Moduli Spaces, Virtual Invariants and Shifted

Moduli Spaces, Virtual Invariants and Shifted Symplectic Structures by Young-Hoon Kiem

• Moduli Spaces, Virtual Invariants and Shifted Symplectic Structures
• Young-Hoon Kiem
• Page: 254
• Format: pdf, ePub, mobi, fb2
• ISBN: 9789819782482
• Publisher: Springer Nature Singapore

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Enumerative geometry is a core area of algebraic geometry that dates back to Apollonius in the second century BCE. It asks for the number of geometric figures with desired properties and has many applications from classical geometry to modern physics. Typically, an enumerative geometry problem is solved by first constructing the space of all geometric figures of fixed type, called the moduli space, and then finding the subspace of objects satisfying the desired properties. Unfortunately, many moduli spaces from nature are highly singular, and an intersection theory is difficult to make sense of. However, they come with deeper structures, such as perfect obstruction theories, which enable us to define nice subsets, called virtual fundamental classes. Now, enumerative numbers, called virtual invariants, are defined as integrals against the virtual fundamental classes. Derived algebraic geometry is a relatively new area of algebraic geometry that is a natural generalization of Serre’s intersection theory in the 1950s and Grothendieck’s scheme theory in the 1960s. Many moduli spaces in enumerative geometry admit natural derived structures as well as shifted symplectic structures. The book covers foundations on derived algebraic and symplectic geometry. Then, it covers foundations on virtual fundamental classes and moduli spaces from a classical algebraic geometry point of view. Finally, it fuses derived algebraic geometry with enumerative geometry and covers the cutting-edge research topics about Donaldson–Thomas invariants in dimensions three and four.

Department Colloquia | Department of Mathematics
Many moduli problems of interest, such as moduli spaces of local systems, come equipped with a natural symplectic structure. shifted symplectic structures .
[PDF] Stability structures, motivic Donaldson-Thomas invariants and .
We can associate with X several moduli spaces which have the virtual dimension zero: a) moduli of holomorphic curves in X with fixed genus and .
Recent Progress on the Donaldson-Thomas Theory: Wall-Crossing .
These moduli spaces admit shifted symplectic structures and the associated d-critical structures . invariants and related topics, with a .
AMS :: Journal of the American Mathematical Society
A Darboux theorem for derived schemes with shifted symplectic structure. HTML articles powered by AMS MathViewer. by Christopher Brav, Vittoria Bussi and .
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[4] Borisov, Dennis; Joyce, Dominic Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. Topol., Volume 21 (2017) no. 6 .
Symplectic Field Theorist | because math is hard, so we need .
structure theorems about the moduli space of J-holomorphic curves. In fact . {mathcal J} is a space of smooth tame/compatible almost complex structures on some .

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