Birational Invariants Research Articles

On Some Birational Invariants of Hyper-Kähler Manifolds

Abstract We study in this article three birational invariants of projective hyper-Kähler manifolds: the degree of irrationality, the fibering gonality, and the fibering genus. We first improve the lower bound in a recent result of Voisin saying that the fibering genus of a Mumford–Tate very general projective hyper-Kähler manifold is bounded from below by a constant depending on its dimension and the second Betti number. We also study the relations between these birational invariants for projective $K3$ surfaces of Picard number $1$ and study the asymptotic behaviors of their degree of irrationality and fibering genus.

The birational invariants of Lins Neto’s foliations

Lins Neto [Ann. Sci. École Norm. Sup. (4) 35 (2002), pp. 231–266] constructed families of foliations which are counterexamples to Poincaré’s Problem and Painlevé’s Problem. We will determine the minimal models of these families of foliations, calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. We prove that the slopes of Lins Neto’s foliations are at least 6, and their limits are bigger than 7 7 .

Motivic integration and birational invariance of BCOV invariants

Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi--Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called the BCOV invariant, was constructed by Fang--Lu--Yoshikawa and Eriksson--Freixas i Montplet--Mourougane. The BCOV invariant is conjecturally related to the Gromov--Witten theory via mirror symmetry. Based upon the previous work of the second author, we prove the conjecture that birational Calabi--Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi--Yau varieties with Kawamata log terminal singularities and prove its birational invariance for Calabi--Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration.

Birational invariants of toric orbifold surfaces

We study birational invariants of toric orbifold surfaces.

Universal birational invariants and -homology

We prove that the functor of zeroth -homology is universal as a functorial birational invariant of smooth proper varieties taking values in a category enriched by abelian groups, when the ground field admits resolution of singularities. Communicated by Tony Varilly-Alvarado

Combinatorial Burnside groups

We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.

On the cohomology of reciprocity sheaves

AbstractIn this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.

The Hodge ring of varieties in positive characteristic

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}(X) = \dim H^j(X,\Omega_X^i)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the $h^{i,0}(X)$ and the $h^{0,j}(X)$ (and their duals $h^{i,n}(X)$ and $h^{n,j}(X)$). The corresponding statements for compact K\"ahler manifolds were proven by Kotschick and Schreieder.

Higher dimensional elliptic fibrations and Zariski decompositions

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian.

Birational types of algebraic orbifolds

We introduce a variant of the birational symbols group of Kontsevich, Pestun and the second author, and use this to define birational invariants of algebraic orbifolds. Bibliography: 20 titles.

Morel homotopy modules and Milnor-Witt cycle modules

We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis. As a corollary, we recover a theorem of Ananyevskiy and Neshitov, and we prove that the Milnor-Witt K-theory groups are birational invariants.

Cycle Class Maps and Birational Invariants

AbstractWe introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps. © 2020 Wiley Periodicals LLC

On Severi type inequalities

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least $$\frac{3}{8}$$ . We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least $$2(\dim X)!$$ . Moreover, if $${{\,\mathrm{vol}\,}}(X)=2(\dim X)!$$ , the canonical model of X is a double cover of a principally polarized abelian variety $$(A, \Theta )$$ branched over some divisor $$D\in |2\Theta |$$ .

Crystalline aspects of geography of low dimensional varieties I: numerology

This is a modest attempt to study, in a systematic manner, the structure of low dimensional varieties in positive characteristics using $p$-adic invariants. The main objects of interest in this paper are surfaces and threefolds. It is known that there are many (counter) examples of `pathological' or unexpected behavior in surfaces and even of threefolds. Our focus has been on, obtaining systematically, general positive results instead of counter examples. There are many results we prove in this paper and not all can be listed in this abstract. Here are some of the results we prove: We show that $c_1^2\leq 5c_2+6b_1$ holds for a large class of surfaces (of general type). We prove that for a smooth, projective, Hodge-Witt, minimal surface of general type (with additional assumptions such as slopes of Frobenius on $H^2_{cris}(X)$ are $\geq\frac{1}{2}$) that $$c_1^2\leq 5c_2.$$ Novelty of our method lies in our use of slopes of Frobenius (and Hodge-Witt assumption which allows us to control the slope spectral sequence). We also construct new birational invariants of surfaces. Applying our methods to threefolds, we characterise Calabi-Yau threefolds with $b_3=0$.

Degrees of iterates of rational maps on normal projective varieties

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh and Sibony (Ann. Sci. Éc. Norm. Supér. (4) 37 (2004) 959–971), and by Truong (J. Reine Angew. Math. 758 (2020) 139–182). Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao (Ann. Inst. Fourier (Grenoble) 65 (2015) 1367–1369) and Popovici (Math. Ann. 364 (2016) 649–655), which generalizes Siu's inequality (see Trapani, Math. Z 219 (1995) 387–401) to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in X.

Effective algebraic integration in bounded genus

We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results in order to investigate the effective algebraic integration of foliations on the projective plane. In particular, we describe the Zariski closure of the set of foliations on the projective plane of degree d admitting rational first integrals with fibers having geometric genus bounded by g.

Birational invariance in logarithmic Gromov–Witten theory

Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Abstract Let 𝕂 {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over 𝕂 {\mathbb{K}} . Let f : X ⊢ X {f:X\vdash X} and g : Y ⊢ Y {g:Y\vdash Y} be dominant correspondences, and π : X ⇢ Y {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that π ∘ f = g ∘ π {\pi\circ f=g\circ\pi} . We define relative dynamical degrees λ p ( f | π ) ≥ 1 {\lambda_{p}(f|\pi)\geq 1} for any p = 0 , … , dim ⁡ ( X ) - dim ⁡ ( Y ) {p=0,\dots,\dim(X)-\dim(Y)} . These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy ( φ , ψ ) {(\varphi,\psi)} from π 2 : ( X 2 , f 2 ) → ( Y 2 , g 2 ) {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to π 1 : ( X 1 , f 1 ) → ( Y 1 , g 1 ) {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have λ p ( f 1 | π 1 ) ≥ λ p ( f 2 | π 2 ) {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when 𝕂 = ℂ {\mathbb{K}=\mathbb{C}} . Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Irrationality issues for projective surfaces

This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces.

Multiplicity and invariants in birational geometry

The multiplicity of a point on a variety is a fundamental invariant to estimate how the singularity is bad. It is introduced in a purely algebraic context. On the other hand, we can also attach to the singularity the log canonical threshold and the minimal log discrepancy, which are introduced in a birational theoretic context. In this paper, we show bounds of the multiplicity by functions of these birational invariants for a singularity of locally a complete intersection. As an application, we obtain the affirmative answer to Watanabe's conjecture on the multiplicity of canonical singularity of locally a complete intersection up to dimension 32.