Bettadapura, Kowshik
Description
The interplay between geometry and supergeometry, from an
algebraic point of view, sets the theme guiding the
considerations in this thesis. In the smooth setting there is a
sense in which these geometries can be identified and, in the
complex (i.e., holomorphic) setting, such an identification no
longer holds. As such, for at least this reason, complex
algebraic supergeometry can find interest in its own right. It is
the subject of this thesis and we study it...[Show more] here under two broad
headings: obstruction theory and deformation theory.
Under the umbrella of obstruction theory, we focus largely on
foundational aspects of supermanifolds and their description by
means of supersymmetric thickenings. We start from the general
principle that: any supermanifold will define a supersymmetric
thickening but not necessarily conversely. One of the key
objectives in this part of the thesis is in precisely formulating
and proving this principle by elementary methods. We complement
the proof given with examples of obstructed thickenings on the
complex projective plane. To illustrate obstruction theory more
generally for complex supermanifolds, we include and comment on a
collection of examples from the literature, in addition to
providing some new examples. Moreover, we will also consider the
splitting problem for complex supermanifolds. Upon obtaining a
characterisation of the obstruction classes to splitting via the
grading vector field, we present a new proof of the Koszul
splitting theorem for supermanifolds.
Regarding deformation theory, we concern ourselves with the
construction of (odd) infinitesimal deformations of
superconformal structures. These are structures on supermanifolds
and, in the one dimensional case, arise under the guise of super
Riemann surfaces. Explicit and elementary constructions of (odd)
deformations are given for N = 1 and N = 2 super Riemann
surfaces. One of the key objectives in this part of the thesis is
on establishing precise relations between (1) the deformation
theory of these super Riemann surfaces and (2) their obstruction
theory as supermanifolds.
We conclude this thesis with a brief sketch on the state of
supermoduli spaces, in both the N = 1 and N = 2 setting, as it
presently stands in the literature. These discussions lead
naturally toward directions for future research and paint a
grander scheme in which this thesis sits.
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