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Complex manifold: Quiz


Question 1: One can define an analogue of a ________ for complex manifolds, called a Hermitian metric.
Differentiable manifoldRiemannian geometryRiemannian manifoldMetric tensor

Question 2: For example, the 6-dimensional sphere S6 has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the ________, but this is not a complex structure.
Cayley–Dickson constructionOctonionQuaternionComplex number

Question 3: The ________ 1-dimensional complex manifolds are isomorphic to either:
TopologySimply connected spaceOrthogonal groupTorus

Question 4: Concretely, this is an endomorphism of the ________ whose square is −I; this endomorphism is analogous to multiplication by the imaginary number i, and is denoted J (to avoid confusion with the identity matrix I).
Differentiable manifoldFiber bundleTangent bundleVector space

Question 5: Similarly, the ________ analogs of these are also complex manifolds.
BiquaternionQuaternionVector spaceComplex number

Question 6: The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a ________, the structure of which remains an area of active research.
Geometric invariant theoryPicard groupModuli spaceScheme (mathematics)

Question 7: In ________, a complex manifold is a manifold with an atlas of charts to the open unit disk[1] in Cn, such that the transition maps are holomorphic.
Differentiable manifoldDifferential geometryCalculusDifferential topology

Question 8: Smooth complex ________ are complex manifolds, including:
Algebraic varietyAlgebraic geometryScheme (mathematics)Projective space

Question 9: ________, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon.
Riemann surfaceAbelian varietyRiemann sphereProjective space

Question 10: Examples of Kähler manifolds include smooth ________ and more generally any complex submanifold of a Kähler manifold.
Algebraic varietyProjective spaceAlgebraic geometryScheme (mathematics)


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