If it is not of general type, then the Iitaka fibration $X\to Z$ is fibered by tori say $F$ (the fibers of the Iitaka fibration have Kodaira dimension 0 and maximal Albanese dimension, and so by a ...

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the ...

$X$ is of general type iff it is birational to its canonical model $X^c={\rm Proj}(\oplus _{m\geq 0}H^0(mK_X))$. Here $X^c$ has canonical singularities and $mK_{X^c}$ is a very ample Cartier divisor ...

How about the following example in Debarre's paper "ON COVERINGS OF SIMPLE ABELIAN VARIETIES" available on his web page. On page 5, it says: "...if L is an ample line bundle on an abelian variety A of ...

I don't think it is true that $\mathcal X$ is $\mathbb Q$-Gorenstein. Suppose in fact that $\dim \mathcal X _t=2$ for all $t\in C$ and $\mathcal X \to Z$ is a flipping contraction with exceptional ...

Theorem: Let $A$ be an abelian variety and $f:A\to X$ a dominant morphism to a projective variety of general type, then $\dim X=0$. Proof: Replacing $f$ by a birational model of the Stein ...

I don't have any direct/immediate consequence of the main result of the paper. However the techniques that we develop here are very useful. In work in progress, we plan to use these techniques to ...

Koll\'ar's paper "Moduli of varieties of general type" https://arxiv.org/abs/1008.0621 provides an excellent introduction. We now know that moduli spaces of canonical models of varieties of general ...

Claim: $D_*$ is SV iff for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D_m-H$ is nef. Suppose $D_*$ is SV. Fix $A$ very ample. Since $D_*$ is SV, then $H^i(D_m-H-jA)=0$ ...

I just saw this question (better late than never). The following references come to mind (the first one has an example which somewhat improves on the example of Ein-Lazarsfeld, the others have a ...

Let $X\to \mathbb P ^1$ be a family of smooth quadric surfaces degenerating to $X_0=\mathbb F _2=\mathbb P (\mathcal O _{\mathbb P ^1}\oplus \mathcal O _{\mathbb P ^1} (2))$ where $0\in \mathbb P ^1$. ...

Does Example 2.4 from http://www.math.utah.edu/~hacon/commA.dvi answer your question?

Let's assume that $|kD|=|kM|+kF$ where $|M|$ is base point free and moreover $Sym ^kH^0(M)\to H^0(kM)$ is surjective for any $k>0$. This can be achieved replacing $X$ by an appropriate resolution ...

From "Bounded sets of sheaves on Kähler manifolds" By Matei Toma, J. reine angew. Math. 710 (2016), 77–93 Lemma 4.4. Let X be a Kähler manifold, r be an integer and F be a set of compact reduced ...

If $X$ is $\mathbb P ^2$ blown up at a point $O$ with exceptional divisor $E$ and $D=H+E$ where $H$ is the pull-back of the hyperplane class, then any effective divisor $D'\equiv D$ will vanish along $...

We have $K_X+\Delta +\lambda C=f^*(K_Z+\Delta '+\lambda C')$ by the Base Point Free Thm (3.3 and 3.7(4) in Koll\'ar-Mori 1998). Clearly $K_Z+\Delta '+\lambda C'$ is nef. If $f^+:X^+\to Z $ is the flip,...

If $K_{\tilde B}+\tilde \Delta=f^*(K_B+\Delta)+E$ where $E$ is effective and exceptional, then $h^0(m(K_{\tilde B}+\tilde \Delta))=h^0(m(K_B+\Delta))$ for any $m\geq 0$ and hence also the Kodaira ...

In Prop. 3.13 of http://arxiv.org/pdf/1212.5105.pdf there is an example of a generically finite map $\lambda : T\to A$ where $A$ is an abelian variety and $T$ is a Gorenstein variety with log ...

If $X$ is smooth (projective over the complex numbers), then $R(K_X)$ is finitely generated by BCHM. We may thus assume that $R(kK_X)$ is generated in degree 1 for some $k>0$. Passing to a log ...

If $(X,D)$ is terminal and the stable base locus of $K_X+D$ contains no components of the support of $D$, then any sequence of steps $f:X\to X'$ of the $K_X+D$ MMP yields a terminal pair $(X',D'=f_*D)$...

If $R(L)=\oplus H^0(mL)$ is not finitely generated, the above surjectivity will fail, however it will hold "asymptotically" for any big line bundle $L$. In fact, by Fujita's approximation of ...

There is a spectacular proof of the ACC for log canonical thresholds in $\mathbb C ^N$ due to Ein, de Fernex and Mustata https://arxiv.org/abs/0905.3775 that relies on taking generic limits of ...

I would proceed as follows. Replacing $S$ by its minimal resolution I have a weak log Fano pair $(S,B)$ which is klt and $-(K_S+B)$ is semiample and big. In particular $-K_S\sim _Q -(K_S+B)+B$ is big ...

I think the answer is (almost, i.e. up to birational modification) yes but non-trivial see Theorem 1.3 of arXiv:1208.4150 "ACC for log canonical thresholds" by Christopher Hacon, James McKernan, ...

I believe that this never happens. The reason is as follows. The morphism $f:X\to Y$ is a small birational morphism (it contracts some divisors on the special fiber) and so by standard arguments $Y$ ...

For any $P\in Pic^0(A)$ consider the map $|\Theta +P|\times |D-P|\to |\Theta +D|\cong \mathbb P ^2$. Since $D$ is effective, there is an abelian subvariety $T\subset Pic ^0(A)$ such that $|D-P'|\ne \...

By Noetherian induction, it suffices to show that indicies and volumes are bounded over an open subset of any irreducible component of $T$. We may assume that $T$ is smooth and there is a dense set $\...

It could easily happen that $X_0$ has log terminal singularities and $X$ is not log terminal. The standard example is if $f:Y\to X$ is a flipping contraction of a 3-fold over a curve $T$ (where the ...

Actually, I think the above argument always works because the minimal model has mild singularities (or one can reason directly on $X$; see ref below). Let $X$ be a smooth variety of general type and $...

It seems to me that $[\frac 1nF_n]$ is exc. and there is a short exact sequence inducing $$W(k)→V(k+1)→H^1((−∑k_iE_i)|_{\sum \eta _iE_i})$$ which has positive degree on a bunch of (non-reduced $\...