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Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)

Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.

Consider the Briesksorn-Pham polynomials Brieskorn-Pham polynomial $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For In particular, for $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.

In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.

(Edited) Question(s): In general, how is Is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$ for $n > 2$? References are welcome.Is there a geometric genus-like object for $n = 2$? If so, how might it relate to invariants of the torus knots?

Thanks!

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Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)

Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.

Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.

In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.

Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$n > 2$? References are welcome. Thanks! 9 added 1 characters in body; [made Community Wiki] Let$f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface$V_{f, \kappa} = f^{-1}(\kappa)$for small$\kappa > 0$. Let$\pi \colon \tilde{V} \to V$be a resolution of$V = V_{f,0}$, and let$p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$denote the corresponding geometric genus of$V$, which is independent of the chosen resolution. (This is the same geomtric geometric genus in the Yau-Durfee Theorem:$\mu \geq n! \ p_{g}$for$n > 2$, See Y93 or Y06.) Define the map$\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where$\epsilon > 0$is sufficiently small. The intersection of$V_{\kappa}$with a small sphere$S_{\epsilon}^{2n-1}$is an algebraic link. Consider the Briesksorn-Pham polynomials$f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$with$a_{i} \geq 2$. For$f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot$T_{p,q}$if$p$and$q$are coprime. In 1968, Milnor conjectured that the unknotting number$u(T_{p,q})$is related to the dimension of the local algebra$A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group$H_{1}(F_{f,0}; \mathbb{Z})$of the corresponding fiber$F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$. In 1928, Brauner proved$\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$for$f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed. Question(s): In general, how is the genus$g$of the fiber$F_{f,0}$related to the geometric genus$p_{g}$of$V$? Are there sufficient and necessary conditions on the geometric genus of$V$which imply the Milnor conjecture for the case$f = z_{1}^{p} + z_{2}^{q}\$? References are welcome.

Thanks!

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