Bumsig Kim

Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians

Abstract In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians G ( k , n ) to some Gorenstein toric Fano varieties P ( k , n ) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections X ⊂ G ( k , n ) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.

Mirror symmetry and toric degenerations of partial flag manifolds

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n1, . . . , nl, n) to a certain Gorenstein toric Fano variety P (n1, . . . , nl, n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n1, . . . , nl, n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n1, . . . , nl, n). Mathematisches Institut, Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany, email address: batyrev@bastau.mathematik.uni-tuebingen.de Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA email address: ciocan@math.okstate.edu Department of Mathematics, University of California Davis, Davis, CA 95616, USA email address: bumsig@math.ucdavis.edu FB 17, Mathematik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany, email address: straten@mathematik.uni-mainz.de

Quantum hyperplane section theorem for homogeneous spaces

Author(s): Kim, Bumsig | Abstract: We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds. For the proof we followed the scheme of Giventals proof of a mirror theorem for toric complete intersections.

Moduli stacks of stable toric quasimaps

Abstract We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Giventals compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov–Witten theory is also included.

Wall-crossing in genus zero quasimap theory and mirror maps

For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing in this case significantly generalizes toric mirror symmetry (the toric cases correspond to abelian groups G). In particular, we give a geometric interpretation of the mirror map as a generating series of quasimap invariants. We prove our wall-crossing formulas for all targets W//G which admit a torus action with isolated fixed points, as well as for zero loci of sections of homogeneous vector bundles on such W//G.

Gromov-witten invariants for abelian and nonabelian quotients

Let X be a smooth projective variety over C with the (linearized) action of a complex reductive group G, and let T ⊂ G be a maximal torus. In this setting, there are two geometric invariant theory (GIT) quotients, X//T and X//G, with a rational map Φ : X//T − −> X//G between them. We will further assume that “stable = semistable” in the GIT and that all isotropy of stable points is trivial, so X//T and X//G are smooth projective varieties, and Φ is a G/T fibration.

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.

Higher genus quasimap wall-crossing for semipositive targets

In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yaus associated to some non-abelian quotients.

Mirror theorem for elliptic quasimap invariants

We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants for those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.

A compactification of the space of maps from curves

We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper Deligne-Mumford stack equipped with a natural virtual fundamental class.