The wheel classes in the locally

A tropical computation of refined

Graded character sheaves, HOMFLY-PT homology,

We say that a tropical subvariety V R n is B-realizable if it can be lifted to an analytic subset of ./ n .When V is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift L V .C / n .We prove that whenever L V has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with L V is nonvanishing.Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever L V is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety V is B-realizable.As an application, we show that the Lagrangian lift of a genus-0 tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-0 tropical curves are B-realizable.We also prove that tropical curves inside tropical abelian surfaces are B-realizable.

Mutations and faces of the Thurston

We calculate the singular instanton homology with local coefficients for the simplest n-strand braids in S 1 S 2 for all odd n, describing these homology groups and their module structures in terms of the coordinate rings of explicit algebraic curves.The calculation is expected to be equivalent to computing the quantum cohomology ring of a certain Fano variety, namely a moduli space of stable parabolic bundles on a sphere with n marked points.

This is essentially the only way in which a one-edge extension can be "reducible".We record a specific consequence of this in the following lemma.Lemma 3.4 Suppose that F.H 1 ;

Virtual classes via

Let C be a curve of genus g.A fundamental problem in the theory of algebraic curves is to understand maps C ! P r of specified degree d .When C is general, the moduli space of such maps is well understood by the main theorems of Brill-Noether theory.Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality.Here we complete such a picture, by proving analogs of all of the main theorems of Brill-Noether theory in this setting.As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on P 1 .14H51 1. Introduction 193 2. Our degeneration 200 3. Limits of line bundles 200 4. Classification of E e-positive limit line bundles 205 5. Combinatorics 210 6. Set-theoretic regeneration 219 7. Reducedness and Cohen-Macaulayness 233 8. Connectedness 239 9. Normality and irreducibility 242 10.Monodromy 245