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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, togetherExpand
Curve counting via stable pairs in the derived category
For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on theExpand
A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
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 aExpand
Hodge theory and derived categories of cubic fourfolds
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics withExpand
An obstruction to the existence of constant scalar curvature K
We prove that polarised manifolds that admit a constant scalar curvature K\"ahler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $\mu$ for a projectiveExpand
Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes
We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstructionExpand
A study of the Hilbert-Mumford criterion for the stability of projective varieties
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 ofExpand
Curves on K3 surfaces and modular forms
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces toExpand
Stable pairs and BPS invariants
We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend'sExpand
Notes on GIT and symplectic reduction for bundles and varieties
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties,Expand