Machine learning line bundle cohomology. (English) Zbl 1535.32024
Summary: We investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully connected networks with logistic sigmoids is reviewed and its main features and shortcomings are discussed. It has been observed recently that line bundle cohomology can be described by dividing the Picard lattice into certain regions in each of which the cohomology dimension is described by a polynomial formula. Based on this structure, we set up a network capable of identifying the regions and their associated polynomials, thereby effectively generating a conjecture for the correct cohomology formula. For complex surfaces, we also set up a network which learns certain rigid divisors which appear in a recently discovered master formula for cohomology dimensions.
© 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
MSC:
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
| 32J17 | Compact complex \(3\)-folds |
| 14J30 | \(3\)-folds |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 68T07 | Artificial neural networks and deep learning |
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