We study the use of machine learning for finding numerical hermitian Yang-Mills connections on line bundles over Calabi-Yau manifolds. Defining an appropriate loss function and focusing examples elliptic curve, a K3 surface quintic threefold, we show that neural networks can be trained to give close approximation connections.

An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this paper, we give a necessary and sufficient condition for a non-trivial line bundle OX(D) on X with |D| 6= ∅ and D 2 ≥ L − 6 to be an ACM and initialized line bundle with ...

We develop an intersection theory for a singular hermitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure associated with the singular hermitian line bundle. Also for any pseudoeffective line bundle on a smooth projective variety, we prove the existence of a na...

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that any ample line bundle on a nonsingular toric waek Fano 3-fold is normally generated. As an application, we see that an ample line bundle on a Calabi-Yau hypersurface in a nonsingular toric Fano 4-fold is normally ge...