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			In mathematics, '''Deligne cohomology''' is the [[hypercohomology]] of the '''Deligne complex''' of a [[complex manifold]]. It was introduced by [[Pierre Deligne]] in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and [[intermediate Jacobian]]s.

			For introductory accounts of Deligne cohomology see {{harvtxt\|Brylinski\|2008\|loc=section 1.5}}, {{harvtxt\|Esnault\|Viehweg\|1988}}, and {{harvtxt\|Gomi\|2009\|loc=section 2}}.

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			The analytic Deligne complex '''Z'''(''p'')<sub>D, an</sub> on a complex analytic manifold ''X'' is
			:<math>0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^{p-1} \rightarrow 0 \rightarrow \dots</math>
			where '''Z'''(''p'') = (2π i)<sup>''p''</sup>'''Z'''. Depending on the context, <math>\Omega^*_X</math> is either the complex of smooth (i.e., ''C''<sup>∞</sup>) [[differential form]]s or of holomorphic forms, respectively.
			The Deligne cohomology {{nowrap\|{{SubSup\|''H''\|D,an\|''q''}}(''X'','''Z'''(''p''))}} is the ''q''-th hypercohomology of the Deligne complex.

			==Properties==

			Deligne cohomology groups {{nowrap\|{{SubSup\|''H''\|D\|''q''}}(''X'','''Z'''(''p''))}} can be described geometrically, especially in low degrees. For ''p'' = 0, it agrees with the ''q''-th singular cohomology group (with '''Z'''-coefficients), by definition. For ''q'' = 2 and ''p'' = 1, it is isomorphic to the group of isomorphism classes of smooth (or holomorphic, depending on the context) [[principal bundle\|principal '''C'''<sup>×</sup>-bundles]] over ''X''. For ''p'' = ''q'' = 2, it is the group of isomorphism classes of '''C'''<sup>×</sup>-bundles with [[connection (fiber bundle)\|connection]]. For ''q'' = 3 and ''p'' = 2 or 3, descriptions in terms of [[gerbe]]s are available ({{harvtxt\|Brylinski\|2008}}). This has been generalized to a description in higher degrees in terms of iterated [[classifying space]]s and connections on them ({{harvtxt\|Gajer\|1997}}).

			==Applications==
			Deligne cohomology is used to formulate [[Beilinson conjectures]] on [[special values of L-functions]].

			==References==

			*{{Citation \| last1=Brylinski \| first1=Jean-Luc \| title=Loop spaces, characteristic classes and geometric quantization \| origyear=1993 \| publisher=Birkhäuser Boston \| location=Boston, MA \| series=Modern Birkhäuser Classics \| isbn=978-0-8176-4730-8 \| doi=10.1007/978-0-8176-4731-5 \| mr=2362847 \| year=2008}}
			*{{Citation \| last1=Esnault \| first1=Hélène \| last2=Viehweg \| first2=Eckart \| title=Beĭlinson's conjectures on special values of L-functions \| url=http://books.google.com/books?id=CwTvAAAAMAAJ \| publisher=[[Academic Press]] \| location=Boston, MA \| series=Perspect. Math. \| mr=944991 \| year=1988 \| volume=4 \| chapter=Deligne-Beĭlinson cohomology \| chapterurl=http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf \| pages=43–91 \| isbn=978-0-12-581120-0}}
			*{{Citation \| last1=Gajer \| first1=Pawel \| title=Geometry of Deligne cohomology \| doi=10.1007/s002220050118 \| year=1997 \| journal=[[Inventiones Mathematicae]] \| issn=0020-9910 \| volume=127 \| issue=1 \| pages=155–207}}
			*{{Citation \| last1=Gomi \| first1=Kiyonori \| title=Projective unitary representations of smooth Deligne cohomology groups \| doi=10.1016/j.geomphys.2009.06.012 \| mr=2541824 \| year=2009 \| journal=Journal of Geometry and Physics \| issn=0393-0440 \| volume=59 \| issue=9 \| pages=1339–1356 \| arxiv=math/0510187}}

			[[Category:Sheaf theory]]
			[[Category:Homological algebra]]
			[[Category:Cohomology theories]]

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