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Register a new userLangtons type Theorem on Algebraic Orbifolds
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Authors
Yonghong Huang
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In this paper, we show Langtons type theorem on separatedness and properness of moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k
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The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $mathbb{C}^{ times } G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$,of $X$ in the metric topology which is dense with complement of measure $0$ such that if $x ,y in W$ then $left (mathbb{C}^{ times } Gright )_{x}$ is conjugate to $left (mathbb{C}^{ times } Gright )_{y}$. Furthermore, if $G x$ is a closed orbit of maximal dimension and if $x$ is a smooth point of $X$ then there exists $y in W$ such that $left (mathbb{C}^{ times } Gright )_{x}$ contains a conjugate of $left (mathbb{C}^{ times } Gright )_{y}$. The proof involves using the Kempf-Ness theorem to reduce the result to the principal orbit type theorem for compact Lie groups.
We investigate chiral zero modes and winding numbers at fixed points on $T^2/mathbb{Z}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula $n_+-n_-=(-V_++V_-)/2N$, where $n_{pm}$ are the numbers of the $pm$ chiral zero modes and $V_{pm}$ are the sums of the winding numbers at the fixed points on $T^2/mathbb{Z}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M eq 0$, consistently with substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ + V_-)/2N$.
The main purpose of this paper is to make Nakayamas theorem more accessible. We give a proof of Nakayamas theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayamas theorem on algebraic varieties over any algebraically closed field of arbitrary characteristic although Nakayamas original statement is formulated for complex analytic spaces.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $mathbb{C}$ and passing to spaces of $mathbb{C}$-valued points, we obtain new proofs of the topologic
We give a proof of the hard Lefschetz theorem for orbifolds that does not involve intersection homology. This answers a question of Fulton. We use a foliated version of the hard Lefschetz theorem due to El Kacimi.
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