Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.
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 WangCenter for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 106
This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base.
In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models.
In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.
Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Hui-Wen LinDepartment of Mathematics, National Taiwan University, Taipei 106Chin-Lung WangDepartment of Mathematics, National Taiwan University, Taipei 106
For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space.
For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.
In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.
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 completed F-crystals on the absolute prismatic site of a smooth p-adic formal scheme. We define a functor from the category of completed prismatic F-crystals to that of crystalline étale Zp-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.
The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold.
In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality.
Y. IwaoDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Hui-Wen LinDepartment of Mathematics, National Taiwan University, Taipei 106C.-L WangDepartment of Mathematics, National Taiwan University, Taipei 106
We show that the generating functions of Gromov--Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [LLW].
The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications are also indicated.
The main goal of this paper is to prove the following two conjectures for genus up to two:
1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy.
2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds.
The main technique used in the proof is the invariance of tautological equations under loop group action.
In this paper, we extend our previous work to construct (0, 2) Toda-like
mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2)
mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with
deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the
corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
Taiwang DengYau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, ChinaBin XuYau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, China
In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
We study the non-klt locus of singularities of pairs. We show that
given a pair (X, B) and a projective morphism X → Z with connected fibres such
that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected
components near each fibre of X → Z. This was conjectured by Hacon and Han.
In a different direction we answer a question of Mark Gross on connectedness
of the non-klt loci of certain pairs. This is motivated by constructions in Mirror
We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.
Nathan PriddisInstitut für Algebraische Geometrie, Universität Hannover, D-30060 HannoverYuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, UT 84112Mark ShoemakerDepartment of Mathematics, Colorado State University Fort Collins, CO 80523
We establish a new relationship (the MLK correspondence) between twisted FJRW theory and local Gromov-Witten theory in all genera. As a consequence, we show that the Landau-Ginzburg/Calabi-Yau correspondence is implied by the crepant transformation conjecture for Fermat type in genus zero. We use this to then prove the Landau-Ginzburg/Calabi-Yau correspondence for Fermat type, generalizing the results of A. Chiodo and Y. Ruan.