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- R P Thomas
- 2000

We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain… (More)

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope µ for a projective manifold and for each of its subschemes, and show that if X is cscK then µ(Z) ≤ µ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes… (More)

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties (X, L); in particular for K-and Chow stability. For each type of stability this leads to a concept of slope µ for varieties and their subschemes; if (X, L) is semistable then µ(Z) ≤ µ(X) for all Z ⊂ X. We give examples such as… (More)

- Dominic Joyce, Robert Bryant, +5 authors Karen Uhlenbeck
- 2001

1 Introduction Calabi–Yau m-folds are compact Ricci-flat Kähler manifolds (M, J, g) of complex dimension m, with trivial canonical bundle K M. Taken together, the complex structure J, Kähler metric g, and a holomorphic section Ω of K M make up a rich, fairly rigid geometrical structure with very interesting properties — for instance, Calabi–Yau m-folds… (More)

- R P Thomas, S.-T Yau
- 2001

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on that of [Th]. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians,… (More)

- R. P. THOMAS, Daniel Huybrechts, Julien Keller, Dmitri Panov
- 2008

- R. P. Thomas
- 2002

We find stability conditions ([D2], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the… (More)

- R P Thomas
- 2001

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian sub-manifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians, and can be proved for the simple case… (More)

- R P Thomas
- 2000

After outlining the conjectural relationship between the conjec-tural mirror symmetry programmes of Kontsevich and Strominger-Yau-Zaslow, I will describe some natural consequences of this which are proved from scratch in joint work with Mikhail Khovanov and Paul Seidel. Namely, actions of braid groups are found on derived categories of coherent sheaves,… (More)

- R. P. Thomas
- 2005

The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.