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I added a more general definition to Calabi-Yau variety, in fact, the one I had in mind when I recently linked to it. I’m wondering if it is standard to call the first given definition a Calabi-Yau “variety” rather than Calabi-Yau “manifold” since it seems to assume the topology is the analytic one rather than Zariski which is not what the word variety evokes in my mind.
Also, there should be an exact algebraic definition so that when you take the analytification you get exactly the analytic definition. The conventions in the algebraic geometry literature vary a lot. For instance, I mostly see the vanishing cohomology condition that I wrote, but others merely assume trivial canonical bundle with no cohomological conditions. Some assume it must be simply connected. Some assume projective rather than proper. I’ve always been curious what the actual conditions to get a correspondence would be.
I have added some hyperlinks.
Concerning the question on whether to make the analytic meaning explicit: yes, please say explicitly once how the language is to be read. On a wiki, as opposed to in a textbook, we need to make sure that every entry can safely be read on its own.
Just a general remark on hyperlinks:
we all need to be aware that even when a page for some keyword does already exist, it may not actually explain the term in the sense used in a given entry.
For instance Calabi-Yau variety links to holonomy but what is meant is “holonomy” in the sense of “spaces with special holonomy”, whereas the entry currenty just explains the notion of holonomy of a connection along a curve itself.
Also the entry dimension probably does not really do its topic justice yet in the algebraic context.
Just something to keep in mind.
I’m wondering if it is standard to call the first given definition a Calabi-Yau “variety” rather than Calabi-Yau “manifold” since it seems to assume the topology is the analytic one rather than Zariski
A compact Kaehler manifold is automatically a projective algebraic variety over complex numbers. It is standard in complex algebraic geometry to use transcendental tools and the power of complex topology. By Weierstrass theorems you can get the algebraic essence from some analytic expressions, poles and so on. Griffiths-Harris is a standard textbook on the subject in analytic approach.
Thanks, that makes me feel better about the use of the word variety. Maybe it is just because I almost never work over C, but I just feel like the definition of Insert Name variety should work for any variety over any field, and then as a special case you get the version over C which can be translated into analytic language.
Unfortunately, algebraic geometry consists of several specialist communities which ares somewhat lightly socially connected. One expert in foundations of AG told me after witnessing conference on low dimensional complex algebraic geometry: “It felt very weird. I knew all the words they are using and I had still no idea on the content they talked about.”
c_1(X) (X compact Kähler) dices not in general imply the canonical bundle is holomorphically trivial (hyperelliptic surfaces are a counterexample). However if X is simply connected, by a standard argumento using the short exact sequence of sheaves 0->Z-> \mathcal{O} -> \mathcal{O}^{*} -> 0, one can see that if c_1(X)=0, then K_M=0 \in Pic(X).
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