Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements. The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements . The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. Acknowledgements This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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A Mirror Theorem for Non-split Toric Bundles: Toric Bundles

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A Mirror Theorem for Non-split Toric Bundles: Abstract and Intro

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A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

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A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for a Product of Projectives Bundle

A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

A Mirror Theorem for Non-split Toric Bundles: Toric Bundles

A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for Toric Bundles

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for a Product of Projectives Bundle

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