Related papers: Taming symplectic forms and the Calabi-Yau equatio…
In the quests to uncovering the fundamental structures that underlie some of the asymptotic Swampland conjectures we initiate the general study of asymptotic period vectors of Calabi- Yau manifolds. Our strategy is to exploit the…
We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition…
We obtain the vacuum solutions for M-theory compactified on eight-manifolds with non-vanishing four-form flux by analyzing the scalar potential appearing in the three-dimensional theory. Many of these vacua are not supersymmetric and yet…
We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen's Weyl tensor…
We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the…
A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view…
We develop formalism for computing the kinetic terms of 4d fields in string compactifications, particularly with warping. With the help of the Hamiltonian approach, we identify a gauge dependent inner product on the compactification…
M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone.…
We apply results on inducing stability conditions to local Calabi-Yau threefolds and obtain applications to Donaldson-Thomas (DT) theory. A basic example is the total space of the canonical bundle of $Z=\mathbb{P}^1\times \mathbb{P}^1$. We…
Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle,…
We use Joyce's theory of motivic Hall algebras to prove that reduced Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds coincide with stable pair invariants, and that the generating functions for these invariants are…
We show that "non-polynomial" deformations of semiample (minimal) nondegenerate Calabi-Yau hypersurfaces in complete simplicial toric varieties can be realized as quasismooth complete intersections in higher dimensional simplicial toric…
We consider a conjecture of Kontsevich and Soibelman which is regarded as a foundation of their theory of motivic Donaldson-Thomas invariants for non-commutative 3d Calabi-Yau varieties. We will show that, in some certain cases, the answer…
For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic…
In this paper we study a special class of Calabi-Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of `superpotential descriptions' in motivic…
We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized K\"ahler structures. By considering the commutator $Q$ of the two associated almost complex structures…
This note is a survey of the enumerative geometry of rational curves on Calabi-Yau threefolds, based on a talk given by the author at the May 1991 Workshop on Mirror Symmetry at MSRI. An earlier version appeared in "Essays on Mirror…
In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local…
This thesis is concerned with a realisation of topological theories in terms of statistical models known as Calabi-Yau crystals. The thesis starts with an introduction and review of topological field and string theories. Subsequently…
We study the relation between Donaldson-Thomas theory of Calabi-Yau threefolds and a six-dimensional topological Yang-Mills theory. Our main example is the topological U(N) gauge theory on flat space in its Coulomb branch. To evaluate its…