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The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ ...

Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product $E_{1}\...

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take \begin{equation} S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{...

Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...

Let $M^4$ be a closed orientable smooth manifold and $$ I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z $$ its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive ...