Convex bodies associated to linear series.

*(English)*Zbl 1182.14004In the paper under review, the authors initiate a systematic development of the theory of convex bodies associated to linear series (by a construction of Okounkov). Let \(X\) be a smooth irreducible \(d\)-dimensional projective variety defined over an uncountable algebraically closed field \(\mathbb K\) and
\[
Y_\bullet :X=Y_0\supset Y_1\supset Y_2\supset \dots \supset Y_{d-1} \supset Y_d=\text{pt},
\]
be a fixed flag, i.e., a sequence \(Y_i\) of smooth irreducible subvarieties of codimension \(i\) in \(X\). If \(s\) is a non-zero section of a big divisor \(D\) on \(X\), then one defines \(\nu _1(s)\) to be the order of vanishing of \(s\) along \(Y_1\). Let \(\tilde s_1\) be the corresponding section of the divisor \(D-\nu _1(s)Y_1\) and \(s_1=\tilde s_1|_{Y_1}\) be its restriction to \(Y_1\), then we let \(\nu _2(s)\) be the order of vanishing of \(s_1\) along \(Y_2\). Repeating this procedure one inductively defines valuation like functions \(\nu _1(s),\dots , \nu _d (s)\), i.e., a function \(\nu=\nu _{Y_\bullet,D}:(H^0(\mathcal O _X(D))-\{ 0 \} )\to \mathbb Z ^d\) such that \(\nu (s)=(\nu _1(s),\ldots , \nu _d (s))\). The Okounkov body of \(D\) is the set \(\Delta _{Y_\bullet }(D)\) given by the closed convex hull of
\[
\bigcup _{m\geq 1}\frac 1 m \cdot \text{Im}(\nu _{Y_\bullet}:(H^0(\mathcal O _X(D))-\{ 0 \})\to \mathbb Z ^d ).
\]

As shown by Okounkov, the Euclidean volume of \(\Delta (D)\) is related to the volume of the divisor \(D\) by the formula \[ \frac 1 {d!}\text{vol} _X(D):=\lim _{m\to \infty}\frac {h^0(\mathcal O _X(mD))}{m^d}. \] Note that the volume of big divisors is an invariant that plays an important role in higher dimensional geometry.

The authors show that many basic facts about asymptotic invariants of linear series can be understood in terms of the properties of the corresponding Okounkov body. In particular they show that for any big divisor \(D\), the Okounkov body \(\Delta (D)\) only depends on the numerical class of \([D]\in N^1(X)_{\mathbb R}\). Moreover, there exists a closed convex cone \(\Delta (X)\subset \mathbb R^d\times N^1(X)_{\mathbb R}\) such that for any big class \(\xi \in N^1(X)_{\mathbb R}\), \(\Delta (X)\cap \mathbb R^d\times \{ \xi\}=\Delta (\xi )\). In particular, one recovers the fact that for big classes, the volume defines a continuous function and \(\Delta (\xi )+\Delta (\xi ')\subset \Delta (\xi +\xi ')\). The Brunn-Minkowski theorem then yields that the volume function is log concave, i.e., \[ \text{vol}_X(\xi+\xi ')^{1/d}\geq \text{vol}_X(\xi)^{1/d}+\text{vol}_X(\xi ')^{1/d}. \] Another consequence [first observed in S. Boucksom, C. Favre and M. Jonsson, J. Algebr. Geom. 18, No. 2, 279–308 (2009; Zbl 1162.14003)] is that \(\text{vol}_X\) defines a \(\mathcal C ^1\) function on \(\text{Big }(X)\). The authors then prove a version of the Fujita approximation theorem for (incomplete) graded linear series. Finally it is shown that if we replace \(X\) by its blow up at a very general point \(x\in X\) and we consider a complete flag corresponding to the projectivizations of (very general) linear subspaces of \(T_xX\), then the Okounkov body \(\Delta _{F(x; V_\bullet )}(D)\) is uniquely determined.

As shown by Okounkov, the Euclidean volume of \(\Delta (D)\) is related to the volume of the divisor \(D\) by the formula \[ \frac 1 {d!}\text{vol} _X(D):=\lim _{m\to \infty}\frac {h^0(\mathcal O _X(mD))}{m^d}. \] Note that the volume of big divisors is an invariant that plays an important role in higher dimensional geometry.

The authors show that many basic facts about asymptotic invariants of linear series can be understood in terms of the properties of the corresponding Okounkov body. In particular they show that for any big divisor \(D\), the Okounkov body \(\Delta (D)\) only depends on the numerical class of \([D]\in N^1(X)_{\mathbb R}\). Moreover, there exists a closed convex cone \(\Delta (X)\subset \mathbb R^d\times N^1(X)_{\mathbb R}\) such that for any big class \(\xi \in N^1(X)_{\mathbb R}\), \(\Delta (X)\cap \mathbb R^d\times \{ \xi\}=\Delta (\xi )\). In particular, one recovers the fact that for big classes, the volume defines a continuous function and \(\Delta (\xi )+\Delta (\xi ')\subset \Delta (\xi +\xi ')\). The Brunn-Minkowski theorem then yields that the volume function is log concave, i.e., \[ \text{vol}_X(\xi+\xi ')^{1/d}\geq \text{vol}_X(\xi)^{1/d}+\text{vol}_X(\xi ')^{1/d}. \] Another consequence [first observed in S. Boucksom, C. Favre and M. Jonsson, J. Algebr. Geom. 18, No. 2, 279–308 (2009; Zbl 1162.14003)] is that \(\text{vol}_X\) defines a \(\mathcal C ^1\) function on \(\text{Big }(X)\). The authors then prove a version of the Fujita approximation theorem for (incomplete) graded linear series. Finally it is shown that if we replace \(X\) by its blow up at a very general point \(x\in X\) and we consider a complete flag corresponding to the projectivizations of (very general) linear subspaces of \(T_xX\), then the Okounkov body \(\Delta _{F(x; V_\bullet )}(D)\) is uniquely determined.

Reviewer: Christopher Hacon (Salt Lake City)