A note on the noncommutative Hodge conjecture for graded matrix factorizations

Shizhuo Zhang; Xun Lin

REVIEW

A note on the noncommutative Hodge conjecture for graded matrix factorizations

Xun Lin, Shizhuo Zhang

Algebraic Geometry · Submitted 3 Sep 2026 · 22 references

Pith's verdict: Solid, significant new result—rational Hodge conjecture for a non-Fermat family of split hypersurfaces in all even dimensions at least four—with a real but patchable gap where the paper asserts rather than verifies the polarizable VHS structure on the identity sector.

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arXiv ID2609.03784 (version 1)

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CitationShizhuo Zhang, Xun Lin. "A note on the noncommutative Hodge conjecture for graded matrix factorizations." arXiv:2609.03784 (2026). https://pith.science/paper/ZUIYT5M2

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Editor's note

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Core of the paper: it proves the rational Hodge conjecture for a genuinely new infinite family of hypersurfaces—sums of independently varying squarefree binary forms, in every even dimension at least four—and the main mechanism is a braid-monodromy exhaustion argument via the reduced Burau representation. The result is new, the family is non-Fermat (Section 9's Hessian-multiplicity argument settles that cleanly), and the proof largely holds up. The one soft spot is real but patchable: the claim that the identity sector carries a polarizable variation of Hodge structures is asserted, not verified.

What is actually new: explicit rank-one matrix factorizations for each binary factor, whose tensor products generate a rank-(d-1)^{m+1} lattice in the identity sector, together with the point-sector generators from grading shifts of the stabilized residue field. Proposition 6.2 is the heart: the rational finite-monodromy subspace of each binary state space is exactly the algebraic sector. The irreducibility and signature analysis of the specialized Burau representation is careful; Lemma 6.1, the Galois-orbit input, is a neat elementary argument; and the CDK-based exhaustion in Section 7 is sound once its PVHS hypothesis is granted. The arithmetic in Appendix B checks out.

The soft spot is exactly where the stress-test points: Section 7's opening paragraph simply asserts that the twisted de Rham model, the singularity pairing, and Thom–Sebastiani combine to make the identity sector a polarizable rational VHS over the full parameter space. The identity sector is a direct summand of the vanishing cohomology, not the whole space, and the paper never shows the restricted pairing stays nondegenerate, is compatible with the connection and filtration, and survives the transfer to the dg-categorical structure. My own reading says the claim is true: the scalar monodromy commutes with Gauss–Manin, distinct eigenspaces are orthogonal for the polarization, nondegeneracy restricts to each eigenspace, and Hodge–Riemann holds eigenspace-wise. So I'd call this a rigor gap, not a load-bearing flaw—but a referee should ask the authors to spell it out in a few sentences or cite one precise theorem instead of three suggestive ones. The rest of the proof I checked is consistent; the Burau conventions are dense but coherent, and I found no circularity. The Lin/Perry categorical framework is used as established theory, which is legitimate here.

Who it is for: singularity theorists, noncommutative Hodge people, and anyone tracking the rational Hodge conjecture. It deserves serious peer review, with the polarization point as the requested revision. I would not desk-reject this, and I would not block it on the gap either—but I want the gap closed in the published version.

What would settle it

Take one binary factor of degree $d \geq 7$ and the cyclic cover $y^d = \prod_{j=1}^d (x - z_j)$; compute numerically the braid-group action on the $\zeta^r$-eigenspace of its first cohomology for any $1 \leq r \leq d-1$. Proposition 6.2 predicts this representation is irreducible over $\mathbb{C}$, has infinite image, and carries a nondegenerate invariant Hermitian form of signature $(d-r-1, r-1)$ (an alternating form with unipotent transvections when $r = d/2$); finding a nonzero braid-invariant rational class outside the augmentation lattice $A_F$, or a finite image, would break the exclusion argument and the rank formula $\dim_{\mathbb{Q}} \mathrm{Hdg}(G_b,\mathbb{Q}) = (d-1)^{m+1} + d-1$. At the level of the hypersurface, the octic fourfold case $m=2$, $d=8$ predicts exactly $343$ rational classes inside the $9331$-dimensional $(2,2)$ summand, a count that could be checked by direct monodromy computation.

Referee report

Significance

If the result stands, it is a substantial advance: it establishes the rational noncommutative Hodge conjecture for a genuinely non-Fermat family of graded matrix factorization categories in every even ambient dimension at least four, with exact Hodge and topological K-theory ranks. The paper gives explicit algebraic generators for the full rational Hodge-class space rather than an existence statement, and Section 9 confirms that the examples are not linearly equivalent to Fermat hypersurfaces. The proof is carefully structured: the boundary-bulk Chern character calculation, the sector decomposition, the reduced-Burau irreducibility and signature argument, and the semiorthogonal transfer are all clearly separated and mutually consistent. The stress-test concern about the identity-sector PVHS does not, in my reading, land as a correctness issue: Section 7 explicitly invokes the twisted de Rham model for the flat connection and filtration, and the identity-sector projector is a Hodge-compatible flat projector, so the restricted local system inherits the structure of a polarizable sub-variation by standard sub-VHS facts.

Minor objections (4)

[§7] The Hodge-locus argument is load-bearing, and the paragraph after Proposition 4.1 asserts that the identity sector carries a polarizable rational variation. Please add one sentence explaining that the identity-sector projector is a flat morphism of the underlying variation compatible with the Hodge filtration and the polarization, so that the identity sector is a polarizable sub-VHS and the restricted singularity pairing remains nondegenerate. The cited references cover the absolute case, and the sub-VHS inheritance is standard, but this is exactly the point that the current text leaves implicit.
[§6.2 and §7] The equality G := pi_1(B,b) = prod_i B_d is not literally correct as stated, because U_d is an open subset of Sym^d(V*) and its fundamental group also contains the C* scaling factor. The proof only needs the braid subgroup coming from the slice with fixed leading coefficient, and this slice is enough for the very general statement. Please reformulate by either restricting the family to that slice or explicitly noting that the additional scalar factors act through the finite-order automorphism tau and therefore do not affect the finiteness conclusions.
[§6.4] The notation B_i(t)^* = B_i(t)^T in the invariance theorem is nonstandard when h_t is called a Hermitian form, since the adjoint with respect to a Hermitian form is normally the conjugate transpose. Please clarify whether this equality is a convention inherited from [Ven14] or whether the form is being treated as a complex bilinear form; the subsequent Schur-lemma argument requires this convention to be unambiguous.
[§7, Lemma 7.2] The definition L_Z := L_Q cap V_Z should be justified as giving an integral local system. If the intersection can jump, one should instead pass to a finite-index sublattice contained in L_Q after a finite base change; the orthogonal-group finiteness argument then still applies. This is a small gap in presentation rather than in the mathematical strategy.

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The paper

The paper claims

All rational Hodge classes are algebraic for graded matrix factorizations of sums of very general binary forms: dimension (d−1)^(m+1) + d−1, transferred to the rational Hodge conjecture for hypersurfaces.

The reading

The paper proves the rational noncommutative Hodge conjecture for the dg category of graded matrix factorizations of $f = F_0(u_0,v_0) + \cdots + F_m(u_m,v_m)$ (the categorical model of the graded singularity $\{f = 0\}$), where the $F_i$ are independently very general squarefree binary forms of degree $d \geq 7$ and $m \geq 2$. The claim is that every rational Hodge class of the category is the Hochschild Chern character of an explicitly constructed object: $(d-1)^{m+1}$ classes built by Thom–Sebastiani tensor products of rank-one factorizations of the binary factors, plus $d-1$ classes from grading shifts of the stabilized residue field. Because the factorization category and the derived category of the smooth projective hypersurface $X = \{f = 0\} \subset \mathbb{P}^{2m+1}$ sit in semi-orthogonal decompositions whose additional pieces are exceptional, additivity transfers the theorem to the classical rational Hodge conjecture for $X$, in every even dimension at least four. The general members of the family are shown not to be linearly equivalent to Fermat hypersurfaces, so the result covers new examples beyond the classical Fermat and cubic-fourfold cases.

What carries the argument

Four pieces of machinery carry the argument. (1) The equivariant Hochschild formula decomposes $\mathrm{HH}_*(G_f)$ into the identity sector, isomorphic to the scalar-invariant twisted de Rham (Jacobian) state space of dimension $((d-1)^{2m+2}+d-1)/d$, and $d-1$ one-dimensional point sectors coming from the scalars that fix only the origin. (2) For each binary factor, $L_r = (\ell_r, h_r)$ is an explicit rank-one matrix factorization (a two-periodic pair of maps whose compositions multiply to $F$), and its boundary–bulk Chern characters span the augmentation lattice $A_F$ of rank $d-1$ inside the Jacobian algebra; Thom–Sebastiani tensor products $E_I = L_{i_0} \boxtimes \cdots \boxtimes L_{i_m}$ multiply these into $(d-1)^{m+1}$ independent algebraic classes. (3) The engine is the reduced Burau representation: the braid-group action on each nonidentity scalar eigenspace $W_r$ ($1 \leq r \leq d-1$) of a binary factor equals the Burau representation specialized at $\zeta^r$, and complex-reflection formulas, the signature of the invariant Hermitian form, and unipotent transvections at $t = -1$ show every $W_r$ is irreducible with infinite image. (4) The algebraicity of Hodge loci turns a rational Hodge class at a very general parameter into a rational type-$(0,0)$ sub-local-system, which is integral, polarized, and therefore has finite monodromy; Proposition 6.2 then forces every such class into the identity multi-sector $A_{b,\mathbb{Q}}$ spanned by the explicit factorizations.

Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1.1: for $m \geq 2$, $d \geq 7$ and very general $b$, the rational noncommutative Hodge conjecture holds for $G_b = \mathrm{MF}^{\mathrm{gr}}(f_b)$, meaning the algebraic Chern character $ch \colon K_0(G_b)_{\mathbb{Q}} \to \mathrm{HH}_0(G_b)$ maps onto the rational Hodge-class space $\mathrm{Hdg}(G_b,\mathbb{Q})$. That space splits as $A_{b,\mathbb{Q}} \oplus T_{b,\mathbb{Q}}$, with $\dim_{\mathbb{Q}} A_{b,\mathbb{Q}} = (d-1)^{m+1}$ generated by boundary–bulk images of explicit rank-one matrix factorizations and their tensor products, and $\dim_{\mathbb{Q}} T_{b,\mathbb{Q}} = d-1$ generated by grading shifts of the stabilized residue field; the rational topological ranks are $\mathrm{rk}\, K_0^{\mathrm{top}}(G_b)_{\mathbb{Q}} = ((d-1)^{2m+2} + d-1)/d + d-1$ and $K_1^{\mathrm{top}}(G_b)_{\mathbb{Q}} = 0$. The decisive step is a braid-monodromy rigidity statement: the nonidentity scalar sectors of each binary factor are irreducible braid modules identified with the reduced Burau representation specialized at $d$-th roots of unity, whose images are infinite, so a very-general Hodge class — forced by the algebraicity of Hodge loci to generate a rational type-$(0,0)$ sub-local-system, hence a finite-monodromy subspace — can only live in the identity multi-sector spanned by the explicit factorizations. Additivity of the conjecture for semi-orthogonal decompositions, applied to the decompositions for Fano, Calabi–Yau, and general-type hypersurfaces, then yields Corollary 1.2: $X_b$ satisfies the rational Hodge conjecture, with primitive middle Hodge rank $(d-1)^{m+1}$ and full middle rank $(d-1)^{m+1}+1$.

Load-bearing premise

The load-bearing premise is that the identity sector of the graded factorization category carries a polarizable variation of Hodge structures, built from the twisted de Rham model of cyclic homology with its Gauss–Manin connection, singularity pairing, and rational lattice; if that Hodge-theoretic framework did not apply to graded matrix factorizations, the Hodge-locus argument that converts rational Hodge classes into finite-monodromy subspaces would collapse, and the braid-monodromy calculation could no longer exclude any classes.

Editorial extensions

If this is right

The classical rational Hodge conjecture holds for the smooth hypersurface $X_b \subset \mathbb{P}^{2m+1}$ in every even dimension $2m \geq 4$ covered by the theorem, with primitive middle Hodge rank exactly $(d-1)^{m+1}$ and full middle rank $(d-1)^{m+1}+1$.
The categorical ranks are exact: $\dim_{\mathbb{Q}} \mathrm{Hdg}(\mathrm{MF}^{\mathrm{gr}}(f_b),\mathbb{Q}) = (d-1)^{m+1}+d-1$ and $\mathrm{rk}\, K_0^{\mathrm{top}}(G_b)_{\mathbb{Q}} = ((d-1)^{2m+2}+d-1)/d + d-1$, with $K_1^{\mathrm{top}}(G_b)_{\mathbb{Q}} = 0$; the paper tabulates the octic fourfold (343 primitive classes), the septic sixfold (1296), and the nonic eightfold (32768).
The classes obtained are not divisor classes: weak Lefschetz shows products of $(1,1)$-classes span only $h^m$, so the $(d-1)^{m+1}$ classes are additional codimension-$m$ classes carried by explicit matrix factorizations.
The general members of the family are not linearly equivalent to Fermat hypersurfaces (detected by Hessian divisor multiplicities), so the theorem extends the classical diagonal-Fermat results to new hypersurfaces.
The transfer covers all three curvature types at once — Fano ($d < 2m+2$), Calabi–Yau ($d = 2m+2$), and general type ($d > 2m+2$) — because the relevant semi-orthogonal decompositions and the additivity theorem treat the three cases uniformly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The monodromy-control strategy should generalize: for any split singularity whose factor monodromy is irreducible with infinite image, the same Hodge-locus-to-finite-monodromy exclusion would compute the very general Hodge classes, so the conjecture for such families looks like a statement about braid-like monodromy rather than about constructing cycles case by case.
The rank formulas expose a quantitative gap between topology and algebraicity: the rational topological $K$-rank $((d-1)^{2m+2}+d-1)/d + d-1$ vastly exceeds the Hodge-class rank $(d-1)^{m+1}+d-1$ for large $d$, so most rational topological classes fail the type-$(0,0)$ condition, and the family is a natural testing ground for the integral (not just rational) noncommutative Hodge conjecture, whose
The very-general hypothesis leaves the exceptional loci open: sums of binary forms carrying intermediate symmetries could in principle support extra rational Hodge classes, and the braid argument makes no prediction for those special parameters.
A computational check could isolate the key claim from the surrounding Hodge theory: for $d = 7, 8, 9$, evaluate the specialized reduced Burau matrices on explicit braid words and verify irreducibility and infinite order numerically before invoking the projective consequences.

Original abstract

Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $m\geq 2$ and $d\geq 7$.	background 0.87	0.09	0.01	unclear 0.56
s1	We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$.	contribution 0.44	0.37	0.02	asserted 0.63
s2	We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations.	result 0.68	0.96	0.86	asserted 1.00
s3	Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors.	result 1.00	0.83	0.05	asserted 1.00
s4	Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors.	result 0.93	0.89	0.06	asserted 1.00
s5	A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter.	method 0.63	0.90	0.10	asserted 1.00
s6	Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$.	result 1.00	0.90	0.26	asserted 1.00
s7	Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.	result 0.71	0.81	0.39	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

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full text
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P abstract

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References

Works cited (22)

[1]

N. Aoki, Some new algebraic cycles on Fermat varieties, J. Math. Soc. Japan 39 (1987), no. 3, 385--396

View on Pith 1987
[2]

M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes \`Etudes Sci. 120 (2014), 1--111

View on Pith 2014
[3]

A. Blanc, Topological K -theory of complex noncommutative spaces, Compos. Math. 152 (2016), no. 3, 489--555

View on Pith 2016
[4]

E. Cattani, P. Deligne, and A. Kaplan, On the locus of Hodge classes, J. Amer. Math. Soc. 8 (1995), no. 2, 483--506

View on Pith 1995
[5]

I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012

View on Pith 2012
[6]

T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), no. 2, 223--274

View on Pith 2011
[7]

A. I. Efimov, Cyclic homology of categories of matrix factorizations, Int. Math. Res. Not. IMRN (2018), no. 12, 3834--3869

View on Pith 2018
[8]

L. Funar and T. Kohno, On Burau representations at roots of unity, Geom. Dedicata 169 (2014), 145--163

View on Pith 2014
[9]

D. Kaledin, Spectral sequences for cyclic homology, in Algebra, Geometry, and Physics in the 21st Century, Progr. Math., vol. 324, Birkh\"auser/Springer, Cham, 2017, 99--129

View on Pith 2017
[10]

X. Lin, Noncommutative Hodge conjecture, J. Noncommut. Geom. 17 (2023), no. 4, 1367--1389

View on Pith 2023
[11]

D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II, Progr. Math., vol. 270, Birkh\"auser Boston, Boston, MA, 2009, 503--531

View on Pith 2009
[12]

A. Perry, The integral Hodge conjecture for two-dimensional Calabi--Yau categories, Compos. Math. 158 (2022), no. 2, 287--333

View on Pith 2022
[13]

A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch--Riemann--Roch formula for matrix factorizations, Duke Math. J. 161 (2012), no. 10, 1863--1926

View on Pith 2012
[14]

A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1--122

View on Pith 2016
[15]

A. Preygel, Thom--Sebastiani and duality for matrix factorizations, arXiv:1101.5834, 2011

View on Pith Pith review arXiv 2011
[16]

Z. Ran, Cycles on Fermat hypersurfaces, Compos. Math. 42 (1980), no. 1, 121--142

View on Pith 1980
[17]

T. Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), no. 2, 175--184

View on Pith 1979
[18]

C. C. Squier, The Burau representation is unitary, Proc. Amer. Math. Soc. 90 (1984), no. 2, 199--202

View on Pith 1984
[19]

J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 525--563

View on Pith 1977
[20]

T. N. Venkataramana, Image of the Burau representation at d -th roots of unity, Ann. of Math. (2) 179 (2014), no. 3, 1041--1083

View on Pith 2014
[21]

C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002

View on Pith 2002
[22]

S. Zucker, The Hodge conjecture for cubic fourfolds, Compos. Math. 34 (1977), no. 2, 199--209. 5

View on Pith 1977

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

N. Aoki, Some new algebraic cycles on Fermat varieties, J. Math. Soc. Japan 39 (1987), no. 3, 385--396

View on Pith 1987

[2] [2]

M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes \`Etudes Sci. 120 (2014), 1--111

View on Pith 2014

[3] [3]

A. Blanc, Topological K -theory of complex noncommutative spaces, Compos. Math. 152 (2016), no. 3, 489--555

View on Pith 2016

[4] [4]

E. Cattani, P. Deligne, and A. Kaplan, On the locus of Hodge classes, J. Amer. Math. Soc. 8 (1995), no. 2, 483--506

View on Pith 1995

[5] [5]

I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012

View on Pith 2012

[6] [6]

T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), no. 2, 223--274

View on Pith 2011

[7] [7]

A. I. Efimov, Cyclic homology of categories of matrix factorizations, Int. Math. Res. Not. IMRN (2018), no. 12, 3834--3869

View on Pith 2018

[8] [8]

L. Funar and T. Kohno, On Burau representations at roots of unity, Geom. Dedicata 169 (2014), 145--163

View on Pith 2014

[9] [9]

D. Kaledin, Spectral sequences for cyclic homology, in Algebra, Geometry, and Physics in the 21st Century, Progr. Math., vol. 324, Birkh\"auser/Springer, Cham, 2017, 99--129

View on Pith 2017

[10] [10]

X. Lin, Noncommutative Hodge conjecture, J. Noncommut. Geom. 17 (2023), no. 4, 1367--1389

View on Pith 2023

[11] [11]

D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II, Progr. Math., vol. 270, Birkh\"auser Boston, Boston, MA, 2009, 503--531

View on Pith 2009

[12] [12]

A. Perry, The integral Hodge conjecture for two-dimensional Calabi--Yau categories, Compos. Math. 158 (2022), no. 2, 287--333

View on Pith 2022

[13] [13]

A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch--Riemann--Roch formula for matrix factorizations, Duke Math. J. 161 (2012), no. 10, 1863--1926

View on Pith 2012

[14] [14]

A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1--122

View on Pith 2016

[15] [15]

A. Preygel, Thom--Sebastiani and duality for matrix factorizations, arXiv:1101.5834, 2011

View on Pith Pith review arXiv 2011

[16] [16]

Z. Ran, Cycles on Fermat hypersurfaces, Compos. Math. 42 (1980), no. 1, 121--142

View on Pith 1980

[17] [17]

T. Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), no. 2, 175--184

View on Pith 1979

[18] [18]

C. C. Squier, The Burau representation is unitary, Proc. Amer. Math. Soc. 90 (1984), no. 2, 199--202

View on Pith 1984

[19] [19]

J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 525--563

View on Pith 1977

[20] [20]

T. N. Venkataramana, Image of the Burau representation at d -th roots of unity, Ann. of Math. (2) 179 (2014), no. 3, 1041--1083

View on Pith 2014

[21] [21]

C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002

View on Pith 2002

[22] [22]

S. Zucker, The Hodge conjecture for cubic fourfolds, Compos. Math. 34 (1977), no. 2, 199--209. 5

View on Pith 1977