2 deleted 18 characters in body; added 1 characters in body; deleted 4 characters in body

Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the set of connected components of the fixed point locus $X^{\mathbb{C}^*}.$ Rougly speaking, $X^+(N)$ is related to the positive part of the weight decomposition of $T_p X$, for $p\in N$ (see Theorem 4.1 in [2] for a more precise definition).

Define the following relation on the set of connected components of $X^{\mathbb{C}^*}$:

Relation $\prec$ 1: $N_1 \prec N_2 \Leftrightarrow \mbox{ the boundary of } X^+(N_1) \mbox{ intersects } X^+(N_2).$

Question 1: Is this relation antisymmetric?

I know that the answer to this question is affermative in the following case I shall explain in the following.

First, I need to recall some Morse theory. Let $M$ be a compact Riemannian Manifold and $f\colon M\rightarrow \mathbb{R}$ a smooth function.

Definition. ([1]) Let $N \subset M$ be a connected submanifold of $M.$ It shall be called a non-degenerate critical manifold for $f$ if and only if

1. the differential $df$ of $f$ is identically zero along $N$,
2. the Hessian $H_N f$ of $f$ is non-degenerate on the normal bundle $\nu(N)$ of $N.$

Assume that the critical set of $f$ is a union of non-degenerate critical manifolds. In this case, we shall call $f$ Morse function. To any non-degenerate critical manifold $N$ one can associate the so-called stable manifold $M^+(N)$, that is, the set of points of $M$ belonging on trajectories of the gradient flow of $f$ converging to $N.$ One has the so-called Morse decomposition $$M=\bigcup_N M^+(N).$$

In the paper [1], Atiyah and Bott define a relation on the set of critical manifolds in the following way:

Relation $2:$N_1< N_2 \Leftrightarrow \mbox{ the boundary of } M^+(N_1 ) \mbox{ intersects } M^+(N_2 ).$Moreover, they state that$N_1 < N_2 \Rightarrow f(N_1 ) < f(N_2 )$, hence$ Relation 2 is antisymmetric.

Let $X$ be a compact Kahler manifold endowed with an action of a compact holomorphic one-dimensional torus $T.$ One can define a Morse function $f$ associated to the torus action.

Facts (Chapter 3 in [3]): The non-degenerate critical manifolds coincide with the connected components of $X^{T}.$ Moreover, the stable manifolds of the Morse decomposition associated to $f$ coincide with the stable manifolds of the Bialynicki-Birula decomposition associated to the $T$-action.

Thus, in this case the relation $\prec$ Relation 1 coincides with the relation $Relation 2, hence Relation 1 is antisymmetric. Question 2: Do we really need Morse theory to prove the antisymmetric property of$\prec$?Relation 1? References: [1] Atiyah, Bott. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615. [2] Bialynicki-Birula. Some theorems on actions of algebraic groups. The Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480-497. [3] Nakajima. Lectures on Hilbert schemes of points on surfaces, American Math. Society, Providence RI, 1999. 1 # Relation on the set of connected components of the$\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition Let$X$be a smooth variety with an action of$\mathbb{C}^*.$One has the so-called Bialynicki-Birula decomposition of$X$given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where$N$varies in the set of connected components of the fixed point locus$X^{\mathbb{C}^*}.$Rougly speaking,$X^+(N)$is related to the positive part of the weight decomposition of$T_p X$, for$p\in N$(see Theorem 4.1 in [2] for a more precise definition). Define the following relation on the set of connected components of$X^{\mathbb{C}^*}$: Relation$\prec$:$N_1 \prec N_2 \Leftrightarrow \mbox{ the boundary of } X^+(N_1) \mbox{ intersects } X^+(N_2).$Question 1: Is this relation antisymmetric? I know that the answer to this question is affermative in the following case I shall explain in the following. First, I need to recall some Morse theory. Let$M$be a compact Riemannian Manifold and$f\colon M\rightarrow \mathbb{R}$a smooth function. Definition. ([1]) Let$N \subset M$be a connected submanifold of$M.$It shall be called a non-degenerate critical manifold for$f$if and only if 1. the differential$df$of$f$is identically zero along$N$, 2. the Hessian$H_N f$of$f$is non-degenerate on the normal bundle$\nu(N)$of$N.$Assume that the critical set of$f$is a union of non-degenerate critical manifolds. In this case, we shall call$f$Morse function. To any non-degenerate critical manifold$N$one can associate the so-called stable manifold$M^+(N)$, that is, the set of points of$M$belonging on trajectories of the gradient flow of$f$converging to$N.$One has the so-called Morse decomposition $$M=\bigcup_N M^+(N).$$ In the paper [1], Atiyah and Bott define a relation on the set of critical manifolds in the following way: Relation$: $N_1< N_2 \Leftrightarrow \mbox{ the boundary of } M^+(N_1 ) \mbox{ intersects } M^+(N_2 ).$

Moreover, they state that $N_1 < N_2 \Rightarrow f(N_1 ) < f(N_2 )$, hence $Let$X$be a compact Kahler manifold endowed with an action of a compact holomorphic one-dimensional torus$T.$One can define a Morse function$f$associated to the torus action. Facts (Chapter 3 in [3]): The non-degenerate critical manifolds coincide with the connected components of$X^{T}.$Moreover, the stable manifolds of the Morse decomposition associated to$f$coincide with the stable manifolds of the Bialynicki-Birula decomposition associated to the$T$-action. Thus, in this case the relation$\prec$coincides with the relation$

Question 2: Do we really need Morse theory to prove the antisymmetric property of $\prec$?

References:

[1] Atiyah, Bott. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615.

[2] Bialynicki-Birula. Some theorems on actions of algebraic groups. The Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480-497.

[3] Nakajima. Lectures on Hilbert schemes of points on surfaces, American Math. Society, Providence RI, 1999.