Saturday, 27 November 2010

vector bundles - What are some open problems in algebraic geometry?

There's also Fujita's conjecture.



Conjecture: Suppose X is a smooth projective dimensional complex algebraic variety with ample divisor A. Then



  1. H0(X,mathcalOX(KX+mA)) is generated by global section when m>dimX.

  2. KX+mA is very ample for m>dimX+1

It's also often stated in the complex analytic world.



Also there are many refinements (and generalizations) of this conjecture. For example, the assumption that X is smooth is probably more than you need (something close to rational singularities should be ok). It also might even be true in characteristic p>0.



It's known in relatively low dimensions (up to 5 in case 1. I think?)

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