Abstract
We show various vanishing theorems for the cohomology groups ofcompact Hermitian manifolds for which the Bismut connection has a(restricted) holonomy contained in SU(n) and classify all such manifoldsof dimension four. In this way we provide necessary conditions for theexistence of such structures on compact Hermitian manifolds withvanishing first Chern class of non-Kähler type. Then we apply ourresults to solutions of the string equations and show that such solutionsadmit various cohomological restrictions such as, for example, that undercertain natural assumptions the plurigenera vanish. We also find thatunder some assumptions the string equations are equivalent to thecondition that a certain vector is parallel with respect to the Bismutconnection.
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