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Real Einstein loci

Gabriella Clemente

TL;DR

The paper investigates how anti-holomorphic symmetries on a compact Kähler-Einstein manifold $X$ produce real Einstein submanifolds via fixed point sets. It develops a framework connecting anti-holomorphic maps, Kähler potentials, and real structures, leveraging KE uniqueness and first Chern class sign to deduce Einstein-ness and Lagrangian properties for fixed loci. The main results show that if $f$ is bi-anti-holomorphic and $X^f$ is $n$-dimensional, then $X^f$ is Einstein when $c_1(X)\le 0$, and, for $c_1(X)>0$ with $f$ involutive, a suitable $a\in Aut_0(X)$ yields $X^{a\circ f}$ Einstein; moreover, such fixed sets are totally real and, under isometries, Lagrangian. The work provides concrete constructions of Einstein and Lagrangian submanifolds, with explicit examples on projective spaces and toric varieties, and opens avenues for linking real and complex geometry in KE contexts.

Abstract

The aim of this article is to study the interplay between the complex, and underlying real geometries of a Kaehler manifold. We prove that certain anti-holomorphic automorphisms of a compact Kaehler-Einstein manifold give rise to real Einstein submanifolds.

Real Einstein loci

TL;DR

The paper investigates how anti-holomorphic symmetries on a compact Kähler-Einstein manifold produce real Einstein submanifolds via fixed point sets. It develops a framework connecting anti-holomorphic maps, Kähler potentials, and real structures, leveraging KE uniqueness and first Chern class sign to deduce Einstein-ness and Lagrangian properties for fixed loci. The main results show that if is bi-anti-holomorphic and is -dimensional, then is Einstein when , and, for with involutive, a suitable yields Einstein; moreover, such fixed sets are totally real and, under isometries, Lagrangian. The work provides concrete constructions of Einstein and Lagrangian submanifolds, with explicit examples on projective spaces and toric varieties, and opens avenues for linking real and complex geometry in KE contexts.

Abstract

The aim of this article is to study the interplay between the complex, and underlying real geometries of a Kaehler manifold. We prove that certain anti-holomorphic automorphisms of a compact Kaehler-Einstein manifold give rise to real Einstein submanifolds.

Paper Structure

This paper contains 6 sections, 21 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Anti-holomorphisms and Kähler metrics
  3. Background
  4. Properties of anti-holomorphic maps
  5. Examples of totally geodesic real parts
  6. Submanifolds of Kähler-Einstein manifolds

Key Result

Theorem 1

Let $n\geq 1,$ and $(X,g)$ be a compact Kähler-Einstein manifold of real dimension $2n.$ Let $f$ be a bi-anti-holomorphism of $X,$ and denote its set of fixed points by $X^f.$ If $X^f$ is an $n$-dimensional submanifold and $c_1(X) \leq 0,$ then $(X^f,g|_{X^f})$ is Einstein. If $c_1(X)>0$ and $f$ is

Theorems & Definitions (40)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Example 1
  • ...and 30 more