Abstract
ABSTRACTWe collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.
Highlights
In this paper we take the first step towards implementing a program, laid out in [11], to find and classify four-dimensional Fano manifolds using mirror symmetry
Four-dimensional Fano manifolds with index r > 1 have been classified
A precise definition can be found in [12, §B], but roughly speaking cd is the ‘virtual number’ of degree-d rational curves C in X that pass through a given point and satisfy certain constraints on their complex structure. (The degree of a curve C here is the quantity −KX, C .) The quantum period is discussed in detail in [11, 12]; one property that will be important in what follows is that the regularized quantum period
Summary
In this paper we take the first step towards implementing a program, laid out in [11], to find and classify four-dimensional Fano manifolds using mirror symmetry. §6.2.4, where new tools for computing Gromov–Witten invariants (twisted I-functions for toric complete intersections [13] and an improved Quantum Lefschetz theorem [10]) make a big practical difference to the computation of quantum periods. This should be contrasted with [12, §19], where the new techniques were not available. This suggests in particular that, for each four-dimensional Fano manifold X with Fano index r > 1, the regularized quantum differential equation of X is either extremal or of low ramification. This will allow the reader to verify the calculations presented here, or to perform similar computations
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