Homological Algebra of Mirror Symmetry

Abstract

Abstract. One important problem arising in algebraic geometry is the computation of effective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundle L on a projective manifold X, can one compute explicitly an integer m 0 such that mL is very ample for m m 0 ? It turns out that the answer is much easier to obtain in the case of adjoint line bundles 2(K X + mL), for which universal values of m 0 exist. We indicate here how such bounds can be derived by a combination of powerful analytic methods: theory of positive currents and plurisubharmonic functions (Lelong), L 2 estimates for ∂ (Andreotti-Vesentini, Hörmander, Bombieri, Skoda), Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities.

D-brane monodromies from a matrix-factorization perspective

“…Each cohomology element in Ext 1 (X[n], P ), or equivalently each cohomology element in Ext p (X, P ), gives rise to a tachyonic open-string state along the path around the conifold monodromy and triggers a condensation. The presented heuristic arguments motivate the formula for the conifold monodromy as proposed by Kontsevich [45,46] …”

On the boundary coupling of topological Landau-Ginzburg models

“…Such objects are generalizations of the more familiar superconnections of type (0, 1). Graded rather than standard superconnections appear for the sigma model because topological D-branes are Z-graded in that context [7,34,12,15,20]. As explained in [2,5], turning on the Landau-Ginzburg potential will generally break the integral grading to a Z 2 group.…”

“…This is potentially important since it seems to allow for a simple realization of the framework of [7,8,9,10,11,12,13,14,15,16,17] in such theories.…”

Anomalies and graded coisotropic branes

“…On physical grounds, mirror symmetry exchanges the A-model on X with the B-model on its mirrorX, and therefore must exchange the sets of A-branes and Bbranes. One promising proposal to understand this mirror phenomenon in mathematical terms is the Homological Mirror Symmetry (HMS) conjecture [7], which interprets mirror symmetry as the equivalence of two triangulated categories: the bounded derived category of coherent sheaves D b (X) on the one hand, and the derived Fukaya category DF (X) on the other hand. It was later argued by Douglas [2] (see also [1]) that the derived category D b (X) corresponds to the category of topological B-branes.…”