arXiv subjects

Paul Seidel

Publications and source records attributed to Paul Seidel.

60 records · Page 4Linked to original sources

arXiv2000

Braid group actions on derived categories of coherent sheaves

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

math.AG↗

arXiv2000

Lagrangian two-spheres can be symplectically knotted

This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian submanifolds. The examples are constructed using a special class of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer homology. Revised version: one footnote removed, one reference added

math.DG↗

arXiv2000

Graded Lagrangian submanifolds

In the usual setup, the grading on Floer homology is relative: it is unique only up to adding a constant. "Graded Lagrangian submanifolds" are Lagrangian submanifolds with a bit of extra structure, which fixes the ambiguity in the grading. The idea is originally due to Kontsevich. This paper contains an exposition of the theory. Several applications are given, amongst them: (1) topological restrictions on Lagrangian submanifolds of projective space, (2) the existence of "symplectically knotted" Lagrangian spheres on a K3 surface, (3) a result about the symplectic monodromy of weighted homogeneous hypersurface singularities. Revised version: minor modifications, journal reference added.

math.SG↗

arXiv1998

Symplectic automorphisms of T^*S^2

Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. π_0(S) is generated by the class of the standard "generalized Dehn twist". As a consequence, we show that there are different connected components of S which lie in the same connected component of the corresponding group of diffeomorphisms.

math.DG↗

arXiv1998

On the group of symplectic automorphisms of $\C P^m \times \C P^n$

Let M be the product of \C P^m and \C P^n, with the standard integral symplectic form. We prove that the inclusion map from the group of symplectic automorphisms of M to its diffeomorphism group is not surjective on homotopy groups. More precisely, it is not surjective on π_j for all odd j \leq \max\{2m-1,2n-1\}. This is a weak higher-dimensional analogue of Gromov's results for \C P^1 \times \C P^1. The proof uses parametrized Gromov-Witten invariants in a new (?) way. We also give some information about the symplectic automorphism groups of M with differently weighted product symplectic structures.

math.DG↗

arXiv1997

$π_1$ of symplectic automorphism groups and invertibles in quantum homology rings

We define a homomorphism from (a certain extension of) the fundamental group of the Hamiltonian automorphism group of a symplectic manifold to the group of invertibles in its quantum cohomology ring. The manifold must satify a technical condition similar to weak monotonicity. This revised version has been entirely rewritten. It is more general and contains new examples and applications.

dg-ga↗