Abstract
In this final chapter, we would like to introduce the reader to the complex version of Morse theory that has proved to be very useful in the study of the topology of complex projective varieties, and more recently in the study of the topology of symplectic manifolds.
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Notes
- 1.
To be accurate, what we call a linear system is what algebraic geometers refer to as an ample linear system.
- 2.
E.g., (X,A) is a compact ENR pair if X is a compact CW-complex and A is a subcomplex.
- 3.
The are called vanishing because they “melt” when pushed inside \(\hat{X}\).
- 4.
The orientation of the disk is determined by a linear ordering of the variables \({u}_{1},\ldots,{u}_{n}\).
- 5.
Note that while in the definition of the bundle orientation we tacitly used a linear ordering of the variables u i , the bundle orientation itself is independent of such a choice.
- 6.
This sign is different from the one in [AGV2] due to our use of the fiber-first convention. This affects the appearance of the Picard-Lefschetz formulæ. The fiber-first convention is employed in [Lam] as well.
- 7.
The choices of Δ and ∇ depended on linear orderings of the variables u i . However, the intersection number ∇ ∘ Δ is independent of such choices.
- 8.
Given an oriented submanifold S ⊂ X ∗ its Poincaré dual should satisfy either \({\int }_{S}\omega ={ \int }_{{X}_{{_\ast}}}\omega \wedge {\delta }_{S}\) or \({\int }_{S}\omega ={ \int }_{{X}_{{_\ast}}}{\delta }_{S} \wedge \omega \), \(\forall \omega \in {\Omega }^{\dim S}({X}_{{_\ast}})\), dω = 0. Our sign convention corresponds to first choice. As explained in [Ni1, Prop. 7.3.9] this guarantees that for any two oriented submanifolds S 1, S 2 intersecting transversally we have \({S}_{1} {_\ast} {S}_{2} ={ \int }_{{X}_{{_\ast}}}{\delta }_{{S}_{1}} \wedge {\delta }_{{S}_{2}}\).
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Nicolaescu, L. (2012). Basics of Complex Morse Theory. In: An Invitation to Morse Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1105-5_5
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DOI: https://doi.org/10.1007/978-1-4614-1105-5_5
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