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The weighted projective superspace with weights $+1, -1$ and an analog of the Fubini--Study form

Ekaterina Shemyakova, Theodore Voronov

Abstract

As a by-product of our work on super Plücker embedding, we came to the notion of a weighted projective superspace $P_{+1,-1}(V\oplus W)$ with weights $+1,-1$. The construction is not in itself super and makes sense in ordinary (purely even) framework. Unlike the familiar weighted projective spaces with positive weights, the (super)space $P_{+1,-1}(V\oplus W)$ is a smooth (super)manifold. We describe its structure and show that it possesses an analog of the Fubini--Study form.

The weighted projective superspace with weights $+1, -1$ and an analog of the Fubini--Study form

Abstract

As a by-product of our work on super Plücker embedding, we came to the notion of a weighted projective superspace with weights . The construction is not in itself super and makes sense in ordinary (purely even) framework. Unlike the familiar weighted projective spaces with positive weights, the (super)space is a smooth (super)manifold. We describe its structure and show that it possesses an analog of the Fubini--Study form.
Paper Structure (3 sections, 5 theorems, 14 equations)

This paper contains 3 sections, 5 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Definition and structure
  3. Analog of Fubini--Study form

Key Result

Theorem 1

For arbitrary $V$ and $W$, the space $P_{+1,-1}(V\oplus W)$ has natural fiber bundle structures over $P(V)$ and $P(W)$ : \begin{tikzcd} & P_{+1,-1}(V\oplus W) \arrow[dl, "\pi_V" above left] \arrow[dr,"\pi_W"] \\ P(V) && P(W)\,. \end{tikzcd}The bundle $\pi_V\colon P_{+1,-1}(V\oplus W)\to P(V)$ is

Theorems & Definitions (11)

  • Definition 1
  • Example 1
  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2: symplectic form on $P_{1,-1}(V\oplus W$
  • ...and 1 more