We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of Bøgvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.

我们研究了在奇特征域上定义的有限维李超代数的上同调支持变体。验证我们之前工作中的一个猜想，我们展示了有限维超模的支持多样性可以实现为底层李超代数的奇零锥的显式子集。我们还证明了有限维超模的支持度为零当且仅当超模是有限投影维的。因此，我们获得了 Bøgvad 定理的正特征版本，表明如果在奇特征场上的有限维李超代数是绝对无扭的，那么它的包络代数是有限全局维的。