \omega \iff \Lambda (\partial \omega ) - \partial (\Lambda \omega ) = -i{{\bar{\partial }}}^\star \omega \iff \Lambda (\partial \omega ) = -i{{\bar{\partial }}}^\star \omega , \end{aligned}$$

the last of which is (ii) of (97), where for the first equivalence we used the identity ${{\bar{\tau }}}^\star \omega = -2\,{{\bar{\partial }}}^\star \omega$ proved above as (i) in (97), while for the second equivalence we used the fact that $\Lambda \omega = n$, hence $\partial (\Lambda \omega )=0$. $\square$