On the atomic decomposition of complete intersection in flag varieties
Abstract
We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let X be a Fano fourfold with h3,1(X)=1 arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and X is Hodge general, then X is irrational; (b) if every eigenvalue has Jordan defect at most one and X is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in LG(3,6); the very general smooth quadric section in G2/P2; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.
Related
- Good stratifications of non-degenerate locally tame complete intersection varieties
- Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties
- The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad
- Free-Monodromic Mixed Tilting Sheaves on Flag Varieties
- Independence of the Grothendieck classes of twisted symplectic flag varieties
BibTeX
@misc{alexeev2026atomic,
title = {{On the atomic decomposition of complete intersection in flag varieties}},
author = {Boris Alexeev and Leonardo F. Cavenaghi and Giovane Galindo and Bogdan Georgiev and Ludmil Katzarkov and Pedro Antonio Muniz Martins},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21914},
eprint = {2609.21914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21914},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the atomic decomposition of complete intersection in flag varieties
AU - Boris Alexeev
AU - Leonardo F. Cavenaghi
AU - Giovane Galindo
AU - Bogdan Georgiev
AU - Ludmil Katzarkov
AU - Pedro Antonio Muniz Martins
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.21914
UR - https://arxiv.org/abs/2609.21914
AB - We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let X be a Fano fourfold with h^3,1(X)=1 arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and X is Hodge general, then X is irrational; (b) if every eigenvalue has Jordan defect at most one and X is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in LG(3,6); the very general smooth quadric section in G_2/P_2; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.
ER -