[Paper Review] Special Hermitian metrics and Lie groups

Nicola Enrietti, Anna Fino|arXiv (Cornell University)|Apr 8, 2011

Geometry and complex manifolds22 references4 citations

TL;DR

This paper constructs new 4n-dimensional strong Kähler with torsion (SKT) Lie algebras from 2n-dimensional SKT Lie algebras using a Hermitian flat connection, providing a systematic method to generate SKT examples in dimension 8. It classifies symplectic forms taming complex structures on 4-dimensional Lie algebras and identifies Kähler metrics within these SKT structures.

ABSTRACT

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form $ω$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfrak g$ {and using} a Hermitian flat connection on $\mathfrak g$ we construct a $4n$-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.

Motivation & Objective

• To develop a general construction method for generating higher-dimensional SKT Lie algebras from lower-dimensional SKT Lie algebras.
• To classify symplectic forms that tame complex structures on 4-dimensional Lie algebras.
• To identify conditions under which SKT metrics on 4-dimensional Lie algebras admit associated Kähler metrics.
• To explore the relationship between SKT geometry and generalized Kähler structures via the Bismut connection and closed 3-forms.
• To provide explicit examples of SKT and generalized Kähler structures in dimension 8 using known 4-dimensional SKT Lie algebras.

Proposed method

• Utilizes a Hermitian flat connection on a 2n-dimensional SKT Lie algebra to induce a 4n-dimensional SKT Lie algebra structure.
• Applies the construction to specific 4-dimensional SKT Lie algebras: $\mathfrak{r}_{4,\lambda,0}$, $\mathfrak{d}_{4,2}$, $\mathfrak{d}_{4,\frac{1}{2}}$, $\mathfrak{d}'_{4,\lambda}$, and $\mathfrak{aff}_{\mathbb{R}}\times\mathfrak{aff}_{\mathbb{R}}$.
• Employs the condition $\partial\overline{\partial}\omega = 0$ to define SKT metrics, ensuring the fundamental 2-form $\omega$ is $\partial\overline{\partial}$-closed.
• Analyzes the existence of Kähler metrics by checking whether certain (1,1)-forms $\omega_k$ are closed and positive definite.
• Uses the equivalence between $\partial\omega = \overline{\partial}\beta$ and the existence of a symplectic form taming the complex structure $J$.
• Verifies that all $d$-closed 3-forms are exact in the considered Lie algebras by computing $b_3^{\text{inv}} = \dim H^3(\mathfrak{g}) = 0$.

Experimental results

Research questions

• RQ1 Can a 2n-dimensional SKT Lie algebra be used to construct a 4n-dimensional SKT Lie algebra via a Hermitian flat connection?
• RQ2 Which 4-dimensional Lie algebras admit symplectic forms that tame their complex structures?
• RQ3 Under what conditions does an SKT metric on a 4-dimensional Lie algebra admit a Kähler metric in the same complex structure?
• RQ4 How do the Bismut connection and the torsion 3-form $c$ relate to the SKT condition $dc = 0$ in the constructed examples?
• RQ5 Can the generalized Kähler structure from [15] be recovered using this construction method on known 4-dimensional SKT Lie algebras?

Key findings

• The construction yields new 8-dimensional SKT Lie algebras from known 4-dimensional SKT Lie algebras, including $\mathfrak{d}_{4,\frac{1}{2}}$ and $\mathfrak{d}'_{4,\lambda}$.
• For $\mathfrak{aff}_{\mathbb{R}}\times\mathfrak{aff}_{\mathbb{R}}$, the SKT metric $\omega = e^{12} + e^{34} + t(e^{13} + e^{24})$ satisfies $\partial\omega = \overline{\partial}(a\alpha^{12})$ with $a = -\frac{t}{2} + i\frac{u_3 - y_2}{2(x_3 + v_2)}$.
• In $\mathfrak{r}_{4,\lambda,0}$, the Kähler metric $\omega_k = ne^{12} + me^{34} + m\frac{y_1y_3}{x_1^2 + y_3^2}(e^{14} - e^{23}) - m\frac{y_1x_1}{x_1^2 + y_3^2}(e^{13} + e^{24})$ exists when $n > m\frac{y_1}{x_1^2 + y_3^2}(y_3 - x_1)$ and $n,m > 0$.
• For $\mathfrak{d}_{4,2}$, the Kähler metric $\omega_k = ne^{12} + me^{34} + m\frac{2u_1}{3x_1}(e^{14} - e^{23})$ is valid when $n > m\frac{4u_1^2}{9x_1^2}$ and $m > 0$.
• In $\mathfrak{d}'_{4,\lambda}$, the Kähler metric $\omega_k = m\frac{1+q^2}{r^2}e^{12} + me^{34} + m\frac{q}{r}(e^{14} - e^{23})$ is closed and positive for $m > 0$.
• The construction recovers the generalized Kähler example from [15] using the $\mathfrak{d}_{4,\frac{1}{2}}$ Lie algebra, confirming consistency with prior results.