Journal article
A classical model for derived critical loci
- Abstract:
- 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-fold over $\mathbb K$, are also $-1$-shifted symplectic derived schemes. Thus, their theory may have applications in algebraic symplectic geometry, and in Donaldson-Thomas theory of Calabi-Yau 3-folds. This paper defines and studies a new class of spaces we call "algebraic d-critical loci", which should be regarded as classical truncations of $-1$-shifted symplectic derived schemes. They are simpler than their derived analogues. We also give a complex analytic version of the theory, "complex analytic d-critical loci", and an extension to Artin stacks, "d-critical stacks". In the sequels arXiv:1305.6302, arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090 we will define truncation functors from $-1$-shifted symplectic derived schemes or stacks to algebraic d-critical loci or d-critical stacks, and we will apply d-critical loci to motivic and categorified Donaldson-Thomas theory, and to intersections of (derived) complex Lagrangians in complex symplectic manifolds. We will show that the important structures one wants to associate to a derived critical locus -- virtual cycles, perverse sheaves and mixed Hodge modules of vanishing cycles, and motivic Milnor fibres -- can be defined for oriented d-critical loci and oriented d-critical stacks.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
Actions
Authors
- Publisher:
- International Press
- Journal:
- Journal of Differential Geometry More from this journal
- Volume:
- 101
- Issue:
- 2
- Pages:
- 289-367
- Publication date:
- 2015-09-16
- Acceptance date:
- 2014-06-24
- EISSN:
-
1945-743X
- ISSN:
-
0022-040X
- Language:
-
English
- Keywords:
- Pubs id:
-
pubs:396010
- UUID:
-
uuid:267b8993-772c-46b0-87de-81cc8a736cd2
- Local pid:
-
pubs:396010
- Source identifiers:
-
396010
- Deposit date:
-
2015-05-04
- ARK identifier:
Terms of use
- Copyright holder:
- Dominic Joyce
- Copyright date:
- 2015
- Notes:
- This article has been accepted for publication in Journal of Differential Geometry, published by International Press.
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