Mixed Hodge structures

Given an n-dimensional variety Z with rational singularities, we conjecture that if f : Y → Z is a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, thenWe prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization OX ( * D) is at most n − 3.

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“…. Moreover, the spectral sequence associated to this filtration degenerates at E 1 (see e.g., [5,Theorem 3.3]), which implies:…”

Section: The Map βmentioning

confidence: 98%

Local Vanishing and Hodge Filtration for Rational singularities

Mustaţă

Olano²,

Popa³

2018

J. Inst. Math. Jussieu

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“…In the late 1960's and early 1970's Deligne has developed the mixed Hodge theory -a complete theory of weight filtration of complex algebraic varieties (see [31] for an introduction). One of the applications of this theory is the global monodromy of families of projective smooth varieties (given a smooth projective map X Ñ S of smooth complex varieties for any s P S the monodromy representation of π 1 pS, sq on all the H i pX s , Cq is completely reducible).…”

Section: Overview Of Lefschetz Theorymentioning

confidence: 99%

Weil Conjectures Exposition

Goncharov

2018

Preprint

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In this paper we provide a full account of the Weil conjectures including Deligne's proof of the conjecture about the eigenvalues of the Frobenius endomorphism.Section 1 is an introduction into the subject. Our exposition heavily relies on the Etale Cohomology theory of Grothendieck so I included an overview in Section 2. Once one verifies (or takes for granted) the results therein, proofs of most of the Weil conjectures are straightforward as we show in Section 3.Sections 4-8 constitute the proof of the remaining conjecture. The exposition is mostly similar to that of Deligne in [7] though I tried to provide more details whenever necessary. Following Deligne, I included an overview of Lefschetz theory (that is crucial for the proof) in Section 6.Section 9 contains a (somewhat random and far from complete) account of the consequences. Numerous references are mentioned throughout the paper as well as briefly discussed in Subsection 1.4.

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“…We use the notations in [D] freely. The basic references on this topic are [D,Section 8], [E1,Part II], [E2, Chapitres 2 and 3], and the book [PS].…”

Section: Mhs For Cohomology With Compact Supportmentioning

confidence: 99%

“…with suitable shifts of complexes and weight filtrations (for the details, see, for example, [E1, I.3], [E2, 3.7.14], [EL,Section 3.3.4] or [PS,Theorem 3.22]), where φ and ψ are induced by the natural restriction map. More precisely, we define a complex…”

Section: Mhs For Cohomology With Compact Supportmentioning

confidence: 99%