The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.
We study the quantum sheaf cohomology of flag manifolds with deformations of the
tangent bundle and use the ring structure to derive how the deformation transforms under
the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted twodimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the
notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have
been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs
describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations
for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated
with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2)
We introduce the notion of pseudo-N´eron model and give new examples of varieties admitting pseudo-N´eron models other than Abelian varieties. As an application of pseudo-N´eron models, given a scheme admitting a finite morphism to an Abelian scheme over a positive-dimensional base, we prove that for a very general genus-0, degree-d curve in the base with d sufficiently large, every section of the scheme over the curve is contained in a unique section over the entire base.
Karl ChristBen-Gurion University of the Negev and Leibniz University HannoverXiang HeEinstein Institute of Mathematics and Yau Mathematical Sciences CenterIlya TyomkinBen-Gurion University of the Negev
Journal für die reine und angewandte Mathematik, 2022.4
Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent
progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.
Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Hui-Wen LinTaida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei 106Chin-Lung WangTaida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei 106
For projective conifold transitions between Calabi-Yau threefolds X and Y, with X close to Y in the moduli, we show that the combined information provided by the A model (Gromov--Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corresponding combined information on Y. Similar result holds in the reverse direction when linked with the exceptional curves.
A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is proved. A new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pull-backs of line bundles from the target via different markings is used for the reconstruction result. Examples of calculations in quantum cohomology and quantum K-theory are given.