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Let $f\colon X\to Y$ be a flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remarks. (1) If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.

(2) If $Y$ is a curve, then the question is known to have positive answer as it is a part of a topological approach to the construction of the vanishing cycle.

Let $f\colon X\to Y$ be a faithfully flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remarks. (1) If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.

(2) If $Y$ is a curve, then the question is known to have positive answer as it is a part of a topological approach to the construction of the vanishing cycle.

Let $f\colon X\to Y$ be a flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remark. If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.

Let $f\colon X\to Y$ be a flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remarks. (1) If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.

(2) If $Y$ is a curve, then the question is known to have positive answer as it is a part of a topological approach to the construction of the vanishing cycle.

Let $f\colon X\to Y$ be a flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remarks. 1) If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.

2. For non-flat morphism $f$ it is easy to construct a counter-example to the question: take the blow up of a surface at a point. In this case some intersections will consist of a single point, some will be empty.

Let $f\colon X\to Y$ be a flat morphism of smooth projective varieties over $\mathbb{C}$. Assume that the generic fiber of $f$ is smooth. Then there exists a non-empty Zariski open subset $U\subset Y$ such that the morphism $f\colon f^{-1}(U)\to U$ is smooth.

For a closed point $x\in X$ we denote by $B(x,\varepsilon)$ the open ball in $X$ of radius $\varepsilon$ centered at $x$ with respect to a fixed Kahler metric.

Question. Fix $x\in X$. Does there exist $\varepsilon_0>0$ such that for any $\varepsilon \in (0,\varepsilon_0)$ the following is satisfied:

for any $z\in B(x,\varepsilon)\cap f^{-1}(U)$ the intersection $f^{-1}(f(z))\cap B(x,\varepsilon)$ has Betti numbers independent of $z$ and $\varepsilon$?

Remark. If anything like that is true, the above Betti numbers should not depend on the Kahler metric for small $\varepsilon_0$.