Threefolds of general type with no holomorphic forms?

Question

In relation to this question, I would like to ask for examples of (complex) threefolds of general type with no (nontrivial) holomorphic form.

Answer 1

One simply connected example arises as the total space of a family of Godeaux surfaces over $\mathbb{P}^1_k$ with sufficiently positive discriminant.

Let $k$ be a field of characteristic different from $5$. Let $\mathbb{P}^1_k$ be $\text{Proj}\ k[R,S]$. Let $\mathbb{P}^3_k$ be $\text{Proj}\ k[T_0,T_1,T_2,T_3]$. Let $e$ be a nonnegative integer. Let $b_0,b_1,b_2,b_3 \in k[R,S]_e$ be a $4$-tuple of homogeneous polynomials of degree $e$ on $\mathbb{P}^1_k$.

Hypothesis 1. The integer $e$ is $\geq 3$. Every zero in $\mathbb{P}^1_k$ of each polynomial $b_i$ is a simple zero, and there is no simultaneous zero of two or more of the polynomials $b_i,b_j$.

Notation 2. Denote by $f$ be the section of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(5)$, $$f = b_0 T_0^5 + b_1 T_1^5 + b_2 T_2^5 + b_3 T_3^5.$$ Denote by $Y$ be the zero scheme of $f$ as a hypersurface in $\mathbb{P}^1_k\times_{\text{Spec}\ k} \mathbb{P}^3_k$. For each $i=0,1,2,3$, denote by $p_i$ the $k$-point of $\mathbb{P}^3_k$ where $T_j$ vanishes for every $j\neq i$. Also denote by $Y_i\subset Y$ the product $\text{Zero}(b_i)\times\{p_i\}$. Finally, denote the restrictions of $\text{pr}_{\mathbb{P}^1}$ and $\text{pr}_{\mathbb{P}^3}$ to $Y$ as follows, $$\pi:Y\to \mathbb{P}^1_k, \ \ \rho:Y\to \mathbb{P}^3_k.$$

Let $\mu_5$ denote the group $k$-scheme of $5^{\text{th}}$ roots of unity in $\mathbb{G}_{m,k}$. Since $\text{char}(k)$ is not $5$, this is $k$-smooth. Let $\mu_5$ act on $\mathbb{P}^1_k\times_{\text{Spec}\ k}\mathbb{P}^3_k$ as follows, $$\zeta \bullet([S_0,S_1],[T_0,T_1,T_2,T_3]) = ([S_0,S_1],[T_0,\zeta T_1, \zeta^2 T_2, \zeta^3 T_3]).$$ The polynomial $f$ is invariant for this action. Thus, there is an induced action on $Y$.

Notation 3. Denote the quotient of this $\mu_5$-action on $Y$ by $\nu:Y\to X'$. Denote by $\phi:X\to X'$ any strong desingularization of $X'$ that is projective.

Proposition 4. The projective $k$-scheme $X$ is simply connected and of general type. Moreover, for $r=1,2,3$, the only global section of $\Omega^r_{X/k}$ is the zero section.

This will be proved in stages.

Lemma 5. The hypersurface $Y$ is smooth. The singular locus of the projection $\pi$ equals $Y_0\cup Y_1\cup Y_2 \cup Y_3$. The projection $\rho$ is smooth over $p_i$ for every $i=0,1,2,3$. The dualizing sheaf of $Y$ equals the restriction of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e-2)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(1)$.

Proof. By the Jacobian criterion, the projection $\pi$ is smooth away from $\cup_i Y_i$. The fiber of $\rho$ over $p_i$ is the zero scheme of $b_i$, and this is reduced (hence smooth) by hypothesis. Thus, the projection morphism $\rho$ is smooth at every point of $\cup_i Y_i$. For every point of $Y$, either $\pi$ or $\rho$ is smooth, and hence $Y$ is everywhere smooth. The remaining computations are straightforward. QED

Lemma 6. The action of $\mu_5$ on $Y\setminus \cup_i Y_i$ is free. For every point of $\cup_i Y_i$, the age (in the sense of Reid -- Shepherd-Barron -- Tai) is $> 1$.

Proof. The fixed points of the action on $\mathbb{P}^3_k$ are the points $p_i$. Since the projection $\rho$ is equivariant, every fixed point in $Y$ maps to $p_i$ for some $i$. Thus, the fixed locus is contained in $\cup_i Y_i$.

Since the projection $\rho$ is étale at every point of $\cup_i Y_i$, the age of that fixed point in $Y$ equals the age of the image fixed point, say $p_i$, in $\mathbb{P}^3_k$. By the choice of action of $\mu_5$ on $\mathbb{P}^3_k$, every eigenvalue that occurs in $T_{\mathbb{P}^3,p_i}$ has multiplicity $1$. Moreover, since the fixed points are isolated, none of these eigenvalues equals $1$. Thus, for each choice of primitive generator $\zeta$ of $\mu_5$, the eigenvalues $\zeta^a,\zeta^b,\zeta_c$ that occur have $0<a<b<c <5$, up to rearrangement. Thus, the sum $a+b+c$ is at least $1+2+3=6$. Therefore the age $\alpha(\mathbb{P}^3,p_i) = (a+b+c)/5$ is strictly larger than $1$. QED

Lemma 7. The $k$-scheme $X'$ is projective, normal, $\mathbb{Q}$-Gorenstein, has ample $\mathbb{Q}$-canonical divisor class, and has only terminal finite quotient singularities. Finally, $X'$ is simply connected. Thus, every projective desingularization $X$ of $X'$ is a simply connected, projective $3$-fold of general type.

Proof. The only nontrivial claim is that $X'$ has only terminal singularities. This follows by the Reid--Shepherd-Barron--Tai criterion and the computation of the age above. Since $Y$ is simply connected, also $X'$ is simply connected. QED

Since the morphism $\pi$ is $\mu_5$-invariant, there is a unique $k$-morphism, $$\pi':X'\to \mathbb{P}^1_k,$$ such that $\pi'\circ \nu$ equals $\pi$. By Lemma 3, the singular locus of $\pi'$ equals $X'_0\cup X'_1\cup X'_2 \cup X'_3$, for $X'_i:= \nu(Y_i)$. Thus, the smooth locus of $\pi$ equals the open complement $U$ of $\cup_i X'_i$ in $X'$. On $U$, the transitivity exact sequence of $\pi$ gives a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}|_U \to \Omega_{U/k} \to \Omega_{\pi}|_U \to 0.$$ This induces a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}\otimes\Omega_{\pi}|_U \to \Omega^2_{U/k} \to \Omega^2_{\pi}|_U \to 0.$$ This also induces an isomorphism, $$\Omega^3_{U/k} \cong \pi^*\Omega_{\mathbb{P}^1/k}\otimes \Omega^2_{\pi}|_U.$$

Lemma 8. For $r=1,2$, the only global section of $\Omega^r_\pi$ on $U$ is the zero section.

Proof. These coherent sheaves are locally free sheaves on the smooth $k$-scheme $U$. Every nonzero section on $U$ restricts to a nonzero section of the restriction on the generic fiber $F$ of $\pi$. The restriction of $\Omega^r_\pi$ to the generic fiber $F$ equals $\Omega^r_{F/k(\mathbb{P}^1)}$. The generic fiber $F$ of $\pi$ is a Godeaux surface over $k(\mathbb{P}^1)$. Thus, $\Omega^r_{F/k(\mathbb{P}^1)}$ has only the zero global section for $r=1$ and $r=2$. QED

Proof of Proposition 4. It only remains to prove that every global section of $\Omega^r_{X/k}$ is the zero section for $r=1,2,3$. Since $\phi$ is an isomorphism over $U$, it suffices to prove that every global section of $\Omega^r_{U/k}$ is the zero section.

Since the global sections of $\Omega^r_\pi$ on $U$ are zero for $r>0$, the only possibility for a nonzero global section of $\Omega^r_{U/k}$ is a global section of $\pi^*\Omega_{\mathbb{P}^1_k/k}$ with $r=1$. Of course $\pi^*\Omega_{\mathbb{P}^1_k/k}$ equals $\pi^*\mathcal{O}(-2)$. Since $X'$ is normal and since $\cup_i X'_i$ consists of finitely many points, every global section of $\pi^*\mathcal{O}(-2)$ on $U$ is the restriction of a unique global section on all of $X'$ of $\pi^*\mathcal{O}(-2)$. By the projection formula, every such global section is the zero global section. QED

Answer 2

Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds with properties 1--3, the list is given by Iano-Fletcher in Section 15 of [1], where he classified all possible cases for $a_d\leq 100$ and $d\leq 2$. One interesting thing is that Iano-Fletcher's list is later proved to be a full classification of $3$-folds with properties 1--3 by Chen--Chen--Chen [2]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$. The reason why it is interesting is that for this $X$, $|mK_X|$ gives a birational map when $m\geq 27$, but $|26K_X|$ is not birational. This is so far the only known example of 3-fold of general type with this property (that $|26K_X|$ is not birational).

References: [1] A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–-173

[2] J-J. Chen, J.A. Chen, M. Chen, On quasismooth weighted complete intersections, J. Algebraic Geom. 20 (2011): 239--262.