# Tagged Questions

**4**

votes

**1**answer

320 views

### The Gauss-Bonnet theorem for Sheaves

Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem
Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage ...

**4**

votes

**1**answer

137 views

### Euler number for base change of a K3 surface

Suppose you have a K3 surface $S$ containing a smooth rational curve $C$ and suppose you have an elliptic fibration $S \rightarrow \mathbb P^1$ that restricts to a morphism $C \rightarrow \mathbb P^1$ ...

**6**

votes

**1**answer

220 views

### Are there perverse sheaves on abelian varieties with small Euler characteristic?

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can ...

**13**

votes

**0**answers

400 views

### Status of the Euler characteristic in characteristic p

In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...

**5**

votes

**2**answers

404 views

### Top chern class under finite, unramified, dominant morphism

Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...

**2**

votes

**1**answer

303 views

### Higher Euler characteristics (possible generalizations)

Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(w_X)$. Hence a "generalization": $\chi(w^{\otimes k}_X)$.
I'd like something of this ...

**3**

votes

**1**answer

174 views

### non degenerate quadratic form on the group of correspondences on an algebraic curve?

Hi,
Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a correspondence to be a line
bundle $L$ on $X\times Y$. A trivial correspondence is a correspondence of the form ...

**6**

votes

**0**answers

614 views

### Euler Characteristic of a Variety

Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results.
In terms of singular homology ...

**12**

votes

**2**answers

936 views

### Spaces that are both homotopically and cohomologically finite

Is it true that every connected space with
1) just finitely many nontrivial homotopy groups, all finite,
and
2) just finitely many nontrivial rational cohomology groups, all finite rank,
is ...

**2**

votes

**0**answers

149 views

### Is there some short formula for the “defect” of Hilbert function

Let $X\subset\Bbb P^n_{\Bbb C}$ be a connected, Cohen Macaulay sub-scheme. (Possibly singular, reducible or non-reduced.) For $k\gg0$ the numbers $h^0(\mathcal{O}_X(k))$ depend polynomially on $k$. ...

**3**

votes

**1**answer

499 views

### Behaviour of euler characteristics in characteristic p for finite etale covers

Let $k$ be an algebraic closure of a finite field of characteristic $p$. Fix an integer $l\neq p$. For a separated $k$-scheme $X$ of finite type, we define the (compactly supported) Euler ...

**7**

votes

**4**answers

1k views

### Is the Euler characteristic a birational invariant

Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...

**18**

votes

**4**answers

1k views

### Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...