We will present an approach to the Bloch-Beilinson filtration via Voevodsky’s triangulated category of mixed motives DM.

We will present an approach to the Bloch-Beilinson filtration via Voevodsky’s triangulated category of mixed motives DM.

Roughly speaking, Shimura varieties are a certain generalisation of modular curves and Siegel modular varieties (moduli spaces of principally polarised abelian varieties), and have been one of the central objects in number theory. Especially important for number theoretic applications is p-adic analytic study of Shimura varieties, which has features analogous to complex analytic geometry and scheme theory.

Recently, there have been many exciting new developments in the study of p-adic geometry and cohomology of Shimura varieties. Underlying many of these developments is the new technique to study “infinite-level” Shimura varieties as p-adic analytic spaces, as well as new developments in p-adic Hodge theory (both of which are built upon P. Scholze’s theory of perfectoid spaces).

For the majority of the talk, we use the examples of modular tower and moduli of principally polarised abelian varieties to illustrate the geometric results on Shimura varieties proved in the recent paper by Caraiani and Scholze, which “decomposes" certain Shimura varieties in a way that gives a meaningful decomposition of the cohomology. We will conclude by describing my work in progress to generalise this result for unramified Hodge-type Shimura varieties. We will not necessarily strive to give a precise definition or construction of each main object, and may settle for giving some simple examples (even at the risk of over-simplifying).

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.

Primitive (birational) automorphisms of projective manifolds, which are irreducible au-
tomorphisms of manifolds, are natural objects in birational algebraic geometry. They are
also very closely related to complex dynamics of several variables. In fact, the dynamical
degrees of birational automorphisms (a kind of renement of a more classical notion of
topological entropy of automorphisms, tting very well with birational geometry), the relative dynamical degrees (their relative version) and the product formula (a kind of Kunneth type formula), introduced by Dinh-Sibony [DS], Dinh-Nyugen [DN], provide very powerful tools also in studying primitive birational automorphisms of manifolds.
In Lecture 1, I would like to give an overview of the basic notions (entropy, dynamical
degrees, relative dynamical degrees etc) and their basic properties with concrete applications for the study of primitive (birational) automorphisms. This lecture is, in some sense, an updated version of [Og].
In Lectures II, III, I would like to prove the well-denedness and the birational invariance
of dynamical degrees, the most basic property of the dynamical degree. The original proof ([DS]) is transcendental being based on some detailed analysis of currents. Here I explain a new purely algebro-geometric proof due to Truong ([Tr]), which is based on a precise form of Chow's moving lemma ([Ro]).
In Lectures IV, V, I would like to prove the product formula, the most fundamental
and useful property of relative dynamical degrees. Here I explain again an algebraic proof
following a guideline explained in [Tr], which is a modication of original analytic proofs
([DN], [DNT]) into an algebraic one again using a precise form of Chow's moving lemma.

References

[DS] Dinh, T.-C., Sibony, N., Une borne superieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005) 1637{1644. arXiv:math/0303271.

[DN] Dinh, T.-C., Nguyen V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011) 817{840. arXiv:0903.2621.

[DNT] Dinh, T.-C., Nguyen V.-A., Truong, T.-T., On the dynamical degrees of meromorphic maps preserving a bration, Commun. Contemp. Math. 14 (2012) 18pp, arXiv: 1108.4792.

[Og] Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians, Seoul 2014 (Invited Lectures) Vol.II (2015), 695{721. arXiv:1404.2982.

[Ro] Roberts, J., Chow's moving lemma, in Algebraic geometry, Oslo 1970, F. Oort (ed.), WoltersNoordhoff, Publ. Groningnen (1972), 89{96.

[Tr] Truong, T.T., (Relative) dynamical degrees of rational maps over an algebraic closed eld, arXiv:1501.01523.