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Introduction to non-Abelian Patchworking

Turgay Akyar, Mikhail Shkolnikov

Abstract

The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian Tropical Geometry makes sense for an arbitrary reductive complex group, the state of the art is that of full understanding of tropicalizations of surfaces within three dimensional groups $PGL_2(\mathbb{C})$ and $SL_2(\mathbb{C}),$ which are closely related via the two-fold covering. We stress our point, that this is an announcement of a framework, taking care of explaining explicitly the input, which is more geometric and less combinatorial than in the original Viro's method, to construct possible types of real algebraic surfaces in the real projective 3-space, and verify that it reproduces all the existing isotopy types of surfaces up to degree three. We obtain two general theorems concerning the topology of primitive PGL2 surfaces, observing in particular that they may have different Euler charteristic for a fixed degree greater than one, not necessarily equal to the signature of the corresponding complex surface, which would be the case for primitive combinatorial patchworking due to a result of Itenberg.

Introduction to non-Abelian Patchworking

Abstract

The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian Tropical Geometry makes sense for an arbitrary reductive complex group, the state of the art is that of full understanding of tropicalizations of surfaces within three dimensional groups and which are closely related via the two-fold covering. We stress our point, that this is an announcement of a framework, taking care of explaining explicitly the input, which is more geometric and less combinatorial than in the original Viro's method, to construct possible types of real algebraic surfaces in the real projective 3-space, and verify that it reproduces all the existing isotopy types of surfaces up to degree three. We obtain two general theorems concerning the topology of primitive PGL2 surfaces, observing in particular that they may have different Euler charteristic for a fixed degree greater than one, not necessarily equal to the signature of the corresponding complex surface, which would be the case for primitive combinatorial patchworking due to a result of Itenberg.
Paper Structure (10 sections, 2 theorems, 25 equations, 2 figures)

This paper contains 10 sections, 2 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $PSL_2(\mathbb{C})$-tropical surfaces
  3. Primitive surface diagrams of even degree
  4. Primitive surface diagrams of odd degree
  5. Real Structures on $PGL_2(\mathbb{C})$
  6. Real structure $I_{T^2}$
  7. Real structure $I_{\emptyset}$
  8. Real structure $I_{S^2}$
  9. Topology of real $PGL_2(\mathbb{C})$-phase diagrams
  10. Discussion

Key Result

Theorem 1

Let $X\subset\mathbb{C}{\mathbb{P}}^3$ be a complex smooth surface of degree $d$ and $D$ be the $PGL_2(\mathbb{C})$-phase diagram of degree $d$, obtained with the initial data described above. If $D$ is ${I_{T^2}}$-real and $d=2k+1,$ then the Euler characteristic is bound to In the even case and $D$ being real ${I_{T^2}}$-real, where $d=2k$, Moreover, for $I_\emptyset$-real $D$ Lastly, for $I_{S

Figures (2)

  • Figure 1: A schematic representation of the real locus of $PGL_2(\mathbb{C})$ with respect to $I_{S^2}.$
  • Figure 2: This diagram represents $D^{I_{T^2}}$ for degree 3 in a way that both ends are shown as points rather than a point and an $S^1$. In this particular example, we have a $(1,1)$ curve (red) and a (3,3) curve (blue) at the critical level, intersecting at four points, together with a choice of signs on each connected component of the complement of the union of the curves.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3