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( \alpha , \beta ) = \int\limits \alpha \wedge \star \beta , where $ \star $

===Divergence and Hodge star=== Let $*$ denote the [[Hodge star]] operator on the oriented Riemannian manifold $\Sigma$ (cf. [[Laplace oper

modules $ H _ { \mathop{\rm dR} } ^ {n} ( X/S) = R ^ {n} f _ \star ( \Omega _ {X/S} ^ {\bullet } ) $, {\mathcal O} _ {S ^ {h} } \otimes _ {\mathbf C} R ^ {n} f _ \star \mathbf C _ {X ^ {h} } ,

...structure thus obtained is considered as a point in the moduli variety of Hodge structures of a given type. are then provided with a pure polarized Hodge structure, which is defined by a homomorphism of real algebraic groups (cf.

Using the [[Hodge structure|Hodge structure]] on the cohomology $ H _ {\textrm{ B } } ^ {i} ( X ) $ H _ {\mathcal M} ^ \bullet ( X, \mathbf Q ( \star ) ) _ {\mathbf Z} =

is equal to $ \delta ^ \nabla = (- 1) ^ {p + 1 } \star d ^ \nabla \star $, where $ \star $

...A } \rangle$ to any connection $A$ on $\xi $; here $*$ refers to the Hodge star operator (cf. also [[Laplace operator|Laplace operator]]) determined by the

the Hodge star operator $ * : \wedge ^ {p} (M) \rightarrow \wedge ^ {2m-p} (M) $, form one has the Hodge–Lepage decomposition $ \omega = \omega _ {0} + \epsilon _ \Omega \ome

...dge decomposition of the cohomology of an analytic manifold and the recent Hodge decomposition of the cohomology of a commutative algebra, [[#References|[a1 ...is a deformation-theoretic approach to quantization based on the so-called star-products, which are non-commutative deformations of the commutative product