Related papers: A mirror theorem for non-split toric bundles
We give a historical presentation of the Grothendieck theorem on the splitting of vector bundles over the Riemann sphere, and explore some of its links with the Riemann-Hilbert-Birkhoff problems and the Birkhoff factorization theorem.
Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the…
We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds $X$ over Hirzebruch surfaces. To determine the representations of these matter multiplets…
We explore the cohomological structure for the (possibly singular) moduli of $\mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the…
We propose a conjectural determination of the Gromov-Witten theory of a root stack along a smooth divisor. We verify our conjecture under an additional assumption.
This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or…
We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete…
For the associative algebra $A(\mathfrak g)$ of an infinite-dimensional Lie algebra $\mathfrak g$, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic…
For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class…
We study the bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors.…
We derive an integral expression for the leading-order type I-I-I three-point functions in the $\mathfrak{su}(2) $-sector of $\mathcal{N}=4$ super Yang-Mills theory, for which no determinant formula is known. To this end, we first map the…
A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of…
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations.…
Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial…
In this work, we study topological properties of surface bundles, with an emphasis on surface bundles with a spin structure. We develop a criterion to decide whether a given manifold bundle has a spin structure and specialize it to surface…
A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of…
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the…
We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants…
From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit computation of the Euler data $Q=\{Q_0, Q_1, ... \}$ for an equivariant concavex bundle ${\cal E}$ over a toric manifold, there are two places the structure of the…
We define Gromov-Witten classes and invariants of smooth projective schemes of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth projective scheme over…