Limits of Calabi–Yau metrics when the Kähler class degenerates
Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties.
Mirror symmetry and Arnold's duality.
Mirror symmetry and toric geometry.
Mirror symmetry in dimension 3
Modularity of a nonrigid Calabi-Yau manifold with bad reduction at 13
We identify the weight four newform of a modular Calabi-Yau manifold studied by Hulek and Verrill. The main obstacle is that this Calabi-Yau manifold is not rigid and has bad reduction at prime 13. Replacing the original fiber product of elliptic fibrations with a fiberwise Kummer construction we reduce the problem to studying the modularity of a rigid Calabi-Yau manifold with good reduction at primes p ≥ 5.
New examples of modular rigid Calabi-Yau threefolds.
The aim of this article is to present five new examples of modular rigid Calabi-Yau threefolds by giving explicit correspondences to newforms of weight 4 and levels 10, 17, 21 and 73.
Noncommutative del Pezzo surfaces and Calabi-Yau algebras
The hypersurface in with an isolated quasi-homogeneous elliptic singularity of type , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra to a noncommutative algebra with generators and the following 3 relations labelled...
On a certain generalization of spherical twists
This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of -curves on threefolds, or deforming -objects introduced by D.Huybrechts and R.Thomas.
On certain rigid fibered Calabi-Yau threefolds.
On equivalences of derived and singular categories
Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → , g:Y → . Assuming that there exists a complex of sheaves on X × Y which induces an equivalence of D b(X) and D b(Y), we show that there is also an equivalence of the singular derived categories of the fibers f −1(0) and g −1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective...
On the curvature of moduli space of special Lagrangian submanifolds
In this paper we study the curvature tensor of the Riemannian metric defined in a natural way on the moduli space of compact special Lagrangian submanifolds of a Calabi-Yau manifold. We state some curvature properties and we prove that the Ricci curvature is non negative under an assumption on the determinant of .
On the height of Calabi-Yau varieties in positive characteristic.
Periodic integrals and tautological systems
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.
Some geometry and combinatorics for the -invariant of ternary cubics.
String theory and duality.
Sur l'application d'Abel-Jacobi des variétés de Calabi-Yau de dimension trois
The mathematics of fivebranes.
The modularity of the Barth-Nieto quintic and its relatives.
The moduli space of special lagrangian submanifolds