Skip to main content
MathOverflow

Return to Question

deleted 4 characters in body
Source Link
Mohan Swaminathan
Mohan Swaminathan
  • 1.8k
  • 8
  • 24

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular neighborhood. The purpose of this question is to understand this better. First, I describe the "plumbing" explicitly.

For $i=1,2$, let $p_i\in C_i$ be a pointed compact complex curve and let $z_i:U_i\to\mathbb D_1$ be an analytic isomorphism from a neighborhood $U_i\subset C_i$ of $p$ to the unit disc $\mathbb D_1 = \{z\in\mathbb C:|z|<1\}$ with $z_i(p_i) = 0$. Letting $\mathbb D_{1/4} = \{t\in\mathbb C:|t|<1/4\}$, define $\tilde C_i\subset C_i\times\mathbb D_{1/4}$ to be the complement of the closed subset $\{(p,t):p\in U_i, t\in\mathbb D_{1/4}, |z_i(p)|\le 3/4\}$. Also define $V = \{(x_1,x_2,t)\in\mathbb C^3:x_1,x_2\in\mathbb D_1, t\in\mathbb D_{1/4}, x_1x_2=t\}$. Now, we form a nonsingular complex surface $S$ by gluing $\tilde C_1$, $\tilde C_2$ and $V$ as follows. Identify the point $(p,t)\in\tilde C_1$ withsatisfying $3/4<|z_1(p)|<1$ with $(z_1(p),t/z_1(p),t)\in V$ and similarly for $\tilde C_2$. There is a natural holomorphic map $\pi:S\to\mathbb D_{1/4}$ given by projecting onto the $t$-coordinate in any of the three patches that were glued. We have an obvious identification $\pi^{-1}(0)$ with the nodal curve $\Sigma:=(C_1\sqcup C_2)/(p_1\sim p_2)$. In particular, we have the complex submanifold $C_1\subset \Sigma\subset S$. We can check that the normal bundle $N_{C_1/S}$ has sheaf of sections given by $\mathcal O_{C_1}(-p_1)$. I claim that, in fact, a neighborhood of $C_1$ in $S$ can be identified holomorphically with a neighborhood of the zero section in $N_{C_1/S}$ (with the induced map on $C_1$ being the identity).

To see this, we will use explicit local trivializations of $N_{C_1/S}$ and define a (germ of a) map $N_{C_1/S}\to S$. Over $U'_1 = C_1\setminus\{p\in U_1,|z_1(p)|\le 3/4\}$, use the tautological section $1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U'_1} = U'_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U'_1}\to\tilde C_1\subset S$ by $(p,a)\mapsto (p,a)$. Over $U_1$, use the section $z_1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U_1} = U_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U_1}\to V\subset S$ by $(p,b)\mapsto (z_1(p),b,b\cdot z_1(p))$. The transition function between the trivializations is given by $b = a/z_1$ over $U_1\cap U_1 '$ which shows that we have produced a well-defined map $N_{C_1/S}\to S$ which is holomorphic, induces the identity on $C_1$ and is an isomorphism in a neighborhood of $C_1$. Thus we have the claimed holomorphic tubular neighborhood.

My question is now the following. From a coordinate-invariant viewpoint, what's special about this family of nodal curves which causes the tubular neighborhood to exist and, in particular, causes the short exact sequence $0\to T_{C_1}\to T_S|_{C_1}\to N_{C_1/S}\to 0$ to holomorphically split? The splitting statement depends only on the Kodaira-Spencer class $\kappa_\pi\in\text{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma)$ of this deformation (i.e. the restriction of the deformation to $\mathbb C[t]/t^2$). How can we characterize the Kodaira-Spencer classes for which this splitting occurs? The only thing about $\kappa_\pi$ which is obvious to me from the construction is that its projection to the local $\mathcal{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma) = T_{p_1}C_1\otimes T_{p_2}C_2$ is nonzero (since the node is smoothed by the local equation $x_1x_2=t$). Also, by making different choices of local analytic coordinates $z_1,z_2$ near $p_1,p_2$, we can presumably obtain many different such Kodaira-Spencer classes which all share this splitting property, so it seems a bit mysterious to me.

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular neighborhood. The purpose of this question is to understand this better. First, I describe the "plumbing" explicitly.

For $i=1,2$, let $p_i\in C_i$ be a pointed compact complex curve and let $z_i:U_i\to\mathbb D_1$ be an analytic isomorphism from a neighborhood $U_i\subset C_i$ of $p$ to the unit disc $\mathbb D_1 = \{z\in\mathbb C:|z|<1\}$ with $z_i(p_i) = 0$. Letting $\mathbb D_{1/4} = \{t\in\mathbb C:|t|<1/4\}$, define $\tilde C_i\subset C_i\times\mathbb D_{1/4}$ to be the complement of the closed subset $\{(p,t):p\in U_i, t\in\mathbb D_{1/4}, |z_i(p)|\le 3/4\}$. Also define $V = \{(x_1,x_2,t)\in\mathbb C^3:x_1,x_2\in\mathbb D_1, t\in\mathbb D_{1/4}, x_1x_2=t\}$. Now, we form a nonsingular complex surface $S$ by gluing $\tilde C_1$, $\tilde C_2$ and $V$ as follows. Identify the point $(p,t)\in\tilde C_1$ with $3/4<|z_1(p)|<1$ with $(z_1(p),t/z_1(p),t)\in V$ and similarly for $\tilde C_2$. There is a natural holomorphic map $\pi:S\to\mathbb D_{1/4}$ given by projecting onto the $t$-coordinate in any of the three patches that were glued. We have an obvious identification $\pi^{-1}(0)$ with the nodal curve $\Sigma:=(C_1\sqcup C_2)/(p_1\sim p_2)$. In particular, we have the complex submanifold $C_1\subset \Sigma\subset S$. We can check that the normal bundle $N_{C_1/S}$ has sheaf of sections given by $\mathcal O_{C_1}(-p_1)$. I claim that, in fact, a neighborhood of $C_1$ in $S$ can be identified holomorphically with a neighborhood of the zero section in $N_{C_1/S}$ (with the induced map on $C_1$ being the identity).

To see this, we will use explicit local trivializations of $N_{C_1/S}$ and define a (germ of a) map $N_{C_1/S}\to S$. Over $U'_1 = C_1\setminus\{p\in U_1,|z_1(p)|\le 3/4\}$, use the tautological section $1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U'_1} = U'_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U'_1}\to\tilde C_1\subset S$ by $(p,a)\mapsto (p,a)$. Over $U_1$, use the section $z_1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U_1} = U_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U_1}\to V\subset S$ by $(p,b)\mapsto (z_1(p),b,b\cdot z_1(p))$. The transition function between the trivializations is given by $b = a/z_1$ over $U_1\cap U_1 '$ which shows that we have produced a well-defined map $N_{C_1/S}\to S$ which is holomorphic, induces the identity on $C_1$ and is an isomorphism in a neighborhood of $C_1$. Thus we have the claimed holomorphic tubular neighborhood.

My question is now the following. From a coordinate-invariant viewpoint, what's special about this family of nodal curves which causes the tubular neighborhood to exist and, in particular, causes the short exact sequence $0\to T_{C_1}\to T_S|_{C_1}\to N_{C_1/S}\to 0$ to holomorphically split? The splitting statement depends only on the Kodaira-Spencer class $\kappa_\pi\in\text{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma)$ of this deformation (i.e. the restriction of the deformation to $\mathbb C[t]/t^2$). How can we characterize the Kodaira-Spencer classes for which this splitting occurs? The only thing about $\kappa_\pi$ which is obvious to me from the construction is that its projection to the local $\mathcal{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma) = T_{p_1}C_1\otimes T_{p_2}C_2$ is nonzero (since the node is smoothed by the local equation $x_1x_2=t$). Also, by making different choices of local analytic coordinates $z_1,z_2$ near $p_1,p_2$, we can presumably obtain many different such Kodaira-Spencer classes which all share this splitting property, so it seems a bit mysterious to me.

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular neighborhood. The purpose of this question is to understand this better. First, I describe the "plumbing" explicitly.

For $i=1,2$, let $p_i\in C_i$ be a pointed compact complex curve and let $z_i:U_i\to\mathbb D_1$ be an analytic isomorphism from a neighborhood $U_i\subset C_i$ of $p$ to the unit disc $\mathbb D_1 = \{z\in\mathbb C:|z|<1\}$ with $z_i(p_i) = 0$. Letting $\mathbb D_{1/4} = \{t\in\mathbb C:|t|<1/4\}$, define $\tilde C_i\subset C_i\times\mathbb D_{1/4}$ to be the complement of the closed subset $\{(p,t):p\in U_i, t\in\mathbb D_{1/4}, |z_i(p)|\le 3/4\}$. Also define $V = \{(x_1,x_2,t)\in\mathbb C^3:x_1,x_2\in\mathbb D_1, t\in\mathbb D_{1/4}, x_1x_2=t\}$. Now, we form a nonsingular complex surface $S$ by gluing $\tilde C_1$, $\tilde C_2$ and $V$ as follows. Identify $(p,t)\in\tilde C_1$ satisfying $3/4<|z_1(p)|<1$ with $(z_1(p),t/z_1(p),t)\in V$ and similarly for $\tilde C_2$. There is a natural holomorphic map $\pi:S\to\mathbb D_{1/4}$ given by projecting onto the $t$-coordinate in any of the three patches that were glued. We have an obvious identification $\pi^{-1}(0)$ with the nodal curve $\Sigma:=(C_1\sqcup C_2)/(p_1\sim p_2)$. In particular, we have the complex submanifold $C_1\subset \Sigma\subset S$. We can check that the normal bundle $N_{C_1/S}$ has sheaf of sections given by $\mathcal O_{C_1}(-p_1)$. I claim that, in fact, a neighborhood of $C_1$ in $S$ can be identified holomorphically with a neighborhood of the zero section in $N_{C_1/S}$ (with the induced map on $C_1$ being the identity).

To see this, we will use explicit local trivializations of $N_{C_1/S}$ and define a (germ of a) map $N_{C_1/S}\to S$. Over $U'_1 = C_1\setminus\{p\in U_1,|z_1(p)|\le 3/4\}$, use the tautological section $1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U'_1} = U'_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U'_1}\to\tilde C_1\subset S$ by $(p,a)\mapsto (p,a)$. Over $U_1$, use the section $z_1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U_1} = U_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U_1}\to V\subset S$ by $(p,b)\mapsto (z_1(p),b,b\cdot z_1(p))$. The transition function between the trivializations is given by $b = a/z_1$ over $U_1\cap U_1 '$ which shows that we have produced a well-defined map $N_{C_1/S}\to S$ which is holomorphic, induces the identity on $C_1$ and is an isomorphism in a neighborhood of $C_1$. Thus we have the claimed holomorphic tubular neighborhood.

My question is now the following. From a coordinate-invariant viewpoint, what's special about this family of nodal curves which causes the tubular neighborhood to exist and, in particular, causes the short exact sequence $0\to T_{C_1}\to T_S|_{C_1}\to N_{C_1/S}\to 0$ to holomorphically split? The splitting statement depends only on the Kodaira-Spencer class $\kappa_\pi\in\text{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma)$ of this deformation (i.e. the restriction of the deformation to $\mathbb C[t]/t^2$). How can we characterize the Kodaira-Spencer classes for which this splitting occurs? The only thing about $\kappa_\pi$ which is obvious to me from the construction is that its projection to the local $\mathcal{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma) = T_{p_1}C_1\otimes T_{p_2}C_2$ is nonzero (since the node is smoothed by the local equation $x_1x_2=t$). Also, by making different choices of local analytic coordinates $z_1,z_2$ near $p_1,p_2$, we can presumably obtain many different such Kodaira-Spencer classes which all share this splitting property, so it seems a bit mysterious to me.

Source Link
Mohan Swaminathan
Mohan Swaminathan
  • 1.8k
  • 8
  • 24

Unexpected holomorphic tubular neighborhood

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular neighborhood. The purpose of this question is to understand this better. First, I describe the "plumbing" explicitly.

For $i=1,2$, let $p_i\in C_i$ be a pointed compact complex curve and let $z_i:U_i\to\mathbb D_1$ be an analytic isomorphism from a neighborhood $U_i\subset C_i$ of $p$ to the unit disc $\mathbb D_1 = \{z\in\mathbb C:|z|<1\}$ with $z_i(p_i) = 0$. Letting $\mathbb D_{1/4} = \{t\in\mathbb C:|t|<1/4\}$, define $\tilde C_i\subset C_i\times\mathbb D_{1/4}$ to be the complement of the closed subset $\{(p,t):p\in U_i, t\in\mathbb D_{1/4}, |z_i(p)|\le 3/4\}$. Also define $V = \{(x_1,x_2,t)\in\mathbb C^3:x_1,x_2\in\mathbb D_1, t\in\mathbb D_{1/4}, x_1x_2=t\}$. Now, we form a nonsingular complex surface $S$ by gluing $\tilde C_1$, $\tilde C_2$ and $V$ as follows. Identify the point $(p,t)\in\tilde C_1$ with $3/4<|z_1(p)|<1$ with $(z_1(p),t/z_1(p),t)\in V$ and similarly for $\tilde C_2$. There is a natural holomorphic map $\pi:S\to\mathbb D_{1/4}$ given by projecting onto the $t$-coordinate in any of the three patches that were glued. We have an obvious identification $\pi^{-1}(0)$ with the nodal curve $\Sigma:=(C_1\sqcup C_2)/(p_1\sim p_2)$. In particular, we have the complex submanifold $C_1\subset \Sigma\subset S$. We can check that the normal bundle $N_{C_1/S}$ has sheaf of sections given by $\mathcal O_{C_1}(-p_1)$. I claim that, in fact, a neighborhood of $C_1$ in $S$ can be identified holomorphically with a neighborhood of the zero section in $N_{C_1/S}$ (with the induced map on $C_1$ being the identity).

To see this, we will use explicit local trivializations of $N_{C_1/S}$ and define a (germ of a) map $N_{C_1/S}\to S$. Over $U'_1 = C_1\setminus\{p\in U_1,|z_1(p)|\le 3/4\}$, use the tautological section $1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U'_1} = U'_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U'_1}\to\tilde C_1\subset S$ by $(p,a)\mapsto (p,a)$. Over $U_1$, use the section $z_1$ of $\mathcal O_{C_1}(-p_1)$ to get a trivialization $N_{C_1/S}|_{U_1} = U_1\times\mathbb C$. Using this trivialization, define the map $N_{C_1/S}|_{U_1}\to V\subset S$ by $(p,b)\mapsto (z_1(p),b,b\cdot z_1(p))$. The transition function between the trivializations is given by $b = a/z_1$ over $U_1\cap U_1 '$ which shows that we have produced a well-defined map $N_{C_1/S}\to S$ which is holomorphic, induces the identity on $C_1$ and is an isomorphism in a neighborhood of $C_1$. Thus we have the claimed holomorphic tubular neighborhood.

My question is now the following. From a coordinate-invariant viewpoint, what's special about this family of nodal curves which causes the tubular neighborhood to exist and, in particular, causes the short exact sequence $0\to T_{C_1}\to T_S|_{C_1}\to N_{C_1/S}\to 0$ to holomorphically split? The splitting statement depends only on the Kodaira-Spencer class $\kappa_\pi\in\text{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma)$ of this deformation (i.e. the restriction of the deformation to $\mathbb C[t]/t^2$). How can we characterize the Kodaira-Spencer classes for which this splitting occurs? The only thing about $\kappa_\pi$ which is obvious to me from the construction is that its projection to the local $\mathcal{Ext}^1(\Omega_\Sigma,\mathcal O_\Sigma) = T_{p_1}C_1\otimes T_{p_2}C_2$ is nonzero (since the node is smoothed by the local equation $x_1x_2=t$). Also, by making different choices of local analytic coordinates $z_1,z_2$ near $p_1,p_2$, we can presumably obtain many different such Kodaira-Spencer classes which all share this splitting property, so it seems a bit mysterious to me.