A classical model for derived critical loci

@article{Joyce2013ACM,
  title={A classical model for derived critical loci},
  author={Dominic Joyce},
  journal={arXiv: Algebraic Geometry},
  year={2013}
}
  • D. Joyce
  • Published 16 April 2013
  • Mathematics
  • arXiv: Algebraic Geometry
Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call "$-1$-shifted symplectic derived schemes". They show that intersections of algebraic Lagrangians in a smooth symplectic $\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3… 

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References

SHOWING 1-10 OF 38 REFERENCES
A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications
This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi,
Symmetries and stabilization for sheaves of vanishing cycles
Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of
Symmetries and stabilization for sheaves of vanishing cycles
Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of
A theory of generalized Donaldson–Thomas invariants
This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern
The intrinsic normal cone
Abstract.Let $X$ be an algebraic stack in the sense of Deligne-Mumford. We construct a purely $0$-dimensional algebraic stack over $X$ (in the sense of Artin), the intrinsic normal cone ${\frak
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 a
Shifted symplectic structures
This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic
Quantization and Derived Moduli Spaces I : Shifted Symplectic Structures
This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-symplectic structures, a generalization of
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category
From Hag To Dag: Derived Moduli Stacks
These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation
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