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edited Feb 19, 2019 at 23:12

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It is not difficult to build new manifolds out of old in the smooth category, for example

taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth function,
quotienting by the free action of a cpt Lie group,

forming the connected sum and more generally gluing (in different ways) two manifolds

The connected sum construction is very flexible and important for the topological classification of manifolds. Unfortunately, in the complex category it doesn't behave well *: maybe the connected sum doesn't admit a complex structure at all. In my attempt to understand the topology of complex manifolds, it is natural to ask

What are the most common/powerfuluseful constructions that allows us to form a new complex manifold $X = \mathcal{F}(M, N)$ given two complex manifolds $M,N$?

The constructions I am interested in should have a topological interpretation and when meaningful the complex structure on $X$ should be compatible with the one of $M,N$.

*see Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure? and https://math.stackexchange.com/questions/1411694/is-the-connected-sum-of-complex-manifolds-also-complex)

It is not difficult to build new manifolds out of old in the smooth category, for example

taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth function,
forming the connected sum and more generally gluing (in different ways) two manifolds

What are the most common/powerful constructions that allows us to form a new complex manifold $X = \mathcal{F}(M, N)$ given two complex manifolds $M,N$?

The constructions I am interested in should have a topological interpretation and when meaningful the complex structure on $X$ should be compatible with the one of $M,N$.

*see Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure? and https://math.stackexchange.com/questions/1411694/is-the-connected-sum-of-complex-manifolds-also-complex)

It is not difficult to build new manifolds out of old in the smooth category, for example

taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth function,
quotienting by the free action of a cpt Lie group,

forming the connected sum and more generally gluing (in different ways) two manifolds

What are the most common/useful constructions that allows us to form a new complex manifold $X = \mathcal{F}(M, N)$ given two complex manifolds $M,N$?

The constructions I am interested in should have a topological interpretation and when meaningful the complex structure on $X$ should be compatible with the one of $M,N$.

*see Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure? and https://math.stackexchange.com/questions/1411694/is-the-connected-sum-of-complex-manifolds-also-complex)

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asked Feb 19, 2019 at 22:36

Overflowian

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Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example

taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth function,
forming the connected sum and more generally gluing (in different ways) two manifolds

What are the most common/powerful constructions that allows us to form a new complex manifold $X = \mathcal{F}(M, N)$ given two complex manifolds $M,N$?

The constructions I am interested in should have a topological interpretation and when meaningful the complex structure on $X$ should be compatible with the one of $M,N$.

*see Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure? and https://math.stackexchange.com/questions/1411694/is-the-connected-sum-of-complex-manifolds-also-complex)