论文标题
HKT流形：霍奇理论，形式和平衡指标
HKT manifolds: Hodge theory, formality and balanced metrics
论文作者
论文摘要
令$（m，i，j，k，ω）$为紧凑的hkt歧管，用$ \ partial $表示偶联的dolbeault操作员，相对于$ i $，$ i $，$ \ partial_j：= j^{ - 1} \ 1} \ overline \ partline \ partial j $，$ \ partial J $ $ L：=ω\ wedge- $。在适当的假设下，我们研究了复合物的霍奇理论$（a^{\ bullet，0}，\ partial，\ partial_j）$和$（a^{\ bullet，0}，\ partial，\ partial^λ）$，显示出与KählerArporolds相似的行为。特别地，证明了拉普拉斯主义者之间的几个关系，谐波形式的空间和相关的共同体学组以及坚硬的lefschetz属性。此外，我们表明，对于紧凑的hkt $ \ mathrm {sl}（n，\ mathbb {h}）$ - 使差分分级的代数$（a^{\ bullet，0}，\ partial）$正式正式，这会导致HKT $ \ nustrunt this thimentive hkt $ \ mathrm mathrm = hkt $ {n}（ $（i，j，k，ω）$在紧凑型复杂歧管$（m，i）$上。最后，研究了Solvmanifolds上平衡的HKT结构。
Let $(M,I,J,K,Ω)$ be a compact HKT manifold and denote with $\partial$ the conjugate Dolbeault operator with respect to $I$, $\partial_J:=J^{-1}\overline\partial J$, $\partial^Λ:=[\partial,Λ]$ where $Λ$ is the adjoint of $L:=Ω\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^Λ)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$-structure $(I,J,K,Ω)$ on a compact complex manifold $(M,I)$. Finally, balanced HKT structures on solvmanifolds are studied.
